#  Automorphism Ensemble Polar Code
Decoders for 6G URLLC ††thanks:  The authors acknowledge the financial support
by the Federal Ministry of Education and Research of Germany in the project
“Open6GHub” (grant number: 16KISK019 and 16KISK004).

Claus Kestel1, Marvin Geiselhart2, Lucas Johannsen13,
Stephan ten Brink2 and Norbert Wehn1

1 University of Kaiserslautern, 67663 Kaiserslautern, Germany
Email: {kestel<EMAIL_ADDRESS>2 Institute of Telecommunications,
University of Stuttgart, 70569 Stuttgart, Germany
Email<EMAIL_ADDRESS>3 Koblenz University of
Applied Sciences, 56075 Koblenz, Germany
Email<EMAIL_ADDRESS>
###### Abstract

The URLLC scenario in the upcoming 6G standard requires low latency and ultra
reliable transmission, i.e., error correction towards ML performance.
Achieving near-ML performance is very challenging especially for short block
lengths. Polar codes are a promising candidate and already part of the 5G
standard. The Successive Cancellation List (SCL) decoding algorithm provides
very good error correction performance but at the cost of high computational
decoding complexity resulting in large latency and low area and energy
efficiency. Recently, Automorphism Ensemble Decoding (AED) gained a lot of
attention to improve the error correction capability. In contrast to SCL, AED
performs several low-complexity (e.g., SC) decoding in parallel. However, it
is an open question whether AED can compete with sophisticated SCL decoders,
especially from an implementation perspective in state of the art silicon
technologies. In this paper we present an elaborated AED architecture that
uses an advanced path metric based candidate selection to reduce the
implementation complexity and compare it to state of the art SCL decoders in a
12nm FinFET technology. Our AED implementation outperform state of the art SCL
decoders by up to 4.4x in latency, 8.9x in area efficiency and 4.6x in energy
efficiency, while providing the same or even better error correction
performance.

###### Index Terms:

Polar Code, Automorphism, Ensemble Decoding, Path Metric, ASIC, 12 nm,
Implementation

##  I Introduction

The Ultra-Reliable Low Latency Communications (URLLC) scenario will be one of
the key enablers for 6G. Compared to 5G, it requires lower latency and
particular improved error correction [1] towards Maximum Likelihood (ML)
performance. Short block lengths enable low latency, but typically at the cost
of degraded error correction capability [2]. Hence, one of the key challenges
is to find codes, decoding algorithms and efficient decoder architectures that
enable low latency, high energy efficiency and, at the same time, error
correction performance towards ML performance.

Polar codes are promising candidates for this application. They are the first
class of error correction codes proven to achieve the channel capacity for
Binary Symmetric Memoryless Channels (BSMCs) [3] and have been adopted for the
control channel in the enhanced mobile broadband (eMBB) use case of 5G. Polar
codes can be decoded with the Successive Cancellation (SC) algorithm that is a
low-complexity decoding algorithm, but suffers on error correction
performance, especially for short block lengths. Hence, more sophisticated
decoding algorithms were developed. The Successive Cancellation List (SCL)
decoding algorithm [4] considers several decoding paths in parallel and
selects the best one according to a path metric. With a sufficient large list
size $L$, SCL approaches ML performance, but at the cost of a very high
computational complexity.

Recently, the GRAND [5] algorithms got a lot of attention. In contrast to the
algorithms mentioned above, GRAND is a universal decoding algorithm for any
code. The basic idea is to guess noise patterns to find the correct code word.
GRAND, in principle, can achieve ML performance if the number of guesses is
large enough. But since the number of guesses is unknown in advance, latency
and throughput can vary largely between worst case and average. [6] limits
this variance, but at the cost of a decreased error correction performance.
Moreover, it was shown that the GRAND algorithm and its derivatives are only
practical for high Signal-to-Noise-Ratio (SNR) regions and high code rates.

Automorphism Ensemble Decoding (AED) is another new decoding approach that was
recently published and can be applied to any code that has automorphisms. The
automorphisms are used to permute code words. The different code words can
then be decoded in parallel by low-complexity decoding algorithms. While there
exist many automorphisms for Reed-Muller (RM) codes [7], it is challenging to
find useful automorphisms for polar codes. Recently [8] have shown that
automorphisms can also be found for polar codes and that the error-correction
performance can be improved significantly.

AED is also very interesting from an implementation point of view. Instead of
one complex decoders (e. g., SCL), several low-complexity decoders (e. g., SC)
can be used. Each of the low-complexity decoders has a very high
implementation efficiency in terms of latency, high throughput, large energy
and area efficiency. Since the decoders run in parallel without information
exchange, locality is significantly improved. This is very beneficial for a
large implementation efficiency and, as a result, area and energy efficiency
scale almost linear with the number of ensemble decoder instances $M$, where
$M$ is defined by the requirement on the error correction performance. It is
important to mention that the latency in AED is first order independent of $M$
that is unique for advanced decoding.

This work makes the following new contributions:

  * •

A simple data-driven permutation selection method for AED.

  * •

An efficient AED algorithm for very high throughput and low latency that is
based on a Fast Simplified SC (SSC) algorithm with a new path metric based
candidate selection.

  * •

A detailed comparison of our AED architecture and state-of-the-art SCL
decoders with respect to error correction performance and hardware
implementation on a $12\text{\,}\mathrm{nm}\text{/}$ FinFET silicon
technology. This is, to the best of our knowledge, the first implementation
and comparison of AED with traditional decoding algorithms.

The remainder of this paper is structured as follows: We provide the required
background of Polar codes and their decoding algorithms in Section II. The
concept of AED and a new path metric based candidate selection is provided in
Section III. We present a detailed comparison with SCL decoders regarding
error correction performance and implementation costs in Section IV and
conclude this paper in Section V.

##  II Background

###  II-A Polar Codes

Polar codes $\mathcal{P}(N,K)$ are linear block codes with code length
$N=2^{n}$, which encode $K$ information bits. They use the phenomenon of
polarization to derive $N$ virtual channels. Information is then transmitted
over $K$ reliable channels listed in the information set $\mathcal{I}$, while
the remaining (unreliable) channels are set to zero and called frozen bits
(frozen set $\mathcal{F}$) [3]. Throughout the paper, we use the following
notation: $\mathbf{x}\in\\{0,1\\}^{N}$ is the code word and $\hat{\mathbf{x}}$
its estimation after decoding; $\mathbf{y}$ represents the vector of received
channel Log-Likelihood Ratios (LLRs).

###  II-B Successive Cancellation Decoding

SC decoding can be described as depth-first tree traversal of the Polar Factor
Tree (PFT) [9]. The PFT has $\log(N)+1$ stages $s$ and $N$ leaf nodes at stage
$s=0$, representing frozen bits and information bits. Each node $v$ receives a
LLR-vector $\alpha^{v}$ of size $N_{s}$ to first calculate the elements of the
vector $\alpha^{l}$

$\alpha^{l}_{i}=f\left(\alpha^{v}_{i},\alpha^{v}_{i+N_{v}/2}\right)$ (1)

by the hardware-efficient min-sum formulation of the f-function

$f(a,b)=\operatorname{sign}\left(a\right)\operatorname{sign}\left(b\right)\min\left(\left|a\right|,\left|b\right|\right).$
(2)

The resulting vector $\alpha^{l}$ of size $N_{s}/2$ is passed to the left
child. With the bit vector $\beta^{l}$ received from the left child, the
$N_{s}/2$ elements of $\alpha^{r}$ are calculated by the g-function

$\alpha^{r}_{i}=g\left(\alpha^{v}_{i},\alpha^{v}_{i+N_{v}/2},\beta^{l}_{i}\right)$
(3)

with

$g(a,b,c)=\left(1-2c\right)\cdot a+b,$ (4)

and sent to the right child. With the results of both children $\beta^{l}$ and
$\beta^{r}$, the partial sum $\beta^{v}$ is calculated and send to the parent
node. In the leaf nodes, the bit decisions are made. Frozen bits are, per
definition, always $0$, information bit nodes return

$\beta^{v}=\operatorname{sign}\left(a^{v}\right),$ (5)

which is $0$ if $a^{v}\geq 0$ and $1$ otherwise. The decoder output
$\hat{\mathbf{x}}=\operatorname{SC}(y)$ is equal to the output $\beta^{v}$ of
the root node [9].

###  II-C Fast-SSC Decoding

The PFT can be pruned to reduce the number of operations to decode one code
word. Sub-trees containing only frozen bits don’t have to be traversed, since
their decoding result is known to be an all-zero vector in advance. Such
subtrees are replaced by Rate-0 (R0) nodes. Similarly, subtrees without any
frozen bits can be decoded directly by Hard Decision Decoding (HDD), since no
parity information is contained. Therefore, these nodes are called Rate-1
nodes [9].

Further optimizations were proposed by [10], denoted Fast- SSC decoding. If a
sub-tree contains only one information bit, it is considered a Repetition
(REP) code and replaced by a specialized REP node. All bits with index $j$ of
a REP node are decoded according to:

$\beta^{v,\text{REP}}_{j}=\operatorname{sign}\left(\sum_{k=0}^{N_{s}-1}\alpha^{v}_{k}\right).$
(6)

In subtrees containing only one frozen bit, this bit always acts as parity
bit. Thus, the partial sum of this sub-tree represents a Single Parity-Check
(SPC) code. A specialized SPC node performs ML decoding by calculating the
parity $\gamma^{v}\in\\{0,1\\}$ of the input:

$\gamma^{v}=\bigoplus_{j=0}^{N_{s}-1}\operatorname{sign}\left(\alpha^{v}_{j}\right),$
(7)

finding the least reliable bit

$j_{\min}=\operatorname*{arg\,min}_{j\in[0,N_{s})}\left|\alpha^{v}_{j}\right|$
(8)

and setting $\beta^{v}$ to satisfy the single parity constraint:

$\beta^{v,\text{SPC}}_{j}=\begin{cases}\operatorname{sign}\alpha^{v}_{j}\oplus\gamma^{v}&\text{if
}j=j_{\min}\\\ \operatorname{sign}\alpha^{v}_{j}&\text{otherwise.}\end{cases}$
(9)

##  III Automorphism Ensemble Decoding

AED was proposed in [7] as a general method to improve the error-rate
performance by running an ensemble of $M$ independent SC decoders. Each
decoder $m$ operates on a permuted version of the received sequence and each
decoding result is then unpermuted as

$\hat{\mathbf{x}}_{m}=\pi_{m}^{-1}\left(\operatorname{SC}\left(\pi_{m}(\mathbf{y})\right)\right).$
(10)

The permutations $\pi_{m}$ stem from the automorphism group of the code, which
is the set of permutations of the coded bits that map every code word onto a
(not necessarily different) code word. Permuted received sequences therefore
correspond to different noise realizations which may be easier to decode.
Hence, by choosing the most likely candidate from $\hat{\mathbf{x}}_{m}$, the
probability of finding the correct code word is increased.

###  III-A Automorphisms and Code Construction

The automorphism group of polar codes contains the Block Lower-Triangular
Affine (BLTA) group which is defined as

$\mathbf{z}^{\prime}=\mathbf{A}\mathbf{z}+\mathbf{b},$ (11)

where $\mathbf{A}$ is an invertible, block lower triangular binary matrix,
$\mathbf{b}$ an arbitrary binary vector, and $\mathbf{z},\mathbf{z}^{\prime}$
are the binary representations of the bit indices before and after
permutation, respectively [8]. The block profile of $\mathbf{A}$, and thus,
the number of automorphisms, is dependent on the choice of the information set
$\mathcal{I}$. Therefore, for good performance under AED, the polar code
design is critical. In [11], the $\mathcal{P}(128,60)$ code with minimum
information set $\mathcal{I}_{\mathrm{min}}=\\{27\\}$ was found to be
particularly well suited for AED, as its BLTA group has block profile
$\mathbf{s}=(3,4)$.

###  III-B Automorphism Selection

It was shown in [11] that permutations can be partitioned into equivalence
classes. Within each Equivalence Class (EC), a permuted decoding results in
the same code word estimate. Hence, at most one permutation from each EC
should be used in AED. As the $\mathcal{P}(128,60)$ code exhibits 2205 ECs, a
subset of ECs has to be selected for use in AED.

Additionally, it is beneficial to have a single list of $M$ permutations and
smaller ensemble sizes $M^{\prime}<M$ use a subset of this list. As there is
no theory yet on how to select a good subset of ECs, we propose a greedy,
data-driven method that works as follows:

  1. 1.

Generate one permutation from each EC.

  2. 2.

Due to symmetry, select an arbitrary permutation $\pi_{1}$.

  3. 3.

Collect a batch of noisy code words that cannot be correctly decoded with any
of the selected permutations.

  4. 4.

Decode this batch with the permutations that have not been selected yet and
select the one that can correctly decode most of the noisy code words within
the batch.

  5. 5.

Repeat steps 3 to 5 until $M$ permutations $\pi_{1},\dots,\pi_{M}$ have been
selected.

###  III-C Conventional Candidate Selection

To select the most probable code word $\hat{\mathbf{x}}$ of all $M$ decoding
results $\hat{\mathbf{x}}_{m}$, [7] propose the ML-in-the-list method

$\hat{\mathbf{x}}=\operatorname*{arg\,max}_{\hat{\mathbf{x}}_{m},m\in\\{1,...,M\\}}\mathbf{y}^{\mathrm{T}}(1-2\hat{\mathbf{x}}_{m})$
(12)

that chooses the code word $\hat{\mathbf{x}}_{m}$ that correlates most with
the received LLRs.

###  III-D Path Metric

A drawback of the ML-in-the-list method is that the received channel LLRs
$\mathbf{y}$ are necessary to perform the final decision. This requires a
large buffer in high-throughput decoders that are deeply pipelined. This
buffer can account up to 25% of the total decoder area, dependent on $M$. This
memory is also a major source of power consumption. Hence, it is important to
find techniques to avoid this buffer. We propose a new Path Metric (PM) based
candidate selection for AED with SC constituent decoders. This PM technique
replaces the ML-in-the-list selection and, thus, increases the locality in the
AED implementation. Moreover, it makes the buffer redundant.

A PM is used in SCL decoding to rate the different decoding paths. An
efficient LLR-based PM calculation was presented in [12]. In particular, it
was shown that a PM calculated as

$\text{PM}=\sum_{i=0}^{N-1}\ln\left(1+e^{-(1-2\beta^{v}_{i})\alpha^{v}_{i}}\right)$
(13)

is a decreasing function with the likelihood of the code word candidates.
Therefore, the path with the lowest PM corresponds to the most likely
transmitted code word. The terms in (13) have the implementation-friendly
approximation

$\ln(1+e^{x})\approx\max(0,x).$ (14)

In contrast to SCL decoding, SC decoding does not split the decoding path. For
information bits ($i\in\mathcal{I}$), $\hat{u}_{i}$ is equal to the hard
decision on $\alpha^{v}_{i}$ and, therefore, they do not contribute to the PM.
Thus, for SC decoding, it is sufficient to update the PM only for frozen bits,
where only $\alpha^{v}_{i}<0$ indicates a transmission error. This leads to
the following cost function for the PM in the leaf nodes:

$\text{PM}=\sum_{i\in\mathcal{F}}\lvert\min(0,\alpha^{v}_{i})\rvert$ (15)

The code word $\hat{\mathbf{x}}_{m}$ with the smallest PM is then selected as
output of AED.

###  III-E Fast-SSC Path Metric

State-of-the-art high-throughput low-latency decoder architectures are based
on the Fast-SSC decoding algorithm [13]. Since Fast-SSC decoding operates on
higher stages in the PFT, the LLR values of the frozen leaf nodes are not
calculated. Therefore, the PM calculation presented previously is not directly
applicable. Thus, we derive PM calculation methods for the specialized nodes
analogous to Fast Simplified SCL (SSCL) decoding [14]. The value of
$\text{PM}^{(\ell)}$ is accumulated sequentially, where $\ell$ denotes the
index of the leaf node in the pruned PFT and is initialized with
$\text{PM}^{(0)}=0$.

As discussed above, the SC decoder does not split the decoding path and the PM
calculation method is only needed for specialized nodes containing frozen
bits, i. e., R0, REP and SPC nodes:

####  III-E1 Rate-0 nodes

For the R0 node, we follow (15), but execute the calculation directly on the
input LLR vector $\alpha^{v}$ of the node with node size $N_{s}$:

$\text{PM}^{(\ell)}=\text{PM}^{(\ell-1)}+\sum_{j=0}^{N_{s}-1}\lvert\min(0,\alpha^{v}_{j})\rvert.$
(16)

####  III-E2 Repetition nodes

According to (6), the repeated bit decision is based on the sign of the sum of
all LLRs related to the REP node. Thus, corrected bits are located at the
positions $j$, for which
$\operatorname{sign}(\alpha^{v}_{j})\neq\beta^{v}_{j}$. The PM consequently is
calculated by

$\text{PM}^{(\ell)}=\text{PM}^{(\ell-1)}+\sum_{\begin{subarray}{c}j=0\\\
\operatorname{sign}(\alpha^{v}_{j})\neq\beta^{v}_{j}\end{subarray}}^{N_{s}-1}\lvert\alpha^{v}_{j}\rvert.$
(17)

####  III-E3 Single Parity-Check nodes

For the SPC node, the decoding includes the correction of the least reliable
bit to satisfy the single parity constraint ((7)-(9)). Thus, the PM can be
derived as:

$\text{PM}^{(\ell)}=\text{PM}^{(\ell-1)}+\gamma^{v}\cdot\lvert\alpha^{v}_{j_{\min}}\rvert.$
(18)

Simulations verified that this Fast-SSC PM results in identical error
correction performance as the conventional candidate selection based on (12).

###  III-F Architecture

Figure 1: Path metric based AED architecture with $M=4$. In contrast to
conventional AED, the proposed architecture does not require a memory for the
LLRs. The novel path metric based selection is highlighted in green.

Figure 1 shows an AED architecture with $M=4$ that consists of $M$ identical
SC decoders. The permutations $\pi_{m}$ and $\pi_{m}^{-1}$ map the LLRs
$\mathbf{y}$ and code words related to each SC decoder instance, respectively,
according to the automorphisms (Section III-B). This mapping does not require
any logic or registers. All decoders work independently and output their
decoding results as well as the PM. This PM is calculated in each decoder as
described in Section III-E and fed into the selection unit, which then outputs
the code word estimate $\hat{x}$ with the lowest PM value. All decoders are
generated automatically by a framework as fully unrolled and pipelined
architectures.

To illustrate the proposed optimization, Figure 1 also shows the code word
selection of the conventional AED architecture with a memory for LLRs and the
ML-in-the-list method but crossed out in red, while the novel path metric
based selection is highlighted in green.

##  IV Results

To enable a fair comparison, all decoders (AED and SCL decoders) were
optimized for the same throughput, i. e.,
$64\text{\,}\mathrm{Gbit}\text{/}\mathrm{s}$. We investigated different
degrees of parallelism $M\in\\{2,4,8,16\\}$ for the AE decoders and list sizes
$L\in\\{2,4,8,16\\}$ for the SCL decoders. These decoders are denoted AED-$M$
and SCL-$L$, respectively. The SCL decoders use the optimizations from [15]
and are, to the best of our knowledge, the most efficient SCL decoder
implementations for very high throughput.

While this high throughput might not be necessary for the URLLC use case, the
investigated unrolled and pipelined architectures are not only highly area and
energy efficient, but also offer very low latency, which is one of the core
requirements for URLLC. With a reduction in frequency, the throughput could be
decreased to reduce power consumption, but at the cost of increased latency.

All designs were synthesized with Design Compiler and placed and routed with
IC-Compiler, both from Synopsys in a $12\text{\,}\mathrm{nm}\text{/}$ FinFET
technology from Global Foundries under worst case Process, Voltage and
Temperature (PVT) conditions
($125\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$, $0.9\text{\,}\mathrm{V}$
for timing, $1.0\text{\,}\mathrm{V}$ for power).

Error correction performance was simulated for an Additive White Gaussian
Noise (AWGN) channel and Binary Phase Shift Keying (BPSK) modulation. A
minimum of 100 erroneous code words were simulated. To assess the error
correction capability of the codes, we also evaluated the ML performance. For
this, we simulated two bounds: an upper ML bound and a lower ML bound. If both
bounds match, we get ML performance. The upper ML bound was calculated by
performing SCL decoding with $L=128$ and counting the cases where a code word
is closer to the received channel values than the correct code word. The lower
bound is derived in the same way, but additionally the correct code word has
to be in the list of candidates.

###  IV-A Comparison with SCL

We first compare the error correction performance of AED with SCL decoding
(Figure 2) using the same $\mathcal{P}(128,60)$ as described in Section III-A.
Both decoding algorithms perform very similar over the complete range of $L$ /
$M$. They both benefit from a high minimum distance. We therefore omitted an
additional Cyclic Redundancy Check (CRC) since it doesn’t offer any
improvement over the chosen code. For $L=M=16$ both decoding algorithms
approach the ML Bounds.

Table I shows the implementation results. We observe that AED outperforms its
SCL counterpart in all presented metrics. With increasing $L$ / $M$, the gap
between AED and SCL expands, since the AED implementations largely benefit
from locality. For $L=M=16$, a more than 8.9$\times$ better area efficiency, a
4.5$\times$ better energy efficiency and a 5.3$\times$ shorter latency are
observed.

Figure 2: AED vs. SCL decoding for the same $\mathcal{P}(128,60)$ TABLE I:  Implementation results of AED and SCL decoder for the same $\mathcal{P}(128,60)$ | AED-2 | SCL-2 | AED-4 | SCL-4 | AED-8 | SCL-8 | AED-16 | SCL-16
---|---|---|---|---|---|---|---|---
Frequency [MHz] | 498 | 476 | 497 | 462 | 498 | 478 | 495 | 403
Throughput [Gbps] | 63.7 | 61.0 | 63.6 | 59.1 | 63.7 | 61.2 | 63.3 | 51.6
Latency [CC] | 11 | 21 | 11 | 27 | 11 | 39 | 11 | 48
Latency [ns] | 22.1 | 44.1 | 22.1 | 58.5 | 22.1 | 81.5 | 22.2 | 119.1
Area [mm2] | 0.045 | 0.064 | 0.087 | 0.154 | 0.170 | 0.450 | 0.338 | 2.464
Area Eff. [Gbps/mm2] | 1407.9 | 956.0 | 733.8 | 385.0 | 375.1 | 136.2 | 187.0 | 20.9
Utilization [%] | 73 | 73 | 73 | 72 | 74 | 74 | 73 | 47
Power Total [W] | 0.07 | 0.09 | 0.15 | 0.22 | 0.32 | 0.61 | 0.64 | 2.38
Energy Eff. [pJ/bit] | 1.16 | 1.48 | 2.32 | 3.73 | 5.01 | 10.02 | 10.12 | 46.21
Power Density [W/mm2] | 1.63 | 1.42 | 1.70 | 1.44 | 1.88 | 1.36 | 1.89 | 0.97

###  IV-B Comparison with 5G

We also compare our implementations with optimized state-of-the-art CRC-aided
SCL decoders for a 5G $\mathcal{P}(128,60+C)$ with an CRC length of $C=11$. As
shown in Figure 3, AED outperforms CRC-aided SCL decoding for all $L$ / $M$,
while for $M=L=16$ the difference is shrinking for higher SNR. Note that AED-4
provides very similar performance as the CRC-aided SCL-8 and would benefit
from additional hardware savings under this error correction requirement.
While the ML Bounds for the 5G $\mathcal{P}(128,60+C)$ are lower than for the
$\mathcal{P}(128,60)$ code, the SCL decoders cannot benefit from that
advantage.

Analyzing the implementation costs of the AED and 5G SCL decoders (Table II),
implementations show a similar picture as in Section IV-A. However, the
decoders for the 5G code show a slightly better area and energy efficiency
than for the $\mathcal{P}(128,60)$ code without CRC. This is caused by the
different code structure leading to different building blocks from the PFT
traversal.

It is worth mentioning that the 5G code is optimized for a wide range of code
rates and usage scenarios, and not just for error correction performance and
implementation efficiency.

Figure 3: AED for $\mathcal{P}(128,60)$ vs. CRC-aided SCL decoding for 5G $\mathcal{P}(128,60+C)$ TABLE II:  Implementation results of AED for $\mathcal{P}(128,60)$ and CRC-aided SCL decoders for 5G $\mathcal{P}(128,60+C)$ | AED-2 | SCL-2 | AED-4 | SCL-4 | AED-8 | SCL-8 | AED-16 | SCL-16
---|---|---|---|---|---|---|---|---
Frequency [MHz] | 498 | 478 | 497 | 475 | 498 | 478 | 495 | 412
Throughput [Gbps] | 63.7 | 61.2 | 63.6 | 60.8 | 63.7 | 61.1 | 63.3 | 52.8
Latency [CC] | 11 | 21 | 11 | 26 | 11 | 36 | 11 | 47
Latency [ns] | 22.1 | 43.9 | 22.1 | 54.7 | 22.1 | 75.4 | 22.2 | 114.0
Area [mm2] | 0.045 | 0.063 | 0.087 | 0.138 | 0.170 | 0.378 | 0.338 | 1.990
Area Eff. [Gbps/mm2] | 1407.9 | 968.8 | 733.8 | 440.0 | 375.1 | 161.8 | 187.0 | 26.5
Utilization [%] | 73 | 72 | 73 | 73 | 74 | 77 | 73 | 47
Power Total [W] | 0.07 | 0.08 | 0.15 | 0.21 | 0.32 | 0.62 | 0.64 | 1.84
Energy Eff. [pJ/bit] | 1.16 | 1.39 | 2.32 | 3.52 | 5.01 | 10.08 | 10.12 | 34.82
Power Density [W/mm2] | 1.63 | 1.35 | 1.70 | 1.55 | 1.88 | 1.63 | 1.89 | 0.92

###  IV-C Comparison with GRAND

A comparison with the GRAND or  Ordered Reliability Bits GRAND (ORBGRAND)
algorithm is difficult since there exist not many implementations. In [6]
authors present an optimized architecture in a $7\text{\,}\mathrm{nm}\text{/}$
technology for a $\mathcal{P}(128,105)$. Its minimum latency, measured in
clock cycles, is still more than $2\times$ the latency of our proposed AE
decoder. Further comparison is very difficult since the code differs and the
authors provide synthesis results only, while we performed full placement and
routing under worst case PVT conditions.

##  V Conclusion

In this paper, we presented a first AED implementation and comparison with
state-of-the-art SCL decoders. We show that AED decoders outperform SCL
decoders in latency, area and energy efficiency. We also demonstrate that the
error correction performance is comparable with the SCL decoders and, in the
case of the 5G code, even exceeds the concatenation with CRC. AED and the
proposed architecture is, thus, perfectly suited for the 6G URLLC use case,
where low latency and high error correction is required.

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††thanks<EMAIL_ADDRESS>
# Fano enhancement of second harmonic field via dark-bright plasmon coupling

Mehmet Günay  Department of Nanoscience and Nanotechnology, Faculty of Arts
and Science,
Burdur Mehmet Akif Ersoy University, 15030 Burdur, Turkey

###### Abstract

Surface plasmon resonances, the coherent oscillation of free electrons, can
concentrate incident field into small volumes much smaller than the incident
wavelength. The intense fields at these hot spots enhance the light-matter
interactions and may lead to the appearance of nonlinearity. Controlling such
nonlinearities is significant for various practical applications. Here we
report that by coupling dark modes to the first and the generated second
harmonic modes separately, one can gain control over both fields. We find that
by engineering path interferences (Fano resonances) between bright and dark
plasmon modes it is possible to enhance the fundamental mode without
increasing the nonlinear field, enhance the nonlinear field without modifying
the fundamental mode, and enhance the second harmonic field with enhanced
fundamental mode.

##  I Introduction

Controlling the interaction between light and matter is fundamental to
science, as well as to the technological perspective; ranging from probing
entanglement in quantum physics to controlled use of the spectacular
information-carrying capacity. Traditionally light can only be controlled on
length scales down to a little below the wavelength of light. On the other
hand, plasmonic nanoparticles can be used to confine light to subwavelength
dimensions below the diffraction limit to achieve photonic circuit
miniaturization (Bozhevolnyi, ). Fano resonances, an analog of
electromagnetically induced transparency, in these structures are of
importance. Such type of devices can be probe experimentally and can be useful
in slow-light propagation (Huang, ; Wu, ), electromagnetic induced absorption
(Tassin, ; Taubert, ), an anomaly in light transmission (Imura, ), and solar
cell (Kauranen, ) applications.

Recent years have witnessed significant progress in the field of quantum
plasmonics. Unlike plasmonics, quantum plasmonics involves the study of the
quantum properties of light and its interaction with the matter at the
nanoscale. It provides a unique platform for the manipulation of light through
its interaction with electromagnetic fields, well below the diffraction limit
(1, ; 2, ). Placing an optical emitter near a metal nanostructure can lead to
observe light-matter interactions at different levels, i.e., from weak to
ultra-strong coupling limit (16, ). The emitters used can be a quantum dot
(11, ), a molecule (12, ), and a nitrogen-vacancy (NV) center (13, ). Fano
resonances, observed in the weak coupling regime (15, ), can increase the
lifetime of plasmon excitation (24, ), which leads to further enhancement of
the hot spot field. This extra enhancement (21, ) enables the operation of the
plasmonic laser (SPASER) (3, ) and gives rise to the enhancement of nonlinear
processes, such as second-harmonic generation (SHG) (22, ) and four-wave
mixing (FWM) (23, ). Rabi splitting in the strong coupling regime (16, ; 17,
), however, modifies the resonance scheme and creates two distinct modes,
which can be used as an optical switch (20, ).

Enhancement, due to path interference effects, of both linear (18, ) and
nonlinear (19, ) phenomena that emerged from the hot spots of plasmonic
structures, is well studied via coupling plasmonic modes to the two-level
quantum structures. These path interference effects (Fano resonances), besides
the emitter-plasmon coupling, can also appear when the plasmon mode of the
metal nanoparticle is coupled to a long-live dark plasmon mode (14, ). The
latter can be advantageous in device applications since the performance of the
former ones can decay in time. In this paper, we utilize bright-dark plasmon
coupling in which to gain control over both the linear and the nonlinear field
intensity. Different from our previous work (19, ), here, we consider both
linear and nonlinear Fano resonances at the same setup by coupling two
distinct plasmonic dark modes to the metal nanorod having second harmonic
generation property, as shown in Fig. 1.

It is known that enhancement of the second harmonic field can be observed by
enhancing theIn this paper, first harmonic mode resulting in the enhanced
second-harmonic field (linear Fano resonance) or coupling the emitter to the
second harmonic mode (nonlinear Fano resonance) to enhance the second harmonic
field without modifying the first harmonic field (19, ). The existence of the
second long-live mode, as studied here, can be useful to enhance one of the
modes by suppressing the other or to obtain the better enhancement of the
nonlinear field intensity by applying both enhancement schemes at the same
time.

In the following, we first present a basic analytical model in Sec. II, which
is demonstrated to treat Fano resonances realistically and explain how to
control the intensity of each field. It becomes apparent that the path
interferences in the linear and nonlinear Fano resonances can be controlled
separately, where one can enhance or suppress the nonlinear field intensity
while controlling the first harmonic mode by carefully chosen nanoparticles.
In Sec. III, we discuss the possible implementations and summarize the
findings.

Figure 1: Illustration of the metal nano-spheres/nanorod hybrid system
considered. Two nano-spheres (black) support plasmonic dark modes coupled to
the metal nanorod (top). Two $\omega$ plasmons combine to generate 2$\omega$
plasmons. Left (right) dark spheres having resonance at $\omega_{d1}$
($\omega_{d2}$) interacts with the FH (SH) mode of the nanorod (bottom).

##  II The Model

We consider the structure as shown in Fig. 1, where two metal nano-spheres
support plasmonic dark modes coupled to the metal nanorod. We consider the
nanorod has nonlinearity as a second harmonic generation. The dynamics of the
structure as follows. The incident laser of frequency $\omega$ excites the
first harmonic (FH) mode ($\hat{b}_{1}$) of the nanorod and generates second
harmonic (SH) one ($\hat{b}_{2}$). The resonances of the FH and SH modes are
$\omega_{b1}$ and $\omega_{b2}$ respectively. Each of the FH and SH modes
interact with the first ($\hat{d}_{1}$) and second ($\hat{d}_{2}$) dark modes,
which have resonances at $\omega_{d1}$ and $\omega_{d2}$ respectively.

The components of the Hamiltonian for the coupled system in the light of the
above definitions can be expressed as follows. The free oscillations of the
bright and dark plasmon modes $\hat{H}_{0}$ together with the excitation of
the FH mode of the nanorod by the pump source ($\hat{H}_{P}$) can be given as

$\displaystyle\hat{H}_{0}$ $\displaystyle=$
$\displaystyle\hbar\sum_{j=1}^{2}\\{\omega_{bj}\hat{b}_{j}^{\dagger}\hat{b}_{j}+\omega_{dj}\hat{d}_{j}^{\dagger}\hat{d}_{j}\\},$
(1) $\displaystyle\hat{H}_{P}$ $\displaystyle=$ $\displaystyle
i\hbar(\varepsilon_{p}\hat{b}_{1}^{\dagger}e^{-iwt}-\textit{h.c}),$ (2)

where $\hat{b}_{j}^{\dagger}$ ($\hat{b}_{j}$) and $\hat{d}_{j}^{\dagger}$
($\hat{d}_{j}$) define the creation (annihilation) of the $j$th mode of the
bright and dark plasmons respectively with $j=1,2$. $\varepsilon_{p}$ is
proportional to the laser field amplitude. Hamiltonian for the second harmonic
generation process $\hat{H}_{sh}$ and the interaction between the bright and
dark plasmon modes $\hat{H}_{int}$ are given by

$\displaystyle\hat{H}_{sh}$ $\displaystyle=$
$\displaystyle\hbar\chi^{(2)}(\hat{b}_{2}^{\dagger}\hat{b}_{1}\hat{b}_{1}+\hat{b}_{1}^{\dagger}\hat{b}_{1}^{\dagger}\hat{b}_{2}),$
(3) $\displaystyle\hat{H}_{int}$ $\displaystyle=$
$\displaystyle\hbar\sum_{j=1}^{2}g_{j}\\{\hat{b}_{j}^{\dagger}\hat{d}_{j}+\hat{d}_{j}^{\dagger}\hat{b}_{j}\\}.$
(4)

Here the parameters $g_{1}$ and $g_{2}$ , in units of frequency, are the
coupling strengths of the linear and second harmonic responses of the nanorod
to the dark modes of the nanospheres respectively. $\chi^{(2)}$ is the overlap
integral, which is proportional to the second-order susceptibility of the
nanorod. Thus, the total Hamiltonian can be written as:
$\hat{H}=\hat{H}_{0}+\hat{H}_{P}+\hat{H}_{sh}+\hat{H}_{int}$. The dynamics of
the system can be derived by using Heisenberg equation of motion, e.g.,
$i\hbar\dot{\hat{a}}_{j}=[\hat{a}_{j},\hat{H}]$. To investigate only the field
amplitudes, we replace the second-quantized operators with their expectations,
e.g., $\alpha_{j}=\langle\hat{b}_{j}\rangle$ and
$\beta_{j}=\langle\hat{d}_{j}\rangle$ with $j=1,2$. The resulting equations of
motion by including the decay rates can be obtained as

$\displaystyle\dot{{\alpha}}_{1}$
$\displaystyle=-(i\omega_{b1}+\gamma_{b1}){\alpha}_{1}-i2\chi^{(2)}{\alpha}_{1}^{\ast}{\alpha}_{2}-ig_{1}{\beta}_{1}+\varepsilon_{p}e^{-i\omega
t},$ (5a) $\displaystyle\dot{{\alpha}}_{2}$
$\displaystyle=-(i\omega_{b2}+\gamma_{b2}){\alpha}_{2}-i\chi^{(2)}{\alpha}_{1}^{2}-ig_{2}{\beta}_{2},$
(5b) $\displaystyle\dot{{\beta}}_{1}$
$\displaystyle=-(i\omega_{d1}+\gamma_{d1}){\beta}_{1}-ig_{1}{\alpha}_{1},$
(5c) $\displaystyle\dot{{\beta}}_{2}$
$\displaystyle=-(i\omega_{d2}+\gamma_{d2}){\beta}_{2}-ig_{2}{\alpha}_{2},$
(5d)

where $\gamma_{bj}$ and $\gamma_{dj}$ stand for the decay rates of the bright
and dark modes respectively. To obtain the Fano resonance in such a system,
these parameters play an essential role. In definition, Fano resonance can be
observed when a continuum state interacts with the discrete one, where the
dark-bright coupling fits this picture as dark modes have much longer
lifetimes comparing the bright ones, i.e., $\gamma_{dj}\ll\gamma_{bj}$.

Figure 2: The field intensities of the FH (a) and SH (b) when the nanorod
alone (blue-dashed) and with the nanospheres (red-straight) obtained from the
numerical solutions of the Eqs. (5a) - (5d). A transparency window for the FH
can be seen at $\omega=\omega_{d1}$ that the nonlinear contribution from
nonlinear Fano resonance at $\omega=\omega_{d2}/2$ does not change FH
intensity profile. On the other hand, there are two dips for the SH intensity:
one due to nonlinear Fano resonance when $2\omega=\omega_{d2}$ and the other
due to $\tilde{\alpha}_{1}^{2}$ term in the nominator of the Eq. (8). More
details in the text.

As the system is driven by a source proportinal to the term $e^{-i\omega t}$,
one can expect that the solutions at the steady-state oscillate as:

$\displaystyle\alpha_{k}=\tilde{\alpha}_{k}e^{-ik\omega
t},\qquad\beta_{k}=\tilde{\beta}_{k}e^{-ik\omega t}\qquad k:1,2,$ (6)

where amplitudes with tilde are slowly varying terms that determine the
steady-state field amplitudes. By inserting these equations into the equations
of motion, Eqs. (5a) - (5d), we obtain the equations for FH
($\tilde{\alpha}_{1}$) and SH ($\tilde{\alpha}_{2}$) steady-state amplitudes
as

$\displaystyle\tilde{\alpha}_{1}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{p}}{i(\omega_{b1}-\omega)+\gamma_{b1}+\frac{g^{2}_{1}}{i(\omega_{d1}-\omega)+\gamma_{d1}}}+f(\chi^{(2)}),$
(7) $\displaystyle\tilde{\alpha}_{2}$ $\displaystyle=$
$\displaystyle\frac{-i\chi^{(2)}\tilde{\alpha}_{1}^{2}}{i(\omega_{b2}-2\omega)+\gamma_{b2}+\frac{g^{2}_{2}}{i(\omega_{d2}-2\omega)+\gamma_{d2}}},$
(8)

where $f(\chi^{(2)})$ is the function proportional the $\chi^{(2)}$ term,
which does not play essential role in the suppression and the enhancement of
the FH field (see Figure 2). Although, we obtain the Figures 2 and 3 by
solving the Eqs. (5a) - (5d) numerically, we study the steady-state solutions
in Eqs. (7) - (8) to get better insight for the Fano enhancement of the linear
and the nonlinear fields.

The suppression occurs due to the emergence of a transparency window induced
by Fano resonance, which does not allow an excitation at the converted
frequency. It is shown in Figure. 2 that the transparency window can be
obtained both in the linear and the second harmonic field intensities.
Different from the FH intensity, there are two dips in the response of SH
intensity as shown. This can be explained by a closer look at the denominators
of the Eqs. (7) - (8). That is when the pump frequency becomes resonant with
one of the dark modes, i.e., $\omega=\omega_{d1}$ and/or
$2\omega=\omega_{d2}$, the second terms in these denominators become ruling as
long as the interaction strength is sufficient. Unlike FH intensity, in the SH
intensity there is a $\tilde{\alpha}_{1}^{2}$ dependence in the nominator that
when $\omega=\omega_{d1}$ is met, the dip also appears in the SH intensity due
to this dependence. Since the $\chi^{(2)}$ term, in general, is small in most
of the materials comparing the other frequencies the $f(\chi^{(2)})$ term does
not play an essential role in the FH intensity when $2\omega=\omega_{d2}$ is
met. This can be seen in Figure. 2(a). Here we scale all the frequencies with
$\omega_{b1}$ and use the values $\omega_{b2}=2\omega_{b1}$,
$\gamma_{b1}=\gamma_{b2}=0.05\omega_{b1}$,
$\gamma_{d1}=\gamma_{d2}=10^{-4}\omega_{b1}$ for the numerical calculations in
the obtained results. As we focus on the weak coupling regime
($g_{i}<\gamma_{bi},\gamma_{di}$), we take the interaction strengths as
$g_{i}=0.25\gamma_{bi}$

Similarly, the interference effect can be arranged in such a way that the
nonlinear process can be carried out near the resonance. That is by playing
the resonance values of the dark-modes, the denominators of the Eqs. (7) - (8)
can be minimized in which the field amplitudes can attain large values. In
Figure. 3, the enhancement of both FH and SH intensities can be observed.
Similar to the suppression process nonlinear Fano resonance does not play a
role in the FH field intensity. On the other hand, linear Fano resonance
multiplies the nonlinear one that we obtain 3-orders of magnitude enhancement
in the SH field intensity even the FH field intensity gets enhanced only 6
times. By the tidy arrangement of the parameters, one can obtain a better
cancellation scheme and hence a larger enhancement of the nonlinear field
intensity. It can be seen that, in the elliptic regions of the Figure. 3, SH
field intensity is suppressed even the FH gets enhanced, where the nonlinear
Fano resonance cancels linear Fano resonance contribution. This result can be
useful when the nonlinearity is undesired in such a system.

Figure 3: The enhancement of the field intensities of the FH (a) and SH (b)
obtained from the numerical solutions of the Eqs. (5a) - (5d). Here we take
$\omega=1.05\leavevmode\nobreak\ \omega_{b1}$ and the rest of the parameters
are as given in the text. In the elliptic regions, SH field intensity is
suppressed even the FH gets enhanced. In other words, nonlinear Fano resonance
cancels linear Fano resonance contribution.

##  III Conclusion

In summary, we demonstrate that controlling second-harmonic field intensity is
possible by using both linear and nonlinear Fano resonances. Our results are
twofold. On the one hand, one can obtain a larger enhancement of the nonlinear
field, which is at least one order of magnitude larger comparing with the
enhancement obtained via only the nonlinear Fano resonance. On the other hand,
linear field enhancement can be obtained by suppressing the nonlinear fields.
This can be useful when the nonlinearity is undesired. Our results contribute
to the understanding of the amplification of optical nonlinearities via Fano
resonances and can find potential applications from all-optical switch
nonlinear devices to ultrafast spectroscopies.

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