Article
 HaiZhi Song, Wei Zhang, LiBo Yu, Zhiming M. Wang
 Micropillar Cavity Design for 1.55μm QuantumDot SinglePhoton Sources
 Journal of Electronic Science and Technology, 2019, 17(3): 221230
 http://dx.doi.org/10.11989/JEST.1674862X.71027015

Article History
 Manuscript received October. 20, 2017
 revised December. 04, 2017
W. Zhang and L.B. Yu are with Southwest Institute of Technical Physics, Chengdu 610041 (email: wzhangscu@sina.cn; 18600281100@163.com);
Z. M. Wang is with the Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology, Chengdu 610054 (email: zhmwang@uestc.edu.cn)
Optical microcavities are widely studied for their prospects in many fields of research and technology, such as optical communications, nonlinear optics, optoelectronics, and quantum information technology^{[1]}. For solidstate quantum information processing, microcavities containing semiconductor quantum dots (QDs) have been demonstrated to be effective as indispensable devices such as efficient/indistinguishable and coherent single photon sources (SPSs)^{[2],[3]}. Among many cavity types, micropillar cavities are advantageous for fiberbased quantum information processing owing to their high coupling efficiency to fiber and suitability for electrical driving. For the purpose of quantum communications over silicafiberbased networks, 1.55μm InAs/InP QDs are promising as SPSs^{[4]} and thus micropillar cavities containing InAs/InP QDs are strongly required. However, such a satisfactory cavity has not been practically available, so that more efforts have to be devoted. One kind of adiabatic design was found to be effective in improving micropillar cavities of GaAs/AlGaAs^{[5]} and TiO_{2}/SiO_{2}^{[6]}. We are here trying to extend the adiabatic design to find an effective way of constructing efficient 1.55μm micropillar cavities for QD SPSs.
2. Theory and MethodThe adiabatic design on DBR micropillar cavities is usually slowly decreasing the DBR layer thickness as the DBR goes towards the central spacer. This change makes the bandgap of DBR adiabatically shift to higher energy^{[5]}, so that the optical mode shifts away from the optimized position in the bandgap. When the light wave of the cavity mode travels through such thinned DBRs, there will be more components transmitted so that the confinement of the light field becomes gentler. As a result, the envelope of the electromagnetic field varies more slowly in the cavity. By using the Fourier transform, the wavenumber range is getting narrower and the component out of allreflective condition on the sidewall is getting smaller. This will bring about a higher quality (Q) factor.
From the above view, however, there may not be only one way to increase the transmittance of DBR. The thickness changes in different ways might also change the central position of the mode in the bandgap. The change of the bandgap width itself may also have the same effect. We try analyzing this qualitatively using the way of planar DBR. Let the DBR layers stack along Z direction and consider transverse magneticmode (TMmode) planewaves. The electric field of the wave having wavevector along Z direction is
$ E_z^{\left( l \right)} = E_f^{\left( l \right)} + E_b^{\left( l \right)} $  (1) 
where E_{f}(E_{b}) denotes the electric field component travelling forwards (backwards). The boundary condition at each interface is
${{{\text{ε}} ^{\left( l \right)}}\left( {E_f^{\left( l \right)} + E_b^{\left( l \right)}} \right) = {{\text{ε}} ^{\left( {l  1} \right)}}\left( {E_f^{\left( {l  1} \right)} + E_b^{\left( {l  1} \right)}} \right)} $  (2a) 
${k_z^{\left( l \right)}\left( {E_f^{\left( l \right)}  E_b^{\left( l \right)}} \right) = k_z^{l  1}\left( {E_f^{\left( {l  1} \right)}  E_b^{\left( {l  1} \right)}} \right)} $  (2b) 
where ε is the dielectric constant and k_{z} is the wavenumber. It is easy to get the transfer matrix between layers l–1 andl. Considering the phase shift, the matrix connecting the electric fields at the two sides of a pair of DBR layers (material 2 and material 3) is
$ \begin{array}{l} {\text{F}} = \left(\! {\begin{array}{*{20}{c}} {{e^{  jk_z^{\left(\! 3 \!\right)}{d_3}}}}&0\\ 0&{{e^{jk_z^{\left(\! 3 \!\right)}{d_3}}}} \end{array}} \!\right)\!\!\left(\! {\begin{array}{*{20}{c}} {\frac{1}{2}\!\!\left(\! {\frac{{{{\text{ε}} _2}}}{{{{\text{ε}} _3}}} + \frac{{k_z^{\left(\! 2 \!\right)}}}{{k_z^{\left(\! 3 \!\right)}}}} \!\right)}&{\frac{1}{2}\!\!\left(\! {\frac{{{{\text{ε}} _2}}}{{{{\text{ε}} _3}}}  \frac{{k_z^{\left(\! 2 \!\right)}}}{{k_z^{\left(\! 3 \!\right)}}}} \!\right)}\\ {\frac{1}{2}\!\!\left(\! {\frac{{{{\text{ε}} _2}}}{{{{\text{ε}} _3}}}  \frac{{k_z^{\left(\! 2 \!\right)}}}{{k_z^{\left(\! 3 \!\right)}}}} \!\right)}&{\frac{1}{2}\!\!\left(\! {\frac{{{{\text{ε}} _2}}}{{{{\text{ε}} _3}}} + \frac{{k_z^{\left(\! 2 \!\right)}}}{{k_z^{\left(\! 3 \!\right)}}}} \!\right)} \end{array}} \!\right)\!\!\left(\! {\begin{array}{*{20}{c}} {{e^{  jk_z^{\left(\! 2 \!\right)}{d_2}}}}&0\\ 0&{{e^{jk_z^{\left(\! 2 \!\right)}{d_2}}}} \end{array}} \!\right)\!\!\left(\! {\begin{array}{*{20}{c}} {\frac{1}{2}\!\!\left(\! {\frac{{{{\text{ε}} _3}}}{{{{\text{ε}} _2}}} + \frac{{k_z^{\left(\! 3 \!\right)}}}{{k_z^{\left(\! 2 \!\right)}}}} \!\right)}&{\frac{1}{2}\!\!\left(\! {\frac{{{{\text{ε}} _3}}}{{{{\text{ε}} _2}}}  \frac{{k_z^{\left(\! 3 \!\right)}}}{{k_z^{\left(\! 2 \!\right)}}}} \!\right)}\\ {\frac{1}{2}\!\!\left(\! {\frac{{{{\text{ε}} _3}}}{{{{\text{ε}} _2}}}  \frac{{k_z^{\left(\! 3 \!\right)}}}{{k_z^{\left(\! 2 \!\right)}}}} \!\right)}&{\frac{1}{2}\!\!\left(\! {\frac{{{{\text{ε}} _3}}}{{{{\text{ε}} _2}}} + \frac{{k_z^{\left(\! 3 \!\right)}}}{{k_z^{\left(\! 2 \!\right)}}}} \!\right)} \end{array}} \!\right)\\ = \left(\!\! {\begin{array}{*{20}{c}} {{e^{  jk_z^{\left(\! 3 \!\right)}{d_3}}}\left\{ {\cos \left(\! {k_z^{\left(\! 2 \!\right)}{d_2}} \!\right)  j\frac{1}{2}{\rm{s\;\!in}} \left(\! {k_z^{\left(\! 2 \!\right)}{d_2}} \!\right)\left(\! {\frac{{{{\text{ε}} _3}}}{{{{\text{ε}} _2}}}\frac{{k_z^{\left(\! 2 \!\right)}}}{{k_z^{\left(\! 3 \!\right)}}} + \frac{{{{\text{ε}} _2}}}{{{{\text{ε}} _3}}}\frac{{k_z^{\left(\! 3 \!\right)}}}{{k_z^{\left(\! 2 \!\right)}}}} \!\right)} \right\}}&\!\!\!\!{  j{e^{  jk_z^{\left(\! 3 \!\right)}{d_3}}}\frac{1}{2}{\rm{s\;\!in}} \left(\! {k_z^{\left(\! 2 \!\right)}{d_2}} \!\right)\left(\! {\frac{{{{\text{ε}} _3}}}{{{{\text{ε}} _2}}}\frac{{k_z^{\left(\! 2 \!\right)}}}{{k_z^{\left(\! 3 \!\right)}}}  \frac{{{{\text{ε}} _2}}}{{{{\text{ε}} _3}}}\frac{{k_z^{\left(\! 3 \!\right)}}}{{k_z^{\left(\! 2 \!\right)}}}} \!\right)}\\ {j{e^{\,\,jk_z^{\left(\! 3 \!\right)}{d_3}}}\frac{1}{2}{\rm{s\;\!in}}\left(\! {k_z^{\left(\! 2 \!\right)}{d_2}} \!\right)\left(\! {\frac{{{{\text{ε}} _3}}}{{{{\text{ε}} _2}}}\frac{{k_z^{\left(\! 2 \!\right)}}}{{k_z^{\left(\! 3 \!\right)}}}  \frac{{{{\text{ε}} _2}}}{{{{\text{ε}} _3}}}\frac{{k_z^{\left(\! 3 \!\right)}}}{{k_z^{\left(\! 2 \!\right)}}}} \!\right)}&\!\!\!\!{{e^{\,\,jk_z^{\left(\! 3 \!\right)}{d_3}}}\left\{ {\cos \left(\! {k_z^{\left(\! 2 \!\right)}{d_2}} \!\right) + j\frac{1}{2}{\rm{s\;\!in}} \left(\! {k_z^{\left(\! 2 \!\right)}{d_2}} \!\right)\left(\! {\frac{{{{\text{ε}} _3}}}{{{{\text{ε}} _2}}}\frac{{k_z^{\left(\! 2 \!\right)}}}{{k_z^{\left(\! 3 \!\right)}}} + \frac{{{{\text{ε}} _2}}}{{{{\text{ε}} _3}}}\frac{{k_z^{\left(\! 3 \!\right)}}}{{k_z^{\left(\! 2 \!\right)}}}} \!\right)} \right\}} \end{array}} \!\!\!\right) \end{array} $  (3) 
where d_{2} and d_{3} are the thicknesses of material 2 and material 3, respectively. The transmittance rate of one DBR pair x should be the expected value of the above matrix following
${\rm de\;\,\!\!t}\!\left( {{\text{F}}  x{\text{I}}} \right) = {x^2} \left[ {2\cos\!\left( {k_z^{\left( 2 \right)}{d_2}} \right)  {\rm s\;\,\!\!in}\!\left( {k_z^{\left( 2 \right)}{d_2}} \right){\rm s\;\,\!\!in}\!\left( {k_z^{\left( 3 \right)}{d_3}} \right)\!\!\left( {\frac{{{{\text{ε}} _3}}}{{{{\text{ε}} _2}}}\frac{{k_z^{\left( 2 \right)}}}{{k_z^{\left( 3 \right)}}} + \frac{{{{\text{ε}} _2}}}{{{{\text{ε}} _3}}}\frac{{k_z^{\left( 3 \right)}}}{{k_z^{\left( 2 \right)}}}} \right)} \right]\!x + 1 = 0.$  (4) 
Equation (4) has two solutions, larger or smaller than 1. The smaller value meets the physical solution. For the transverse electric (TE) mode, the solution can be obtained by simply replacing the dielectric constant with the permeability. Fig. 1 shows the result of the above calculation, the thickness dependence of the DBR transmittance. The value x means that the light wave becomes x times weaker when it goes through a DBR pair. We see that, the transmittance is the smallest, i.e., the reflectivity is the most as the layer thicknesses satisfy 1 or 3 times of a quarter of the mode wavelength, as usual DBRs are setup. To get a higher transmittance, one can simply change the DBR layer thickness deviated from the minimumtransmittance points. There are thus many ways to realize adiabatic DBRs, e.g., decreasing both layers, increasing both layers, and increasing one while decreasing another.
In the following, we will study how these ways work for realizing efficient micropillar cavities for 1.55μm QD SPS. Simulation on any cavity structure is performed using the finitedifference timedomain method. The indices of materials are cited from or deduced based on [7]. By launching a polarized light impulse from the light source, the time evolution of the light intensity can be obtained at some monitors. A Fourier transform gives a spectrum of the electric field intensity, showing some peaks representing the cavity modes. By setting the light source as a narrowband emission around a mode wavelength λ, we obtain the intensity decay with time t and the steady state distribution, i.e., the mode profile. The Q factor can be obtained by fitting the exponential light intensity envelope to
${F_{P}} = \frac{{3Q{{\text{λ}} ^3}}}{{4{{\text{π}} ^2}Vn^3}} $  (5) 
where n is the effective refractive index at the maximum light intensity, and the mode volume is
$V = \frac{{\int {{\text{ε}} ({\text{r}}){\text{E}}{{({\text{r}})}^2}{\rm{d}}{\text{r}}} }}{{{{\text{ε}} _{M}}{E_M}^2}} $  (6) 
where ε(r) is the relative dielectric constant, E(r) is the electric field of the light at the position r, and ε_{M} and E_{M} are the corresponding values at the point of the maximum light intensity.
3. Results and DiscussionThe straight way to construct a 1.55μm micropillar cavity could be using the InPlatticematching DBR structures. Due to the small refractive index contrast (~0.2), the conventional InGaAsP/InP DBR micropillar cavities, with wavelengththick spacer and quarterwavelengththick DBR layers, are not easy to satisfy the requirements of 1.55μm SPSs. An adiabatic design may help to increase the cavity quality. As shown in Fig. 2 (a), besides the top and bottom conventional InP/InGaAsP DBRs, the central spacer is replaced by thicknesschanging InP/InGaAsP pairs, i.e., adiabatic DBRs. In detail, the layer thickness in DBRs is t_{1}
Fig. 2 (b) shows how the cavity quality change as the taper slope is tuned. Usually there are optimization conditions, because the mode profile matching, which is the reason for improving the cavity quality^{[6]}, needs some specific configuration. As the cavity has 16/28 pairs of conventional top/bottom DBR layers, the Q factor shows no obvious improvement, presenting the difficulty of this material system. The cavity with 30/50 DBR pairs does have cavity quality improved. With 4 segments of adiabatic DBR layers, the Q factor is increased from 1800 to 2300 as ρ is about 0.04. The structure of more DBR layers show more effects of the adiabatic design. The cavity with 40/70 pairs of DBRs and 6 segments of tapered DBRs would exhibit Q factors near 10^{4}, much higher than the conventional counterpart, 6000, as ρ is tuned to be 0.02.
The purcell factor F_{P} is also enhanced by tapered DBRs. The cavity with 30/50 DBR pairs is improved in the maximal F_{P} from 30 to 40, while that with 40/70 DBR pairs is improved from 110 to 130, as shown in Fig. 2 (c). In general, this cavity can in principle satisfy the requirement of efficient 1.55μm SPS. The best single photon generation rate is inversely proportional to the spontaneous lifetime T_{1} of the QD excitons. As such, the above cavity with F_{P}>10 could increase the operation frequency from several hundreds MHz into the GHz band. For photon indistinguishability, there should be a coherence timeT_{2} comparable or longer than 2T_{1}. InAs/InP QDs are reported to have T_{1}≈1.2 ns without a cavity^{[8]} and have T_{2}≈130 ps^{[9]}. It turns out that a microcavity with F_{P}>2T_{1}/T_{2}≈20 would be required. Obviously, it could be expected that highly indistinguishable single photons could be produced from InAs/InP QDs, using the above adiabatically designed InGaAsP/InP DBRmicropillar cavities. The effect of adiabatic design is not outstanding, however, probably because of the limited reflective index contrast.
Inspired by the use of SiO_{2}/Ta_{2}O_{5}InP DBR micropillar cavities^{[10]}, the Si/SiO_{2}InP hybrid DBR micropillar cavity was proposed^{[11]} to overcome the problem of the InGaAsP/InP DBR micropillar. It has a Q factor of ~3000 and F_{P} of more than 100 at the pillar diameter about 2.3 μm, which satisfies the weak coupling easily. To further improve this hybrid cavity, the adiabatic design might be carried out^{[12]}. As shown in Fig. 3 (a), the top and bottom parts of the pillar are conventional DBRs composed of alternating Si and SiO_{2} layers. Each layer in these DBRs is set quarterwavelength thick, i.e., the thickness t_{3}
With only 2/3.5 DBR pairs and 2 tapered DBR segments, the proposed cavity has quality as good as the previous conventional Si/SiO_{2}InP hybrid micropillar cavity. More significant effects can be seen on a cavity with 4/6.5 DBR pairs. That shown in Fig. 3 (b) is the result of an example structure, with the pillar diameter D
It is no doubt that the above improved cavity quality makes it easier to weakly couple a 1.55μm InAs/InP QD to the micropillar cavity mode and enhance the spontaneous emission rate for 100 times. We can also examine the possibility of strong coupling. The condition of strong coupling^{[14]} usually can be simplified as g>κ/4^{[12]}, where the coupling strength g
The above design is, however, not ideal yet because it has a very thin (~100 nm) InP spacer layer, which is challenging with the fabrication process and subject to the defects at the interface of semiconductor and the dielectric layers. One might consider using 3quarterwavelengththick Si layers in DBRs. Together with the aid of the adiabatic design, these difficulties could be relaxed. As the theoretical analysis suggests, not only the taper structure, other adiabatic designs, e.g., both DBR layers getting thicker and one thicker while another thinner, are also possible to improve the property of Si/SiO_{2}InP hybrid micropillar cavities. Based on a λ/43λ/4 structure, i.e., t_{3}
Arbitrarily tuning ρ_{3} and ρ_{4}, the layers in adiabatic DBRs can change their thicknesses in different ways. One way is to set the Si layer increasing (ρ_{3}<0) and the SiO_{2} layer decreasing (ρ_{4}>0). Taking an example ofρ_{3}
There might be another way related to the adiabatic design to resolve the problem of the thin semiconductor active layer, i.e., inserting adiabatic InGaAsP/InP DBRs. As shown in Fig. 5 (a), besides the Si/SiO_{2} adiabatic DBRs, InP/InGaAsP adiabatic DBRs around the spacer layer are designed. The former is now named as outer adiabatic DBRs, and the latter as inner adiabatic DBRs. The model is described by t_{3}
Calculation shows that, as in Fig. 5 (b), this design does improve the cavity quality. The simplest setting is to take the same taper slope for Si/SiO_{2} and InGaAsP/InP adiabatic DBRs, ρ_{12}
The hybrid micropillar cavities, however, have disadvantages in materialseparated thermal expansion and the complicated fabrication process. Referred to airgap planar DBR cavities^{[15]}, we may use a micropillar consisting of semiconductor layers with partial airgaps^{[16]}. The conventional form of this cavity, however, does not behave well enough, thus the adiabatic design is again required. As shown in Fig. 6 (a), disk shaped (in the XY plane) and coaxially set (in the Z direction) InGaAsP and InP layers with different diameters D and d, respectively, are alternatively stacked on an InP substrate. Effectively, the smallsized InP layers are compassed by surrounding airgaps, or namely with airapertures. Each InP layer in the DBRs is set as thick as t_{1}
Simulation shows that this InGaAsP/InPairaperture micropillar cavity has two useful modes. The fundamental mode O is optimized at D
Adiabatic DBRs are proved effective in creating highquality micropillar cavities for 1.55μm quantum information processing. Conventional InGaAsP/InP DBR micropillar cavities can be improved to be more usable by introducing adiabatically tapered InGaAsP/InP DBRs. On a Si/SiO_{2}InP hybrid micropillar cavity, useful as a weakcoupling SPS, inserting adiabatically tapered Si/SiO_{2} DBRs can enable strong coupling with 1.55μm InAs/InP QDs at smaller diameters. Not only the taper structure, other adiabatic designs, e.g., both DBR layers getting thicker and one thicker while one thinner, also can improve the quality of the Si/SiO_{2}InP hybrid micropillar cavities. The problem of the very thin semiconductor spacer layer, subject to interface defects, and difficult fabrication, can be resolved by inserting adiabatic InGaAsP/InP DBRs and carefully designing their adiabatic matching with Si/SiO_{2} DBRs. On an InGaAsP/InPairaperture micropillar cavity, overcoming the hybrid fabrication by a monolithic process, the adiabatic design brings about a high Q factor (>1×10^{4}), small mode volume, and nanoscaled diameter, and satisfies both weak and strong coupling to 1.55μm InAs/InP QDs. Therefore, micropillar cavities with adiabatic DBR structures are prospective candidates for 1.55μm QD SPSs applied in silicafiberbased quantum information processing.
AcknowledgmentH.Z. Song expresses his appreciation to the Distinguished Experts Program of Sichuan, and Rongpiao Plan of Chengdu for their partial support, and thanks Dr. M. Takatsu who works with Fujitsu Lab. Ltd. for his theoretical help.
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