Spatial mode division de-multiplexing of optical signals has many real-world applications, such as quantum computing and both classical and quantum optical communication. In this context, it is crucial to develop devices able to efficiently sort optical signals according to the optical mode they belong to and route them on different paths. Depending on the mode selected, this problem can be very hard to tackle. Recently, researchers have proposed using multi-objective evolutionary algorithms (MOEAs) ---and NSGA-II in particular--- combined with Linkage Learning (LL) to automate the process of design mode sorter. However, given the very large-search scale of the problem, the existing evolutionary-based solutions have a very slow convergence rate. In this paper, we proposed a novel approach for mode sorter design that combines (1) stochastic linkage learning, (2) the adaptive geometry estimation-based MOEA (AGE-MOEA-II), and (3) an adaptive mutation operator. Our experiments with two- and three-objectives (beams) show that our approach is faster (better convergence rate) and produces better mode sorters (closer to the ideal solutions) than the state-of-the-art approach. A direct comparison with the vanilla NSGA-II and AGE-MOEA-II also further confirms the importance of adopting LL in this domain.

1. A Fast Multi-Objective Evolutionary Approach for
Designing Large-Scale Optical Mode Sorter
Annibale Panichella
a.panichella@tudelf.nl
@AnniPanic
1
Giuseppe Di Domenico
giuseppe.didomenico@attocube.com

2. Background
2

3. Sorters (Demultiplexers)
3
Composite
Signal
Demultiplexer
(Sorter)
Multiplexer
N
input
signals
…
N
output
signals
…
Applications:
• Quantum optics
• Telecommunications
• Quantum memories
• Spatial
fi
ltering
• ….

4. Mode Sorters
4
Hermine-Gauss Modes
A sorter is a device able to discriminate
optical beams based on various signal
properties:
• Wavelength
• Amplitude
• Phase
• Polarization
• …
• Mode (mode sorter)

5. The Optimization Problem
5
Problem: Design a device (on a 2D
surface plasmon polarization platform)
that can distinguish and route the input
plasmonic beams based on their mode
Mode sorter by
Di Domenico et al.,ACS Photonics 2022
The presence of the polymethyl
methacrylate (PMMA) change locally the
beam propagation. Globally it determines
how each mode is routed/redirected

6. Mode
Sorter
The Optimization Problem
6
Input
Location
Target Output
Locations

7. The Optimization Problem
7
Mode
Sorter
Actual Output
Locations
Input
Signal
The objectives to optimize
are the distances between
the target and the actual
output locations
N. objectives = N. Of
modes to redirect

8. Open Challenges
• Very large search space (16K
decision variables)
• Not all decision variables (device
blocks) have the same importance
• The non-dominated fronts show an
hyperbolic geometry
8
Our intuitions:
1. Use a MOEA for large-scale
problem
2. Use a MOEA for hyperbolic
fronts
3. Use linkage learning to learn
which blocks matters

9. Our Solution
9

10. L2-AGE-MOEA
Our framework inherits the key features of
AGE-MOEA-II
• It uses ranking and dominance
• It estimates the geometry of the
fi
rst
non-dominated front
• It updates the diversity and
convergence on the estimated
geometry
10
GECCO ’23, July 15–19, 2023, Lisbon, Portugal A. Panichella and G. Di Domenico
Algorithm 2: L2-AGE-MOEA
Input: ": Number of objectives and #: Population size
Result: Final population %
1 begin
2 % RANDOM-POPULATION(N)
3 F FAST-NONDOMINATED-SORT(%
–
&)
4 FOS INFER-MODEL(F1, ⇤)
5 while not (stop_condition) do
6 & ;
7 forall i in 1..|% | do
8 Parent P8
9 Donor TOURNAMENT-SELECTION(%)
10 O REPRODUCTION(Parent, Donor, FOS)
11 O ADAPTIVE-MUTATION(O)
12 & &
–
{O}
13 F FAST-NONDOMINATED-SORT(%
–
&)
14 F NORMALIZE(F)
15 ? NEWTON-RAPHSON(F1) /* Eq. 3 */
16 3 1 /* First non-dominated rank */
17 while | % | + | F3 |6 # do
18 F? MANIFOLD-PROJECTION(F3 , ?)
19 ⇡ GEODESIC-DIV(F?, ?)
20 SURVIVAL-SCORE(D, F, 3, ?)
21 % %
–
F3
22 3 3 + 1
23 SORT(F3 ) /* by survival scores */
24 % %
–
F3 [1 : (# |% |)]
25 FOS INFER-MODEL(F1, ⇤)
26 return %
The solutions/devices are evaluated using the beam propaga-
tion method [10] and considering the following device and beam
The remainder of Algorithm 2 is identical to the main loop of
the original AGE-MOEA-II. In a nutshell, parent % and o�spring &
populations are combined and sorted using the fast non-dominated
sorting algorithm (line 13) and normalized (line 14). The �rst front
F1 is then used to compute the curvature ? using the Newton-
Raphson method (Equation 3). Finally, in lines 17-22, the survival
score (line 20) is computed iteratively for all fronts using the geo-
desic distance (line 19).
The loop between lines 17 and 24 inserts as many individuals
to the new population % as possible, based on their ranks and
until reaching the population size #. L2-AGE-MOEA �rst selects the
non-dominated solutions from F1; if | F8 |< #, the loop selects
the non-dominated solutions from F8, etc. The loop ends when
inserting all the solutions from the front 3 exceeds the maximum
population size, as shown in line 24. In this case, the algorithm
selects the remaining solutions from the front F3 according to the
descending order of survival score in lines 24-25.
Finally, the linkage model is retrained in line 25 based on the
�rst non-dominated front F1 previously computed on line 13, i.e.,
on the latest population.
3.1 Linkage learning with Clustering Methods
Linkage learning [43] is a category of techniques applied to infer
linkage structures from a population, i.e., groups or clusters of
promising decision variable values that contribute to achieving

11. Linkage Learning
11
• Key new ingredients:
• It uses hierarchical clustering to
infer the linkage structures (LL)
• It applies a stochastic crossover
that leverages the linkage structure
• It uses a linearly-decreasing mutation
rate (bit-
fl
ip mutation)
GECCO ’23, July 15–19, 2023, Lisbon, Portugal A. Panichella and G. Di Domenico
Algorithm 2: L2-AGE-MOEA
Input: ": Number of objectives and #: Population size
Result: Final population %
1 begin
2 % RANDOM-POPULATION(N)
3 F FAST-NONDOMINATED-SORT(%
–
&)
4 FOS INFER-MODEL(F1, ⇤)
5 while not (stop_condition) do
6 & ;
7 forall i in 1..|% | do
8 Parent P8
9 Donor TOURNAMENT-SELECTION(%)
10 O REPRODUCTION(Parent, Donor, FOS)
11 O ADAPTIVE-MUTATION(O)
12 & &
–
{O}
13 F FAST-NONDOMINATED-SORT(%
–
&)
14 F NORMALIZE(F)
15 ? NEWTON-RAPHSON(F1) /* Eq. 3 */
16 3 1 /* First non-dominated rank */
17 while | % | + | F3 |6 # do
18 F? MANIFOLD-PROJECTION(F3 , ?)
19 ⇡ GEODESIC-DIV(F?, ?)
20 SURVIVAL-SCORE(D, F, 3, ?)
21 % %
–
F3
22 3 3 + 1
23 SORT(F3 ) /* by survival scores */
24 % %
–
F3 [1 : (# |% |)]
25 FOS INFER-MODEL(F1, ⇤)
26 return %
The solutions/devices are evaluated using the beam propaga-
tion method [10] and considering the following device and beam
The remainder of Algorithm 2 is identical to the main loop of
the original AGE-MOEA-II. In a nutshell, parent % and o�spring &
populations are combined and sorted using the fast non-dominated
sorting algorithm (line 13) and normalized (line 14). The �rst front
F1 is then used to compute the curvature ? using the Newton-
Raphson method (Equation 3). Finally, in lines 17-22, the survival
score (line 20) is computed iteratively for all fronts using the geo-
desic distance (line 19).
The loop between lines 17 and 24 inserts as many individuals
to the new population % as possible, based on their ranks and
until reaching the population size #. L2-AGE-MOEA �rst selects the
non-dominated solutions from F1; if | F8 |< #, the loop selects
the non-dominated solutions from F8, etc. The loop ends when
inserting all the solutions from the front 3 exceeds the maximum
population size, as shown in line 24. In this case, the algorithm
selects the remaining solutions from the front F3 according to the
descending order of survival score in lines 24-25.
Finally, the linkage model is retrained in line 25 based on the
�rst non-dominated front F1 previously computed on line 13, i.e.,
on the latest population.
3.1 Linkage learning with Clustering Methods
Linkage learning [43] is a category of techniques applied to infer
linkage structures from a population, i.e., groups or clusters of
promising decision variable values that contribute to achieving

12. Linkage Learning
12
A Fast Multi-objective Evolutionary Approach for Designing Large-Scale Optical M
G1 G2 G3 G4 ...
G1, G2
G3, G4
G3, G4, ...
G1, . . .
Hamming
distance
Figure 1: Example of dendrogram or Family Of Subset (FOS
be zero and clustered together in the lower internal nodes of th
dendrogram. Instead, decision variables that never have the sam
values across the di�erent individuals are clustered together but i
the top internal nodes of the dendrogram. The distance betwee
pairs of decision variables is computed using the hamming distanc
a well-known distance function for binary vectors. This is als
the distance recommended in the literature due to its low (linea
computational complexity [10, 24].
Solution/device
Region
with
PMMA
Region
without
PMMA
Linearization Hierarchical Clustering
…
x1 x2 x3 x4 …

13. Stochastic Linkage-Based Crossover
13
…
x1 x2 x3 x4 …
…
x1 x2 x3 x4 …
Donor
Parent
A Fast Multi-objective Evolutionary Approach for Designing Large-Scale Optical Mode Sorters GECCO ’23, July 15–19, 2023, Lis
G1 G2 G3 G4 ...
G1, G2
G3, G4
G3, G4, ...
G1, . . .
Hamming
distance
Figure 1: Example of dendrogram or Family Of Subset (FOS)
be zero and clustered together in the lower internal nodes of the
dendrogram. Instead, decision variables that never have the same
values across the di�erent individuals are clustered together but in
the top internal nodes of the dendrogram. The distance between
pairs of decision variables is computed using the hamming distance,
where 'BC0AC is the initial mutation rate, while '4=3
mutation rate (with 'BC0AC > '4=3); ⇢( denotes the
solution evaluations performed at the generation 8 and
the maximum number of solution evaluations (total sear
4 EMPIRICAL STUDY
To assess the performance of L2-AGE-MOEA in genera
sorters, we performed a computational experiment wit
types of sorters. We describe the setup of our study in
while the results are discussed in Section 4.2.
4.1 Empirical Setup
In this paper, we consider four formulations of the mod
sign problem. First, we consider two-mode and three-m
Linkage Structures
(Family of subsets)

14. Stochastic Linkage-Based Crossover
14
…
x1 x2 x3 x4 …
…
x1 x2 x3 x4 …
Donor
Parent
A Fast Multi-objective Evolutionary Approach for Designing Large-Scale Optical Mode Sorters GECCO ’23, July 15–19, 2023, Lis
G1 G2 G3 G4 ...
G1, G2
G3, G4
G3, G4, ...
G1, . . .
Hamming
distance
Figure 1: Example of dendrogram or Family Of Subset (FOS)
be zero and clustered together in the lower internal nodes of the
dendrogram. Instead, decision variables that never have the same
values across the di�erent individuals are clustered together but in
the top internal nodes of the dendrogram. The distance between
pairs of decision variables is computed using the hamming distance,
where 'BC0AC is the initial mutation rate, while '4=3
mutation rate (with 'BC0AC > '4=3); ⇢( denotes the
solution evaluations performed at the generation 8 and
the maximum number of solution evaluations (total sear
4 EMPIRICAL STUDY
To assess the performance of L2-AGE-MOEA in genera
sorters, we performed a computational experiment wit
types of sorters. We describe the setup of our study in
while the results are discussed in Section 4.2.
4.1 Empirical Setup
In this paper, we consider four formulations of the mod
sign problem. First, we consider two-mode and three-m
Random
Cut-point
Linkage Structures
(Family of subsets)

15. Stochastic Linkage-Based Crossover
15
…
x1 x2 x3 x4 …
…
x1 x2 x3 x4 …
Donor
Parent
A Fast Multi-objective Evolutionary Approach for Designing Large-Scale Optical Mode Sorters GECCO ’23, July 15–19, 2023, Lis
G1 G2 G3 G4 ...
G1, G2
G3, G4
G3, G4, ...
G1, . . .
Hamming
distance
Figure 1: Example of dendrogram or Family Of Subset (FOS)
be zero and clustered together in the lower internal nodes of the
dendrogram. Instead, decision variables that never have the same
values across the di�erent individuals are clustered together but in
the top internal nodes of the dendrogram. The distance between
pairs of decision variables is computed using the hamming distance,
where 'BC0AC is the initial mutation rate, while '4=3
mutation rate (with 'BC0AC > '4=3); ⇢( denotes the
solution evaluations performed at the generation 8 and
the maximum number of solution evaluations (total sear
4 EMPIRICAL STUDY
To assess the performance of L2-AGE-MOEA in genera
sorters, we performed a computational experiment wit
types of sorters. We describe the setup of our study in
while the results are discussed in Section 4.2.
4.1 Empirical Setup
In this paper, we consider four formulations of the mod
sign problem. First, we consider two-mode and three-m
Random
Cut-point
x3, x4, …
x1 x2
Linkage Structures
(Family of subsets)

16. …
x1 x2 x3 x4 …
…
x1 x2 x3 x4 …
Donor
Parent
A Fast Multi-objective Evolutionary Approach for Designing Large-Scale Optical Mode Sorters GECCO ’23, July 15–19, 2023, Lis
G1 G2 G3 G4 ...
G1, G2
G3, G4
G3, G4, ...
G1, . . .
Hamming
distance
Figure 1: Example of dendrogram or Family Of Subset (FOS)
be zero and clustered together in the lower internal nodes of the
dendrogram. Instead, decision variables that never have the same
values across the di�erent individuals are clustered together but in
the top internal nodes of the dendrogram. The distance between
pairs of decision variables is computed using the hamming distance,
where 'BC0AC is the initial mutation rate, while '4=3
mutation rate (with 'BC0AC > '4=3); ⇢( denotes the
solution evaluations performed at the generation 8 and
the maximum number of solution evaluations (total sear
4 EMPIRICAL STUDY
To assess the performance of L2-AGE-MOEA in genera
sorters, we performed a computational experiment wit
types of sorters. We describe the setup of our study in
while the results are discussed in Section 4.2.
4.1 Empirical Setup
In this paper, we consider four formulations of the mod
sign problem. First, we consider two-mode and three-m
Stochastic Linkage-Based Crossover
16
Random
Cut-point
x3, x4, …
x1 x2
Linkage Structures
(Family of subsets)
Each cluster (group of genes) has 50% chance to be copied
from the donor solutions

17. Mutation Rate
17
advanced adaptive schema as part of our future agenda.
Given that we handle a binary problem, L2-AGE-MOEA uses the
bit-�ip mutation, which randomly �ips some decision variable val-
ues from one to zero (or vice versa) based on the mutation rate
'. Instead of using a constant mutation rate, we use a linearly de-
creasing mutation rate, such as it has a large mutation rate in the
initial generation (for better exploration) and a lower mutation rate
in the last generation (for better exploitation). More precisely, the
mutation rate ' at the generation 8 is computed as follows:
'8 = '4=3 +
⇢(<0G ⇢(
⇢(<0G
⇥ ('BC0AC '4=3 ) (6)
Linearly-decreasing mutation Rate (R)
Rate
(R)
Fitness
Evaluations
(FEs)
Rstart
Rend
Mutation rate and
generation i
Maximum number
Of FEs

18. Empirical Evaluation
18

19. Benchmarks
We consider four different mode sorters:
• Two-mode sorter (12350 decision variables)
• Three-mode sorter (16200 decision variables)
• Two-mode + Structure optimization (2+1 objectives)
• Three-mode + Structure optimization (3+1 objectives)
Characteristics of the sorters:
• Plasmonic refractive index
𝑛
0=1.008856
• Plasmonic
fi
eld with wavelength
𝜆
= 1.064 · 10−6/
𝑛
• Vacuum wavevector 2
𝜋
/
𝜆
• …
19

20. Selected MOEAs
• AGE-MOEA-II [A. Panichella 2022]
• L2-NSGA [Olsthoorn and Panichella 2021]
• NSGA-II [Deb et al. 2020]
• L2-AGE-MOEA [Our solution]
20
No Linkage
Learning
With Linkage
Learning
Handles Hyperbolic
Fronts
Does not consider
Hyperbolic fronts
No Linkage
Learning
Does not consider
Hyperbolic fronts
With Linkage
Learning
Handles Hyperbolic
Fronts

21. Overall Results
21
Median and IQR. of the IGD and HV across 30 runs
△ means better than, ▼ worst than L2-AGE-MOEA
A Fast Multi-objective Evolutionary Approach for Designing Large-Scale Optical Mode Sorters GECCO ’23, July 15–19, 2023, Lisbon, Portugal
Table 1: Median and Interquartile Range (IQR) values achieved by the di�erent evolutionary algorithms. " denotes the number
of objectives while ⇡ is the number of decision variables. The best values are highlighted in grey color.
Results for IGD
Problem " ⇡ NSGA-II AGE-MOEA-II L2-NSGA L2-AGE-MOEA
Two-Beams 2 12350 1.4232e+0 (4.85e-1) H 8.6429e-1 (3.13e-1) H 1.0086e+0 (4.65e-1) H 2.6545e-1 (2.19e-1)
Two-Beams + Structure 3 12350 1.2744e+0 (2.40e-1) H 7.3520e-1 (3.44e-1) H 6.2873e-1 (1.86e-1) H 4.1567e-1 (2.02e-1)
Three-Beams 3 16200 1.1844e+1 (1.73e+0) H 2.5116e+0 (9.56e-1) H 6.4420e+0 (3.62e+0) H 3.5408e-1 (2.10e-1)
Three-Beams + Structure 4 16200 6.7468e-1 (2.92e-2) H 3.701e-1 (9.61e-2) H 2.462e-1 (2.48e-2) H 1.9041e-1 (3.86e-3)
Results for HV
Problem " ⇡ NSGA-II AGE-MOEA-II L2-NSGA L2-AGE-MOEA
Two-Beams 2 12350 2.0254e-1 (4.65e-2) H 2.6338e-1 (3.51e-2) H 2.7187e-1 (4.40e-2) H 3.4791e-1 (4.13e-2)
Two-Beam + Structure 3 12350 7.7192e-2 (1.63e-2) H 2.1024e-1 (1.76e-2) H 1.3591e-1 (7.30e-2) H 3.3073e-1 (2.47e-2))
Three-Beams 3 16200 5.2256e-5 (2.17e-3) H 2.2892e-1 (3.84e-2) H 7.7827e-2 (1.10e-1) H 3.7184e-1 (2.34e-2)
Three Beams + Structure 4 16200 2.5452e-2 (1.18e-2) H 2.5838e-1 (2.78e-2) H 1.4414e-2 (4.82e-3) H 3.3585e-1 (1.76e-2)
methods. Instead, AGE-MOEA-II does not use any linkage learn-
ing methods but uses front modeling methods. As we can observe,
Reference R L2-AGE-MOEA L2-NSGA NSGA-II AGE-MOEA-II

22. Results for Two-Beams
22
Results for HV
AGE-MOEA-II L2-NSGA L2-AGE-MOEA
2) H 2.6338e-1 (3.51e-2) H 2.7187e-1 (4.40e-2) H 3.4791e-1 (4.13e-2)
2) H 2.1024e-1 (1.76e-2) H 1.3591e-1 (7.30e-2) H 3.3073e-1 (2.47e-2))
3) H 2.2892e-1 (3.84e-2) H 7.7827e-2 (1.10e-1) H 3.7184e-1 (2.34e-2)
2) H 2.5838e-1 (2.78e-2) H 1.4414e-2 (4.82e-3) H 3.3585e-1 (1.76e-2)
arn-
erve,
imal
also
sum
NSGA
. In-
quiv-
olds
(all
NSGA
er in
EAs
does
two
2 2.5 3 3.5 4
2
2.5
3
3.5
4
51
5
2
Reference R L2-AGE-MOEA L2-NSGA NSGA-II AGE-MOEA-II
Figure 2: Fronts produced by the di�erent MOEAs for
Two-Beams with "=2 and population size #=100. The refer-
ence front R includes all best (non-dominated) solutions
𝑀
=2 and Population Size
𝑁
=100
• Reference front = best of all
non-dominated front from all
EAs and runs
• We plot the fronts with
median HV
• All fronts show a hyperbolic
geometry (p<1 curvature)

23. The Generated Sorter
GECCO ’23, July 15–19, 2023, Lisbon, Portugal A. Panichella and G. Di Domenico
(a) Generated device (right side) and the corresponding
beams propagation (left side)
(b) Target output signal (red line) and generated output
signal (blue line) for the two HG input beams
Figure 3: Example of device generated by L2-AGE-MOEA for the Two-Beams+Structure problem.
depicted in Figure 3. Figure 3a plots the device (right-hand side); manually. Multi-objective evolutionary algorithms (MOEAs) have
52 μm
38
μm
Two-mode sorter
Target Output Signals
Generated Output Signals

24. Full Replication Package (for PlatEMO)
24
https://zenodo.org/record/7851999

25. In Summary
25
Thank you!

26. 26
Mode sorter by Di Domenico et al.,ACS Photonics 2022

27. 27
Mode sorter by Di Domenico et al.,ACS Photonics 2022