# Polarization Modulation in Quantum-Dot Spin-VCSELs for Ultrafast Data
Transmission

Christos Tselios, Panagiotis Georgiou, Christina (Tanya) Politi, Antonio
Hurtado, Dimitris Alexandropoulos Christos Tselios and Christina (Tanya)
Politi are with Department of Electrical and Computer Engineering, University
of Peloponnese, Patra, Achaia, 22100, Greece.Panagiotis Georgiou and Dimitris
Alexandropoulos are with the Department of Materials Science, University of
Patras, Rion, Achaia 26504 Greece.Antonio Hurtado is with the Institute of
Photonics, Department of Physics, University of Strathclyde, Technology and
Innovation Centre, Glasgow G1 1RD, U.K..Antonio Hurtado acknowledges funding
support by the UKRI Turing AI Acceleration Fellowships Programme
(EP/V025198/1).

###### Abstract

Spin-Vertical Cavity Surface Emitting Lasers (spin-VCSELs) are undergoing
increasing research effort for new paradigms in high-speed optical
communications and photon-enabled computing. To date research in spin-VCSELs
has mostly focused on Quantum-Well (QW) devices. However, novel Quantum-Dot
(QD) spin-VCSELs, offer enhanced parameter controls permitting the effective,
dynamical and ultrafast manipulation of their light emission’s polarization.
In the present contribution we investigate in detail the operation of QD spin-
VCSELs subject to polarization modulation for their use as ultrafast light
sources in optical communication systems. We reveal that QD spin-VCSELs
outperform their QW counterparts in terms of modulation efficiency, yielding a
nearly two- fold improvement. We also analyse the impact of key device
parameters in QD spin-VCSELs (e.g. photon decay rate and intra-dot relaxation
rate) on the large signal modulation performance with regard to optical
modulation amplitude and eye-diagram opening penalty. We show that in addition
to exhibiting enhanced polarization modulation performance for data rates up
to $250Gb/s$, QD spin-VCSELs enable operation in dual (ground and excited
state) emission thus allowing future exciting routes for multiplexing of
information in optical communication links.

###### Index Terms:

ultrafast spin-lasers, Quantum-Dot spin-VCSEL, polarization modulation,
polarization dynamics

##  I Introduction

Spintronic devices like, spin-polarized Light-Emitting Diodes (LEDs) [1],
spin-Vertical Cavity Surface Emitting Lasers (VCSELs) [2, 3, 4] and bimodal QD
micropillar cavities [5] are being widely investigated for next-generation
optical communication systems. In these devices the radiative recombination of
spin-polarized carriers leads to luminescence exhibiting circular polarized
emission. There are several advantages of spin devices that have fuelled
recent research. These include reduced lasing threshold [6] compared to
conventional lasers, spin amplification [7, 8] and polarization resolved
output [9, 10, 11]. Spin lasers, including spin VCSELs, exhibit ultrafast
dynamics [6, 12] due to the spin coupling between electrons and photons in
spin-VCSELs thus overcoming the limitation of relaxation oscillation frequency
[13] commonly faced by conventional non-spin lasers.

Harkhoe et al. have used polarization modulation in spin-VCSELs to improve the
processing speed of a photonic reservoir computing [14]. Yokota et al. have
used spin-VCSELs as Analog Radio-over-Fiber transmitters for networks beyond
$5G$ [4]. Lindemann et al. demonstrated the potential to enhance the
modulation bandwidth of optical communication systems up to $240$ $GHz$ and
single channel data transmission rate of $240$ $Gb/s$ [3]. The latter is
nearly an order of magnitude higher than what is achieved with conventional
directly modulated semiconductor lasers. Recently, Huang et al. explored high-
frequency polarization modulation in optically pumped spin-VCSELs over a broad
range of basic device properties and operating conditions [15].

These aforementioned demonstrations employed Quantum-Well (QW) spin-VCSELs.
However, Quantum Dots (QDs) hold much promise for spin devices. Huang et al.
[16] demonstrated electron spin polarization exceeding 90$\%$ at room
temperature by remote spin filtering of InAs QD electrons without a magnetic
field. Giba et al. [17] used QD as gain material in p-doped spin-polarized
Light-Emitting Diodes (LED) to show a two fold improvement in the degree of
circular polarization compared to QW spin-LEDs at zero magnetic field. The
spin optoelectronics literature has been recently enriched with few accounts
on the potential of QD spin-VCSELs for optical communication and processing
applications [18, 19]. Huang and co-workers [18] numerically demonstrated a
high-speed secure key distribution (SKD) that is enabled by the tunability of
the pump polarization ellipticity in QD spin-polarized VCSELs. In [19]
Skontranis et al. exploited emission from two discrete wavebands and two
polarization states to enhance computational efficiency of a dual-state QD
spin-VCSEL based Reservoir Computing (RC). The broad spectral bandwidth of QD
structure can be deployed in wavelength-division multiplexing systems (WDM).

Motivated by these, we study theoretically in detail the polarization
modulation in QD spin-VCSELs for ultrafast optical communication applications.
Our approach demonstrates better performance of QD spin-VCSELs over their QW
counterparts in terms of modulation efficiency. The gauge of evaluation are
the eye diagrams and the Eye Opening Penalty (EOP). The latter reveals the
quality of the dynamic performance of optically modulated signals. To analyse
the modulation efficiency in QD spin-VCSELs, we investigate the amplitude of
the optical modulation as a function of the photon decay rate and intradot
relaxation rate. We use this technique to elucidate the effect of these key
parameters on the evolution of the output polarization ellipticity of QD spin-
VCSELs. Our investigations reveal ultrafast modulation speeds of $250$ $Gb/s$
(and possibly higher) with QD-spin VCSELs with enhanced modulation efficiency
when compared with their QW counterpart. We also show that QD spin-VCSELs
yield ultrafast operation from both their ground states (GS) and excited
states (ES); thus offering great prospects for ultrafast dual wavelength laser
modules.

##  II Theory

The model used here is an elaboration of the SFM coupled rate equations [20]
for the case of dual-state QD spin-VCSELs that accounts for upwards
transitions. It is experimentally demonstrated that carrier escape from the
excited and ground levels into the wetting layer has non-negligible effects on
the carrier density and spin polarization in both levels [16]. The modified
rate equations read:

$\begin{split}\frac{dn_{WL}^{\pm}}{dt}=\gamma_{n}J^{\pm}-\gamma_{0}n_{WL}^{\pm}(1-f_{ES}^{\pm})-\gamma_{n}n_{WL}^{\pm}+\\\
\noindent{+4\gamma_{esc}f_{ES}^{\pm}\mp\gamma_{j}(n_{WL}^{+}-n_{WL}^{-})}\end{split}$
(1)
$\begin{split}\frac{df_{ES}^{\pm}}{dt}&=\frac{1}{4}\gamma_{0}n_{WL}^{\pm}(1-f_{ES}^{\pm})-\gamma_{esc}f_{ES}^{\pm}-\gamma_{n}f_{ES}^{\pm}-\\\
&-\gamma_{21}f_{ES}^{\pm}(1-f_{GS}^{\pm})+\frac{1}{2}\gamma_{12}f_{GS}^{\pm}(1-f_{ES}^{\pm})-\\\
&-\frac{1}{4}\gamma_{n}(2f_{ES}^{\pm}-1)|E_{s_{ES}}^{\pm}|^{2}\mp\gamma_{j}(f_{ES}^{+}-f_{ES}^{-})\end{split}$
(2)
$\centering\begin{split}\frac{df_{GS}^{\pm}}{dt}&=2\gamma_{21}f_{ES}^{\pm}(1-f_{GS}^{\pm})-\gamma_{12}f_{GS}^{\pm}(1-f_{ES}^{\pm})-\gamma_{n}f_{GS}^{\pm}-\\\
&-\frac{1}{2}\gamma_{n}(2f_{GS}^{\pm}-1)|E_{s_{GS}}^{\pm}|^{2}-\gamma_{j}(f_{GS}^{+}-f_{GS}^{-})\end{split}\@add@centering$
(3)
$\frac{dE_{GS}^{\pm}}{dt}=k[h_{1}(2f_{GS}^{\pm}-1)](1+i\alpha)E_{GS}^{\pm}-(\gamma_{a}+i\gamma_{p})E_{GS}^{\mp}$
(4)
$\frac{dE_{ES}^{\pm}}{dt}=k[h_{2}(2f_{ES}^{\pm}-1)](1+i\alpha)E_{ES}^{\pm}-(\gamma_{a}+i\gamma_{p})E_{ES}^{\mp}$
(5)

The dynamical variables $n_{WL}$ and $n_{ES}$ ($n_{GS}$) denote normalized
conduction band carrier concentrations of Wetting Layer (WL) and ES (GS)
level, respectively. Here, superscripts $+$ and $-$ correspond to the Right
Circular Polarized (RCP) and Left Circular Polarized (LCP) components of the
emitted light. The SFM parameters for QD spin-VCSEL are defined as follows:
$k$ is the photon decay rate, $\alpha$ is the linewidth enhancement factor,
$h_{1}$ is the normalized gain coefficient for GS transitions, $h_{2}$ is the
normalized gain coefficient for ES transitions, $\gamma_{n}$ is the carrier
recombination rate, $\gamma_{21}$ is the intradot relaxation rate of spin
polarized carriers relax from ES level at the spin-up(down) GS level,
$\gamma_{0}$ is the capture rate from WL into ES level, $\gamma_{j}$ is the
spin relaxation rate at ES,GS and WL, $\gamma_{p}$ is the birefringence rate
and $\gamma_{a}$ is the dichroism rate. Some carriers can be excited thermally
from ES to WL with escape rate $\gamma_{esc}$ and from GS to ES with escape
rate $\gamma_{12}$. These relations read as follows: $\gamma_{12}$ =
$\gamma_{21}$$e^{-\frac{\Delta{E_{ES,GS}}}{k_{B}T}}$ and $\gamma_{esc}$ =
$\gamma_{0}$$e^{-\frac{\Delta{E_{WL,ES}}}{k_{B}T}}$, where $\Delta{E_{ES,GS}}$
($\Delta{E_{WL,ES}}$) is the energy of carrier excitation from GS (WL) to ES
(WL) and $k_{B}T$ is the product of Boltzmann constant $k_{B}$ and the
temperature $T$.

Lasing occurs via transitions from the ES or GS, to the valence band (VB)
emitting right $(E_{ES}^{+},E_{GS}^{+})$ and left $(E_{ES}^{-},E_{GS}^{-})$
circularly polarized electric fields at two distinct wavelengths. The carrier
injection at ES and GS threshold is respectively denoted by $I_{ES,th}^{\pm}$
and $I_{GS,th}^{\pm}$.

Polarization control can be realized by means of the pump parameters, namely
normalized pump stength $J$, which is equal to $1$ at threshold, and pump
ellipticity $P$. The total pump strength is defined by $J=J_{+}+J_{-}$, where
$J_{+}$ and $J_{-}$ are RCP and LCP normalized pump components and the pump
polarization ellipticity by:

$P=(J_{+}-J_{-})/(J_{+}+J_{-})$ (6)

When, $P=0$ is applied there is no preference on the spin polarization of the
injected carriers $(P=0\rightarrow J_{+}=J_{-})$ and spin-VCSELs are reduced
to conventional VCSELs. The QD spin-VCSEL output is expressed in terms of
circularly polarized intensities
$I_{ES}^{+}=|E_{ES}^{+}|^{2},I_{ES}^{-}=|E_{ES}^{-}|^{2}$ and the polarization
ellipticity $\epsilon$ is defined as:

$\epsilon_{ES}=(I_{ES}^{+}-I_{ES}^{-})/(I_{ES}^{+}+I_{ES}^{-})$ (7)

describing almost circular polarized emission when $\epsilon_{ES}\rightarrow
1$ and linear emission in the case of $\epsilon_{ES}\rightarrow 0$. The same
applies for $I_{GS}^{+}$, $I_{GS}^{-}$ and $\epsilon_{GS}$. Throughout this
paper we use the following parameter values: $k_{B}T=0.025$ $eV$,
$\Delta{E_{ES,GS}}=0.05$ $eV$, $\Delta{E_{WL,ES}}=0.1$ $eV$, $\alpha=3$,
$h_{1}=1.995$, $J=100$, $\gamma_{p}=500$ ${ns^{-1}}$, $\gamma_{n}=1$
${ns^{-1}}$, $\gamma_{0}=$ $800{ns^{-1}}$ and $\gamma_{a}=0$ ${ns^{-1}}$.
Higher values of $J$ can improve polarization modulation performance (see
Appendix II), however, there is a trade-off as growing $J$ will increase the
energy consumption and heating levels in the device.

##  III High speed polarization modulation in dual state QD spin-VCSEL

Figure 1: 2-bit-long eye diagrams of the ES of the QD spin-VCSEL subject to
polarization modulation for bit sequences of $10$ $Gb/s$ $(a)$, $25$ $Gb/s$
$(b)$, $100$ $Gb/s$ $(c)$ and $250$ $Gb/s$ $(d)$. For all panels:
$\gamma_{j}=250$ $ns^{-1}$, $\gamma_{p}=500$ $ns^{-1}$ and $k=250$ $ns^{-1}$.

Optical pumping using circularly polarized light can generate spin-polarized
electron population under optical selection rules. Spin-up electrons are
related to LCP emission, while spin-down electrons result in RCP emission [2].
Based on this, we can modulate the QD spin-VCSEL’s output ellipticity by
dynamically controlling the input ellipticity with fast time-varying optical
spin injection pulses. To do this, we implement a Non-Return-to-Zero (NRZ)
polarization modulation scheme to the QD spin-VCSEL. Bits ’1’ and ’0’ are
defined as RCP and LCP pumping respectively [3]. The bit stream ($127$
pseudorandom bits) is filtered by a raised-cosine filter in order to minimize
the intersymbol interference [21]. The corresponding $\epsilon$, which is
calculated by means of polarized intensities (see equation 7), contains the
information encoded into $P$.

Figure 2: Optical modulation amplitude maps for both ES ($a$) and GS ($e$) as
a function of $\gamma_{21}$ and $k$, evolution of $\epsilon_{ES}$ $(b)$ and
$\epsilon_{GS}$ $(f)$ for a 18-bit word at $250$ $Gb/s$ for three different
pairs of $(k,\gamma_{21})$ and eye diagrams for ES with $k=500$ $ns^{-1}$,
$\gamma_{21}$ = $50$ $ns^{-1}$ $(c)$ and $k=500$ $ns^{-1}$, $\gamma_{21}$ =
$500$ $ns^{-1}$ $(d)$ and GS with $k=500$ $ns^{-1}$, $\gamma_{21}$ = $250$
$ns^{-1}$ $(g)$ and $k=500$ $ns^{-1}$, $\gamma_{21}$ = $500$ $ns^{-1}$ $(h)$.

Spin-lasers revealed the necessity of increasing $\gamma_{p}$ and $\gamma_{j}$
for fast polarization modulation. These can be employed for the case of QD
spin-VCSELs as well. By implementing high-contrast periodic grating, large
birefringence rates can be achieved for QD spin-V(E)CSEL [22]. Despite the
fact that earlier studies promoted the superiority of QD spin VCSELs due to
longer spin relaxation times ($\tau_{s}$) on low temperatures or doped dots,
recent results showed that for carrier lifetime of $1$ ns, $\tau_{s}$ is in
the range between 5 and 8 ps for an optically pumped QD spin-V(E)CSEL lasing
at room temperature [23]. Choosing $\gamma_{j}$ = 250 $ns^{-1}$ and
$\gamma_{p}=500$ $ns^{-1}$, we demonstrate that QD spin-VCSELs can perform
modulation of data for bit rates up to $250$ Gb/s (and possibly higher). This
can be seen in Fig. 1 which shows the eye patterns obtained for the QD spin-
VCSEL subject to polarization modulation for four different bit rates ($10$
$(a)$, $25$ $(b)$ , $100$ $(c)$ and $250$ $(d)$ Gb/s). For the two highest
speed cases investigated of $100$ and $250$ Gb/s, the symmetric shape of the
eye diagrams is disrupted to a certain extent and also reduced vertical eye
diagram openings are obtained. In the cases of $100$ and $250$ Gb/s, although
the achieved extinction ratios are reduced, these permit successful data
modulation even at these ultrafast speed rates. For all cases, GS modulation
efficiency is negligible with respect to the corresponding ES.

The eye diagrams indicate that QD spin-VCSELs deliver better performance to QW
spin-VCSELs [3] in terms of modulation efficiency. Modulation efficiency is
defined as how efficiently a spin-VCSEL cavity converts a $100\%$ input
circular polarized light into another output polarization. For this
application, Lindemann et al. [3] demonstrated single channel ultrafast data
transmission with low amplitude of polarization oscillations, which reaches up
to 1% (0.01) for a 100% injected spin polarization as presented on their
calculated eye diagrams. By considering the maximum value of $\epsilon_{ES}$,
i.e. 5$\%$, we find that it is almost five times larger as that obtained for
QW spin-VCSELs [3]. We also prove that optical pumping is as suitable as
hybrid pumping scheme combining electrical with circularly-polarized optical
pumping for high bit rate modulation.

##  IV Optimization analysis of ultrafast polarization modulation performance

An optimization analysis of different photon lifetimes in the cavity and
intradot relaxation rates in QD spin-VCSELs can lead to even higher values of
modulation efficiency for ultrafast data transmission. The investigation of
the influence of the photon lifetime on the polarization oscillations is
crucial since the rapid emission of the photons from cavity can prevent the
mixing with photons with different ellipticities emitted after the strong
subband mixing. The output ellipticity of both ES and GS is based on the
interplay between fundamental physical processes; the the spin flip processes
tends to equalize the gain for RCP and LCP fields, the intradot relaxation
from ES to GS and the escape of carriers from GS to ES and ES to WL which
couple carriers back and forth between the polarized fields. Hence, we expect
that the variation of $\gamma_{21}$ will enhance the $\epsilon_{GS}$, as GS
will receive carriers emitted from the pumped sublevel before electrons relax
to the density with a different spin. While, the $\epsilon_{ES}$ will be
decreased due to the escaped carriers from GS, which are burdened by strong
subband mixing since spin relaxation is faster than the escape procedure. ES
plays an important role on GS polarization modulation. Rohm et al. [24] and Wu
et al. [25] demonstrated that the increase of excited-to-ground-state
relaxation time strongly limits the GS modulation bandwidth of the QD
conventional laser.

The performance of QD spin-VCSELs as polarization modulation transmitter is
evaluated by the optical modulation amplitude and the eye diagrams. The
optical modulation amplitude is defined as the difference between two levels
of eye diagrams, i.e. output ellipticity values when bits ’1’ and ’0’ are
encoded. To this end, we illustrate optical modulation amplitude for both GS
and ES in the plane of $\gamma_{21}$ and $k$ in Figs. 2 $(a)$ and $(d)$,
respectively. The specific range of $\gamma_{21}$ was chosen from reported
values found in QD lasers literature [26, 27]. In the case of ES the key
parameters seems to have an impact on polarization modulation performance when
$\gamma_{21}$ is lower than $300$ $ns^{-1}$. The increase of $k$ increases the
optical modulation amplitude as we expected, while the increase of
$\gamma_{21}$ leads to lower optical modulation amplitude, a disadvantage
arising from the escape of polarized carriers from GS. The strong subband
mixing yielding in an almost linearly polarized emission ($\epsilon=0$) is
inevitable in the case of GS performance since carrier recombination rate is
low with respect to spin relaxation rate. This can be compensated by the
increase of $\gamma_{21}$ and $k$, where the performance is benefited from the
fast decay of the polarized carriers to the GS.

For high bit rate modulation it is important to investigate the pattern
dependent effects. For presentation purposes we choose to investigate the
impact of three different pairs of $k$ and $\gamma_{21}$, which values provide
close agreement with experimental observations, when the QD spin-VCSEL is
modulated with a 18-bit word at $250$ $Gb/s$. In Fig. 2 $(b)$ the increase of
$k$ results in higher amplitude polarization oscillation (orange curve) with a
clear open eye-diagram (Fig. 2 $(c)$), while the increase of $\gamma_{21}$ up
to $500$ $ns^{-1}$ shows strong pattern dependence and whilst this would
result in a closure of the eye-diagram opening albeit with reduced modulation
efficiency (Fig. 2 $(d)$) yet good extinction ratio between ’1’ and ’0’ bits
is achieved. In Fig. 2 $(f)$ it is shown that all traces have similar
behaviour with increased $\epsilon_{GS}$ in the case of increasing
$\gamma_{21}$ and $k$, the first pulse in a string of pulses is lower than the
ones that follow it. Although both eyes for two pairs of $\gamma_{21}$ and $k$
($k=500$ $ns^{-1}$, $\gamma_{21}$ = $250$ $ns^{-1}$ $(g)$ and $k=500$
$ns^{-1}$, $\gamma_{21}$ = $500$ $ns^{-1}$ $(h)$) are wide open.

The optimization analysis indicates that the use of $k=500$ $ns^{-1}$ and
$\gamma_{21}=500$ $ns^{-1}$ provides enhancement performance, guaranteeing
high polarization extinction ratio between ’1’ and ’0’ bits. The GS response
in the QD spin-VCSEL shows clear open eye at $250$ $Gb/s$ data rates (Fig. 2
$(h)$) with modulation efficiency reaching up to $6\%$, whilst the results
obtained for the QD spin-VCSEL’s ES emission reveal dual ultrafast data
transmission (at $250$ $Gb/s$ in Fig. 2 $(d)$). Therefore, importantly QD
spin-VCSELs in contrast to their QW counterparts permit very high-speed data
modulation from both states, offering promise for dual wavelength light
sources for use as in ultrafast optical communications.

##  V Comparison of polarization modulation performance between QD and QW
spin-VCSELs

Figure 3: $(a-c)$ EOP maps for GS QD spin-VCSEL, ES QD spin-VCSEL and QW spin
VCSEL, respectively. $(d-f)$ The corresponding results for optical modulation
amplitude. The SFM parameters are the same as in Figs. 2 $(d,h)$ but with
$a=5$ for QW spin-VCSEL. The inset eye diagram with the maximum amplitude for
$\gamma_{p}=550$ $ns^{-1}$ and $\gamma_{s}=400$ $ns^{-1}$

The superior modulation performance of QD spin-VCSELs over QW spin-VCSELs is
demonstrated in this section where we compare the polarization modulation
performance of GS (Figs. $\ref{fig:comparison}(a,d))$), ES (Figs.
$\ref{fig:comparison}(b,e)$) and QW spin-VCSEL (Figs.
$\ref{fig:comparison}(c,f)$) by means of optical modulation amplitude and EOP
which is the ratio of the optical modulation amplitude and the vertical eye
opening. This metric is useful for noise free systems since in binary
modulated digital optical systems, waveform distortions represented by eye-
opening penalty may be introduced by nonlinear effects in the transmitter and
can only be manifested through time-domain measurements. We will use this here
in order to find conditions resulting in suitable values of EOP where improved
performance is achieved.

For the simulation of QW spin-VCSELs we employ the generalized SFM from [3].
We use the same SFM parameters as in Figs. 2 $(d,h)$ but with $a=5$ for QW
spin-VCSEL, typical value for linewidth enhancement factor in QWs. Another
difference in the SFM between QW and QD spin-VCSELs is that spin relaxation
rate for the case of QW spin-VCSELs is defined as
$\gamma_{s,QW}=\gamma_{n}+2\gamma_{s,QD}$ [28]. Here, $\gamma_{s,QW}$ is
defined as $\gamma_{s}$ and $\gamma_{s,QD}$ as $\gamma_{j}$ for convenience.
The analysis of optical modulation amplitude’s maps reveal polarization
oscillation with amplitude nearly equal to $0.05$ $(5\%)$ in the case of QW
spin-VCSEL. In terms of single channel data transmission QD spin-VCSELs
outperform their state-of-the-art QW counterparts in terms of modulation
efficiency, yielding a two-fold improvement. Meanwhile, QD spin-VCSELs can
provide two channels ultrafast data transmission with comparable modulation
efficiency of each channel with QW spin VCSEL at its best conditions,
increasing significantly the capacity. It is believed that the proposed high
efficiency dual state QD spin-VCSEL will provide a new device oriented towards
full polarization control at ultrafast regimes.

##  VI Conclusion

In conclusion, we have investigated the ultrafast polarization modulation
properties of QD spin-VCSELs by means of large signal modulation. We provide a
fundamental understanding of their potentials for ultrafast light sources for
use in future optical communication platforms and investigate the effect of
key system parameters for optimised performance. Ultrafast bit rate
polarization modulation analyzed via eye diagrams revealed modulation of data
can be performed for bit rates up to $250$ $Gb/s$ (and possibly higher) with
high modulation efficiencies, exceeding those achievable with QW spin-VCSELs.
Notably, QD-spin-VCSELs allow high speed data modulation from both the GS and
ES; hence offering great promise for ultrafast dual wavelength sources. QD
spin-VCSELs offer therefore an exciting platform for ultrafast sources with
optimised performance and yielding operation at hundreds of Gb/s rates for use
in future optical data communication systems.

## Acknowledgments

The authors acknowledge Prof. Mike Adams of the School of Computer Science and
Electronic Engineering, University of Essex, UK for proofreading the article
and his comments on the chosen parameters.

[Proof of the SFM Rate Equations] The complete set of Rate Equations (REs)
describing the modified SFM model reads as:

$\begin{split}\frac{d(N_{WL}f_{WL}^{\pm})}{dt}=\frac{I}{q}-\gamma_{0}N_{WL}f_{WL}^{\pm}(1-f_{ES}\pm)+\\\
\noindent{+4N_{D}\gamma_{esc}f_{ES}^{\pm}-\gamma_{n}N_{WL}f_{WL}^{\pm}\mp\gamma_{j}N_{WL}(f_{WL}^{+}-f_{WL}^{-})}\end{split}$
(8)
$\begin{split}\frac{d(4N_{D}f_{ES}^{\pm})}{dt}&=\gamma_{0}N_{WL}f_{WL}^{\pm}(1-f_{ES}^{\pm})-4\gamma_{esc}N_{D}f_{ES}\pm-\\\
&-4\gamma_{n}N_{D}f_{ES}^{\pm}-\gamma_{21}4N_{D}f_{ES}^{\pm}(1-f_{GS}^{\pm})+\\\
&+\gamma_{12}2N_{D}f_{GS}^{\pm}(1-f_{ES}^{\pm})\mp\gamma_{j}4N_{D}(f_{ES}^{+}-f_{ES}^{-})-\\\
&-v_{g}\Gamma{a_{ES}}{N_{D}}(2f_{ES}^{\pm}-1)|E_{0_{ES}}^{\pm}|^{2}\end{split}$
(9)
$\centering\begin{split}\frac{d(2N_{D}f_{GS}^{\pm})}{dt}&=\gamma_{21}4N_{D}f_{ES}^{\pm}(1-f_{GS}^{\pm})-\gamma_{12}2N_{D}f_{GS}^{\pm}(1-f_{ES}^{\pm})-\\\
&-\gamma_{n}2N_{D}f_{GS}^{\pm}-v_{g}\Gamma{a_{GS}}{N_{D}}(2f_{GS}^{\pm}-1)|E_{0_{GS}}^{\pm}|^{2}\mp\\\
&\mp\gamma_{j}2N_{D}(f_{GS}^{+}-f_{GS}^{-})\end{split}\@add@centering$ (10)
$\frac{dE_{0_{GS}}^{\pm}}{dt}=k[h_{1}(2f_{GS}^{\pm}-1)](1+i\alpha)E_{0_{GS}}^{\pm}-(\gamma_{a}+i\gamma_{p})E_{0_{GS}}^{\mp}$
(11)
$\frac{dE_{0_{ES}}^{\pm}}{dt}=k[h_{2}(2f_{ES}^{\pm}-1)](1+i\alpha)E_{0_{ES}}^{\pm}-(\gamma_{a}+i\gamma_{p})E_{0_{ES}}^{\mp}$
(12)

The modified SFM REs are:

$\begin{split}\frac{df_{WL}^{\pm}}{dt}=\frac{I}{qA\overline{N}_{WL}}-\gamma_{0}f_{WL}^{\pm}(1-f_{ES}^{\pm})+\frac{4N_{D}}{N_{WL}}\gamma_{esc}f_{ES}^{\pm}-\\\
\noindent{-\gamma_{n}f_{WL}^{\pm}\mp\gamma_{j}(f_{WL}^{+}-f_{WL}^{-})},\end{split}$
(13)

where $\overline{N}_{WL}$=${N_{WL}}/{A}$.

$\begin{split}\frac{df_{ES}^{\pm}}{dt}&=\frac{\gamma_{0}N_{WL}}{4N_{D}}f_{WL}^{\pm}(1-f_{ES}^{\pm})-\gamma_{esc}f_{ES}^{\pm}-\\\
&-\gamma_{n}f_{ES}^{\pm}-\gamma_{21}f_{ES}^{\pm}(1-f_{GS}^{\pm})+\\\
&+\frac{1}{2}\gamma_{12}f_{GS}^{\pm}(1-f_{ES}^{\pm})\mp\gamma_{j}(f_{ES}^{+}-f_{ES}^{-})-\\\
&-\frac{1}{4}v_{g}\Gamma{a_{ES}}(2f_{ES}^{\pm}-1)|E_{0_{ES}}^{\pm}|^{2}\end{split}$
(14)
$\centering\begin{split}\frac{df_{GS}^{\pm}}{dt}&=2\gamma_{21}f_{ES}^{\pm}(1-f_{GS}^{\pm})-\gamma_{12}f_{GS}^{\pm}(1-f_{ES}^{\pm})-\\\
&-\gamma_{n}f_{GS}^{\pm}-\frac{1}{2}v_{g}\Gamma{a_{GS}}(2f_{GS}^{\pm}-1)|E_{0_{GS}}^{\pm}|^{2}\mp\\\
&\mp\gamma_{j}(f_{GS}^{+}-f_{GS}^{-})\end{split}\@add@centering$ (15)

For the normalization of the system of REs we introduce a change in variables
as follows:

$\centering n_{WL}^{\pm}=\frac{N_{WL}}{N_{D}}f_{WL}^{\pm}\@add@centering$ (16)
$\centering
E_{GS}=E_{s_{GS}}\sqrt{\frac{\gamma_{n}}{v_{g}\Gamma{a_{GS}}}}\@add@centering$
(17) $\centering
E_{ES}=E_{s_{ES}}\sqrt{\frac{\gamma_{n}}{v_{g}\Gamma{a_{ES}}}}\@add@centering$
(18)
$\centering\frac{I}{qA\overline{N}_{D}\frac{\tau_{r}}{\tau_{r}}}=\frac{I}{I_{x}\tau_{r}}=\frac{J}{\tau_{r}}=\gamma_{n}J\@add@centering$
(19)

The normalized form of modifed SFM REs (Eqs. 8-12) is:

$\begin{split}\frac{dn_{WL}^{\pm}}{dt}=\gamma_{n}J^{\pm}-\gamma_{0}n_{WL}^{\pm}(1-f_{ES}^{\pm})-\gamma_{n}n_{WL}^{\pm}+\\\
\noindent{+4\gamma_{esc}f_{ES}^{\pm}\mp\gamma_{j}(n_{WL}^{+}-n_{WL}^{-})}\end{split}$
(20)
$\begin{split}\frac{df_{ES}^{\pm}}{dt}&=\frac{1}{4}\gamma_{0}n_{WL}^{\pm}(1-f_{ES}^{\pm})-\gamma_{esc}f_{ES}^{\pm}-\gamma_{n}f_{ES}^{\pm}-\\\
&-\gamma_{21}f_{ES}^{\pm}(1-f_{GS}^{\pm})+\frac{1}{2}\gamma_{12}f_{GS}^{\pm}(1-f_{ES}^{\pm})-\\\
&-\frac{1}{4}\gamma_{n}(2f_{ES}^{\pm}-1)|E_{s_{ES}}^{\pm}|^{2}\mp\gamma_{j}(f_{ES}^{+}-f_{ES}^{-})\end{split}$
(21)
$\centering\begin{split}\frac{df_{GS}^{\pm}}{dt}&=2\gamma_{21}f_{ES}^{\pm}(1-f_{GS}^{\pm})-\gamma_{12}f_{GS}^{\pm}(1-f_{ES}^{\pm})-\gamma_{n}f_{GS}^{\pm}-\\\
&-\frac{1}{2}\gamma_{n}(2f_{GS}^{\pm}-1)|E_{s_{GS}}^{\pm}|^{2}-\gamma_{j}(f_{GS}^{+}-f_{GS}^{-})\end{split}\@add@centering$
(22)
$\frac{dE_{GS}^{\pm}}{dt}=k[h_{1}(2f_{GS}^{\pm}-1)](1+i\alpha)E_{GS}^{\pm}-(\gamma_{a}+i\gamma_{p})E_{GS}^{\mp}$
(23)
$\frac{dE_{ES}^{\pm}}{dt}=k[h_{2}(2f_{ES}^{\pm}-1)](1+i\alpha)E_{ES}^{\pm}-(\gamma_{a}+i\gamma_{p})E_{ES}^{\mp}$
(24)

[Indicative Results for $\gamma_{\alpha}$ and $J$] In this Appendix we focus
on the effect of the parameters on the polarization oscillation amplitude. We
simulate parameter sweeps for a single parameter at a time, keeping all others
at the values given in Figs. 2 $(d)$ and $(h)$.

The increase of $J$ from $100$ to $200$ almost doubles the polarization
oscillation amplitude for both GS and ES. Hence, $J$ is a crucial parameter
toward higher oscillation amplitudes but its increase can cause unwanted
heating in the device. Dichroism rate $\gamma_{a}$ is linked to the magnitude
of birefringence of the cavity and we sweep this parameter in a large region
$(-4ns^{-1}-4ns^{-1})$, but its effect is negligible. So, we choose
$\gamma_{a}=0$ for our simulations.

Figure 4: Optical modulation amplitude for both GS (right column) and ES (left
column) as a function of $J$ $(a)$ and $\gamma_{a}$ $(b)$.

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Christos Tselios received his bachelor’s degree in Physics from the University
of Patras in 2019. Between 2019 and 2021, he studied at the Department of
Materials Science of the University of Patras where he received the Master’s
degree in Applied Optoelectronics. He is currently pursuing his doctoral
degree in Department of Electrical and Computer Engineering of the University
of Peloponnese. Since February 2022, he is also a member of the Industrial
Systems Institute of Athena Research Center. His main research interests are
focused on ultrafast signal transmission in optical communication systems and
simulation of optoelectronic devices.
---
Panagiotis Georgiou received the B.Sc Degree in Materials Science from the
University of Patras, Patras,Greece, in 2017 and the M.Sc degree in Materials
Science from the same institute in 2020. Since 2020 he is working towards his
PhD degree in Optoelectronics at the Department of Materials Science,
University of Patras, Greece. His research interests include Nonlinear
Dynamics in Quantum Dot spin-VCSELs, Photonic Reservoir Computing and Photonic
Spiking Neural Networks.
---
Christina (Tanya) Politi received a B.Sc. in physics from the University of
Athens in 1998 and a M.Sc. degree in the ”Physics of laser communications”
from the University of Essex in 2000. Subsequently, she joined the Photonic
Network Research Group in the Department of Electronic Systems Engineering at
the University of Essex where she worked until early 2005 and obtained her
PhD. In 2005 she joined the National Technical University of Athens –
Telecommunications Laboratory as a research associate where she worked in
various IST and national projects. During this period she was also a visiting
lecturer at the University of Peloponnese, which she joined at the end of 2008
as an Assistant Professor. Currently, she is Assistant Professor at the
Department of Electrical and Computer Engineering , University of Peloponnese.
She has a long experience in designing and providing optoelectronic
components, systems and networks for optical communications as well as in the
area of novel application paradigms and numerous collaboration with Greek and
European organizations. She is a co-author of over 100 international journal
and conference publications in the area of communication systems and
subsystems and networks. Her research interests include optical networking and
optical switching with emphasis on optical switch architectures and
subsystems, optical packet and circuit switched networks, high speed optical
networks and data center interconnection. Recently her work focuses on
innovative technologies in digital infrastructures for IoT applications, high
performance wireless and wired technologies for application areas like smart
cities and future hospitals.
---
Antonio Hurtado received the Ph.D. degree from the Universidad Politécnica de
Madrid (UPM), Madrid, Spain, in December 2006. He is currently a Reader and
Turing Artificial Intelligence (AI) Fellow with the Institute of Photonics,
University of Strathclyde, Glasgow, U.K. He has more than 15 years’ of
International Research Experience in photonics, Universities of Essex and
Strathclyde, U.K., University of New Mexico, USA, and UPM, Spain. His research
interests include neuromorphic photonics, laser nonlinear dynamics, nanolaser
systems, and hybrid nanofabrication. He was the recipient of two Marie Curie
Fellowships by the European Commission: Projects ISLAS (2009–2011) and NINFA
(2011–2014), a Chancellor’s Fellowship by the University of Strathclyde
following which he was appointed as a Lecturer with the Institute of
Photonics, Department of Physics, University of Strathclyde, in 2014, and a
Turing AI Acceleration Fellowship by the U.K. Research and Innovation Office
and the U.K. Government Business, Energy and Industrial Strategy Department to
develop a five-year Research Programme on Photonics for Ultrafast AI, in 2020.
---
Dimitris Alexandropoulos received the B.Sc degree in physics in 1998 from the
University of Athens, Athens, Greece, and the M.Sc. degree in the physics of
laser communications in 1999 from the University of Essex, Colchester, Essex,
U.K., and the PhD degree for work on GaInNAs-based optoelectronic devices,
from the same University in 2003. After one year of postdoctoral work at the
University of Essex he moved in 2004 to the Optical Communications Laboratory,
Department of Informatics and Telecommunications, University of Athens. He is
now an Assistant Professor in nanophotonics at the Department of Materials
Science, University of Patras, Patras, Greece. He is also a collaborating
academic faculty member of the Industrial Systems Institute, Athena Research
Center. He has co-authored more than 80 journal and conference publications.
His research interests include novel photonic materials, design and
fabrication of holographic optical elements, photonic sensors, integrated
photonics and optoelectronic devices.
---