Journal of Electronic Science and Technology  2019, Vol.17 Issue (3): 221-230   DOI: 10.11989/JEST.1674-862X.71027015 PDF
http://dx.doi.org/10.11989/JEST.1674-862X.71027015
51K

#### Article

Hai-Zhi Song, Wei Zhang, Li-Bo Yu, Zhiming M. Wang
Micropillar Cavity Design for 1.55-μm Quantum-Dot Single-Photon Sources
Journal of Electronic Science and Technology, 2019, 17(3): 221-230
http://dx.doi.org/10.11989/JEST.1674-862X.71027015

### Article History

revised December. 04, 2017
Micropillar Cavity Design for 1.55-μm Quantum-Dot Single-Photon Sources
Hai-Zhi Song , Wei Zhang, Li-Bo Yu, Zhiming M. Wang
H.-Z. Song is with the Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology, Chengdu 610054, and also with Southwest Institute of Technical Physics, Chengdu 610041 (e-mail: hzsong@uestc.edu.cn);
W. Zhang and L.-B. Yu are with Southwest Institute of Technical Physics, Chengdu 610041 (e-mail: 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 (e-mail: zhmwang@uestc.edu.cn)
This work was supported by the Sichuan Science and Technology Program under Grant No. 2018JY0084
Abstract: The 1.55-μm quantum-dot (QD) micropillar cavities are strongly required as single photon sources (SPSs) for silica-fiber-based quantum information processing. Theoretical analysis shows that the adiabatic distributed Bragg reflector (DBR) structure may greatly improve the quality of a micropillar cavity. An InGaAsP/InP micropillar cavity is originally difficult, but it becomes more likely usable with inserted tapered (thickness decreased towards the center) distributed DBRs. Simulation turns out that, incorporating adiabatically tapered DBRs, a Si/SiO2-InP hybrid micropillar cavity, which enables weakly coupling InAs/InP quantum dots (QDs), can even well satisfy strong coupling at a smaller diameter. Certainly, not only the tapered structure, other adiabatic designs, e.g., both DBR layers getting thicker and one thicker one thinner, also improve the quality, reduce the diameter, and degrade the fabrication difficulty of Si/SiO2-InP hybrid micropillar cavities. Furthermore, the problem of the thin epitaxial semiconductor layer can also be greatly resolved by inserting adiabatic InGaAsP/InP DBRs. With tapered DBRs, the InGaAsP/InP-air-aperture micro-pillar cavity serves as an efficient, coherent, and monolithically producible 1.55-μm single-photon source (SPS). The adiabatic design is thus an effective way to obtain prospective candidates for 1.55-μm QD SPSs.
Key words: Cavity    distributed Bragg reflectors (DBRs)    micropillar    quantum dot (QD)    single-photon source (SPS)
1. Introduction

Optical microcavities are widely studied for their prospects in many fields of research and technology, such as optical communications, nonlinear optics, opto-electronics, and quantum information technology[1]. For solid-state 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 fiber-based quantum information processing owing to their high coupling efficiency to fiber and suitability for electrical driving. For the purpose of quantum communications over silica-fiber-based networks, 1.55-μm 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 TiO2/SiO2[6]. We are here trying to extend the adiabatic design to find an effective way of constructing efficient 1.55-μm micropillar cavities for QD SPSs.

2. Theory and Method

The 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 all-reflective 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 magnetic-mode (TM-mode) 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 Ef(Eb) 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 kz 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 d2 and d3 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 minimum-transmittance points. There are thus many ways to realize adiabatic DBRs, e.g., decreasing both layers, increasing both layers, and increasing one while decreasing another.

 Fig. 1 Contour diagram of the transmittance of a pair of DBR layers as a function of the layer thicknesses.

In the following, we will study how these ways work for realizing efficient micropillar cavities for 1.55-μm QD SPS. Simulation on any cavity structure is performed using the finite-difference time-domain 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 narrow-band 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 $\exp\!\left(-2{\text{π}} c \, t / Q{\text{λ}}\right)$ , where c is the light velocity in vacuum. It is more meaningful to examine the Purcell factor

 ${F_{P}} = \frac{{3Q{{\text{λ}} ^3}}}{{4{{\text{π}} ^2}Vn^3}}$ (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 εM and EM are the corresponding values at the point of the maximum light intensity.

3. Results and Discussion

The straight way to construct a 1.55-μm micropillar cavity could be using the InP-lattice-matching DBR structures. Due to the small refractive index contrast (~0.2), the conventional InGaAsP/InP DBR micropillar cavities, with wavelength-thick spacer and quarter-wavelength-thick DBR layers, are not easy to satisfy the requirements of 1.55-μm 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 thickness-changing InP/InGaAsP pairs, i.e., adiabatic DBRs. In detail, the layer thickness in DBRs is t1 $=$ λB/(4n1) for InGaAsP and t2 $=$ λB/(4n2) for InP, where n1 and n2 are the refractive indices of InGaAsP and InP, respectively, and λB is the Bragg wavelength usually set to be near 1.55 μm; that of adiabatic DBRs is t1i $=$ t1(1–ρ(2i–1)) for InGaAsP andt2i $=$ t2(1–2ρi) for InP, respectively, where i is the taper segment number and ρ is the adiabatic parameter representing the changed fraction per adiabatic layer; that of the central spacer in between the tapered DBRs is t0 $=$ λB(1–2ρN)/4n1, and N is the total adiabatic segment number. InGaAsP components are chosen to have a 1.3-μm bandgap so that it is completely transparent to 1.55-μm light. Setting ρ to be larger than but close to 0, it corresponds to the taper design, which has been proved to be effective for AlGaAs/GaAs and SiO2/TiO2 micropillar DBR cavities[5],[6].

 Fig. 2 Structure and characters of tapered InGaAsP/InP DBR micropillar cavities: (a) schematic demonstration of the InGaAsP/InP DBR micropillar cavities with adiabatically tapered DBRs, (b) Q factor of these cavities as a function of the tapering parameter, and (c) Purcell factor of such typical cavities as a function of the pillar size.

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 104, much higher than the conventional counterpart, 6000, as ρ is tuned to be 0.02.

The purcell factor FP is also enhanced by tapered DBRs. The cavity with 30/50 DBR pairs is improved in the maximal FP 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 1.55-μm SPS. The best single photon generation rate is inversely proportional to the spontaneous lifetime T1 of the QD excitons. As such, the above cavity with FP>10 could increase the operation frequency from several hundreds MHz into the GHz band. For photon indistinguishability, there should be a coherence timeT2 comparable or longer than 2T1. InAs/InP QDs are reported to have T1≈1.2 ns without a cavity[8] and have T2≈130 ps[9]. It turns out that a microcavity with FP>2T1/T2≈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 DBR-micropillar cavities. The effect of adiabatic design is not outstanding, however, probably because of the limited reflective index contrast.

Inspired by the use of SiO2/Ta2O5-InP DBR micropillar cavities[10], the Si/SiO2-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 FP 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 SiO2 layers. Each layer in these DBRs is set quarter-wavelength thick, i.e., the thickness t3 $=$ λB/(4ne3) for Si and t4 $=$ λB/(4ne4) for SiO2, where ne3(4) is the effective refractive index of Si(SiO2), which is calculated and known to be dependent on the pillar diameter D by using the standard waveguide theory[13]. In between the conventional DBRs, we incorporate more Si/SiO2 segments as adiabatically tapered DBRs on both the top and bottom sides. In detail, the tapered DBRs have layer thicknesses t3i $=$ t3(1–ρ(2i–1)) for Si andt4i $=$ t4(1–2ρi) for SiO2. In between the tapered Si/SiO2 DBRs, an InP layer containing InAs QD, the light source, is inserted as the spacer layer with the thickness t0 $=$ λB(1–2ρN)/4ne2. The whole micropillar is standing on a semi-infinite Si substrate.

 Fig. 3 Structure and characters of tapered SiO2/Si-InP DBR micropillar cavities: (a) schematic cross section of the high-quality tapered hybrid micropillar cavity and (b) Q factors as a function of the total taper segment number for optimized cavities with pillar diameter of 0.8 μm.

With only 2/3.5 DBR pairs and 2 tapered DBR segments, the proposed cavity has quality as good as the previous conventional Si/SiO2-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 $=$ 0.8 μm and 4/6.5 DBRs pairs. It is seen that the Q factor increases monotonically with the total taper segment number N, by in average one order for every additional taper segment. Compared with conventional counterparts, which have (4+N)/(6.5+N) pairs of quarter-wavelength-thick Si/SiO2 DBRs, a wavelength-thick InP spacer, and show Q factor of below 100, the tapered DBRs increase the Q factor for 1 to 3 orders of magnitude. Typically, the Q factor is increased to be ~8×104 by three segments of tapered Si/SiO2 DBRs. With 4 taper segments, there seems saturation effects so that the Q factor reaches 1.4×105, only about twice that of 3 taper segments. This is because that there are no longer enough conventional DBR pairs to take the role of the vertical optical confinement for the would-be higher Q factor. As a matter of fact, when increasing the conventional DBR pairs to 6/9.5, the Q factor can be further increased to be 2.7×105, as shown by a solid square symbol in Fig. 3 (b). Note that this design decreases the pillar diameter from more than 2 μm down to less than 1 μm. What is more, the mode volume V is decreased to be as small as 0.8(λ/n)3.

It is no doubt that the above improved cavity quality makes it easier to weakly couple a 1.55-μm 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 $=\!\sqrt {{e^2}f\!{\text{/}}\!\left( {4{{\text{ε}} _0}{n^2}{m_0}V} \right)}$ and the loss rate of the cavity mode ${\text{κ}}\!=\!2{\text{π}} c$ / $Q{\text{λ}}$ . Using the oscillator strength of QD f $=$ ε0m0cλ2/(2πne2T1), the condition becomes Q/ $\sqrt V \;{\text{>}} \sqrt {{\text{π}} {\text{λ}}{\text{/}}{r_e}nf}$ /2~3000, where re $=$ e2/(4πε0m0c2) is the classical radius of electron. Obviously, this condition can be easily satisfied by the tapered Si/SiO2-InP hybrid micropillar cavity. It is indicative of the feasibility of constructing a coherent SPS or other quantum devices for 1.55-μm band quantum information processing.

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 3-quarter-wavelength-thick 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/SiO2-InP hybrid micropillar cavities. Based on a λ/4-3λ/4 structure, i.e., t3 $=$ 3λB/4ne3 for Si and t4 $=$ λB/4ne4 for SiO2 in the DBRs, let us see a design with adiabatic DBR layers t3i $=$ t3(1–ρ3i) for Si and t4i $=$ t4(1–ρ4i) for SiO2, and the spacer InP layer t0 $=$ 3λB(1–ρ3(N+1))/4ne2.

Arbitrarily tuning ρ3 and ρ4, the layers in adiabatic DBRs can change their thicknesses in different ways. One way is to set the Si layer increasing (ρ3<0) and the SiO2 layer decreasing (ρ4>0). Taking an example ofρ3 $=$ –1/12 and ρ4 $=$ 1/4, the cavity structure appears like that of Fig. 4 (a). Such a cavity, as shown in Fig. 4 (b), presents FP of 20 to 50 for λ≈1.55 μm at the pillar diameter D around 0.8 μm and ~1.4 μm, much smaller than 2.3 μm for the conventional Si/SiO2-InP hybrid micropillar cavity. More importantly, the spacer InP layer is increased to be ~500 nm. This result is consistent with the theoretical analysis above. On a cavity with increasing the Si thickness while fixing the SiO2 thickness, FP can be 20 to 40 at D around 1.15 μm with the spacer InP layer more than 500 nm thick. As the thickness of Si is fixed and that of SiO2 decreases, there can be FP of 20 to 50 at D around 0.9 μm and FP of 20 to 150 at D around 1.5 μm for λ≈1.55 μm, while the spacer InP is ~440 nm thick. These designs remain the optical quality, decrease the diameter, reduce the influence from the interface defects, and degrade the process difficulty for the 1.55-μm Si/SiO2-InP hybrid micropillar cavities.

 Fig. 4 Structure and characters of adiabatic SiO2/Si-InP DBR micropillar cavities: (a) scheme of a Si/SiO2-InP hybrid 3λ/4-λ/4-DBR micropillar cavity in which adiabatic Si and SiO2 change in different ways and (b) Purcell factors of this type of cavity in the cases where adiabatic DBRs are variously constructed. That of previous conventional cavity is shown as a contrast.

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/SiO2 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 t3 $=$ λB/4ne3 for Si and t4 $=$ λB/4ne4 for SiO2 in DBRs, t3i $=$ t3(1–2ρ34i) for Si and t4i $=$ t4(1–ρ34(2i–1)) for SiO2 in the outer adiabatic DBRs, t1i $=$ λB(1–ρ12(2i–1))/4ne1 for InGaAsP and t2i $=$ λB(1–2ρ12i)/4ne2 for InP in the inner adiabatic DBRs, and t0 $=$ λB(1–2ρ12(Nex+Nin))/4ne1 for the central InGaAsP spacer, where Nex(Nin) is the total segment number of the outer (inner) adiabatic DBRs.

 Fig. 5 Structure and characters of adiabatic SiO2/Si-InGaAsP/InP hybrid-DBR micropillar cavities: (a) scheme of the micropillar cavity with hybrid (Si/SiO2 and InGaAsP/InP) adiabatic DBRs and (b) Q factors of such cavities with the same or different adiabatic parameters in the two types of adiabatic DBRs. That of previous conventional cavity is shown as a contrast.

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/SiO2 and InGaAsP/InP adiabatic DBRs, ρ12 $=$ ρ34. In this design, the optimization condition gives good cavity quality: Q factor of 4000 for λ≈1.55 μm at D around 0.8 μm, improved in quality and greatly reduced in size. Better quality can be obtained if two types of adiabatic DBRs are carefully tuned independently, i.e., ρ12ρ34. This design, optimizing the matching between Si/SiO2 and InGaAsP/InP DBRs, brings about Q factor of 1×104 to 4×104 for λ≈1.55 μm at D around 0.9 μm, which is close to the high-quality tapered-Si/SiO2-InP micropillar cavity[13] mentioned above. What is more outstanding, the semiconductor InGaAsP/InP layer becomes more than 1 μm thick, which greatly avoids the effect from the interface defects to the light source.

The hybrid micropillar cavities, however, have disadvantages in material-separated thermal expansion and the complicated fabrication process. Referred to air-gap planar DBR cavities[15], we may use a micropillar consisting of semiconductor layers with partial air-gaps[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 small-sized InP layers are compassed by surrounding air-gaps, or namely with air-apertures. Each InP layer in the DBRs is set as thick as t1 $=$ λB/4, which is actually quarter wavelength in air, while the InGaAsP layers in the DBRs are normally set quarter-wavelength thick, i.e., t2 $=$ λB/4n2. The inserted between the conventional DBRs are more InGaAsP/InP-air-aperture segments as tapered DBRs on both the top and bottom sides, with thicknesses t1i $=$ t1(1–ρ(2i–1)) for InP and t2i $=$ t2(1–2ρi) for InGaAsP. In between the tapered DBRs, an InP layer is inserted as the spacer layer with thickness t0 $=$ t1(1–2ρN). An InAs QD is set in this layer as the light source.

 Fig. 6 Structure and characters of InGaAsP/InP-air-aperture DBR micropillar cavities: (a) schematic demonstration of the cavity structure and (b) Q factors as functions of the adiabatic parameters.

Simulation shows that this InGaAsP/InP-air-aperture micropillar cavity has two useful modes. The fundamental mode O is optimized at D $=$ 0.915 μm and d $=$ 0.260 μm. Adiabatically tuning the taper slope, the Q factor of more than 104 can be obtained for λ≈1.55 μm, as shown in Fig. 6 (b). A higher-order mode A is optimized at D $=$ 0.935 μm and d $=$ 0.265 μm. Fig. 6 (b) demonstrates that the Q factor as high as 105 can be obtained for λ≈1.55 μm if the adiabatic taper slope ρ is suitably set. These two modes both have small mode volumes ~(n/λ)3. It is ready to see that the mode O easily satisfies the requirement of weak coupling, while the mode A effectively enables strongly coupling at λ≈1.55 μm. More importantly, this cavity can be fabricated by a monolithic process, simply including epitaxial growth, dry etching for pillar formation, and selective wet-etching for the air-apertures. Therefore, the adiabatic design makes the InGaAsP/InP-air-aperture micropillar cavity as a promising candidate for efficient/indistinguishable and coherent 1.55-μm QD SPSs applied in silica-fiber-based quantum information processing.

4. Conclusions

Adiabatic DBRs are proved effective in creating high-quality micropillar cavities for 1.55-μm 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/SiO2-InP hybrid micropillar cavity, useful as a weak-coupling SPS, inserting adiabatically tapered Si/SiO2 DBRs can enable strong coupling with 1.55-μm 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/SiO2-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/SiO2 DBRs. On an InGaAsP/InP-air-aperture micropillar cavity, overcoming the hybrid fabrication by a monolithic process, the adiabatic design brings about a high Q factor (>1×104), small mode volume, and nanoscaled diameter, and satisfies both weak and strong coupling to 1.55-μm InAs/InP QDs. Therefore, micropillar cavities with adiabatic DBR structures are prospective candidates for 1.55-μm QD SPSs applied in silica-fiber-based quantum information processing.

Acknowledgment

H.-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.

References
 [1] K. J. Vahala, " Optical microcavities,” Nature, vol. 424, no. 6950, pp. 839-846, Aug. 2003. [2] P. Michler, A. Kiraz, C. Becher,et al., " A quantum dot single-photon turnstile device,” Science, vol. 290, no. 5500, pp. 2282-2285, Dec. 2000. [3] J. P. Reithmaier, G. Sęk, A. Löffler,et al., " Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature, vol. 432, no. 7014, pp. 197-200, Nov. 2004. [4] K. Takemoto, Y. Nambu, T. Miyazawa,et al., " Quantum key distribution over 120 km using ultrahigh purity single-photon source and superconducting single-photon detectors,” Scientific Reports, vol. 5, pp. 14383:1-7, Sept. 2015. [5] M. Lermer, N. Gregersen, F. Dunzer,et al., " Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Physical Review Letters, vol. 108, no. 5, pp. 057402:1-4, Jan. 2012. [6] Y.-N. Zhang and M. Lončar, " Submicrometer diameter micropillar cavities with high quality factor and ultrasmall mode volume,” Optics Letters, vol. 34, no. 7, pp. 902-904, Apr. 2009. [7] E. D. Palik, Handbook of Optical Constants of Solids, San Diego: Academic Press, 1998. [8] K. Takemoto, Y. Nambu, T. Miyazawa,et al., " Transmission experiment of quantum keys over 50 km using high-performance quantum-dot single-photon source at 1.5 μm wavelength,” Applied Physics Express, vol. 3, no. 9, pp. 092802:1-4, Sept. 2010. [9] T. Kuroda, Y. Sakuma, K. Sakoda, K. Takemoto, and T. Usuki, " Decoherence of single photons from an InAs/InP quantum dot emitting at a 1.3 μm wavelength,” Physica Status Solidi, vol. 6, no. 4, pp. 944-947, Apr. 2009. [10] D. Dalacu, D. Poitras, J. Lefebvre, P. J. Poole, G. C. Aers, and R. L. Williams, " InAs/InP quantum-dot pillar microcavities using SiO2/Ta2O5 Bragg reflectors with emission around 1.55 μm,” Applied Physics Letters, vol. 84, no. 17, pp. 3235-3237, Apr. 2004. [11] H.-Z. Song, K. Takemoto, T. Miyazawa,et al., " Design of Si/SiO2 micropillar cavities for Purcell-enhanced single photon emission at 1.55 μm from InAs/InP quantum dots,” Optics Letters, vol. 38, no. 17, pp. 3241-3244, Sept. 2013. [12] H.-Z. Song, K. Takemoto, T. Miyazawa,et al., " High quality-factor Si/SiO2-InP hybrid micropillar cavities with submicrometer diameter for 1.55-μm telecommunication band,” Optics Express, vol. 23, no. 12, pp. 16264-16272, Jun. 2015. [13] A. Yariv, Optical Electronics, 4th ed. San Francisco: Saunders College, 1991. [14] A. Auffèves, D. Gerace, J. M. Gérard, M. F. Santos, L. C. Andreani, and J. P. Poizat, " Controlling the dynamics of a coupled atom-cavity system by pure dephasing,” Physical Review B, vol. 81, no. 24, pp. 245419:1-10, Jun. 2010. [15] C.-K. Lin, D. P. Bour, J.-T. Zhu,et al., " High temperature continuous-wave operation of 1.3- and 1.55-μm VCSELs with InP/air-gap DBRs,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 9, no. 5, pp. 1415-1421, Oct. 2003. [16] H.-Z. Song, M. Hadi, Y.-Z. Zheng,et al., " InGaAsP/InP nanocavity for single-photon source at 1.55-μm telecommunication band,” Nanoscale Research Letters, vol. 12, no. 1, pp. 128:1-7, Dec. 2017.