# Iterated learning and communication jointly explain
efficient color naming systems

Emil Carlsson<EMAIL_ADDRESS>
Devdatt Dubhashi<EMAIL_ADDRESS>
Department of Computer Science and Engineering, Chalmers University of
Technology, Gothenburg, Sweden

Terry Regier<EMAIL_ADDRESS>
Department of Linguistics, UC Berkeley, Berkeley, CA, USA

###### Abstract

It has been argued that semantic systems reflect pressure for efficiency, and
a current debate concerns the cultural evolutionary process that produces this
pattern. We consider efficiency as instantiated in the Information Bottleneck
(IB) principle, and a model of cultural evolution that combines iterated
learning and communication. We show that this model, instantiated in neural
networks, converges to color naming systems that are efficient in the IB sense
and similar to human color naming systems. We also show that iterated learning
alone, and communication alone, do not yield the same outcome as clearly.

Keywords: efficient communication; iterated learning; cultural evolution;
semantic categories; color naming

## Introduction

Semantic categories vary across languages, and it has been proposed that this
variation can be explained by functional pressure for efficiency. On this
view, systems of categories are under pressure to be both simple and
informative (e.g. ?, ?), and different languages arrive at different ways of
solving this problem, yielding wide yet constrained cross-language variation.
There is evidence for this view from semantic domains such as kinship (?, ?),
container names (?, ?), names for seasons (?, ?), and numeral systems (?, ?).
? (?) gave this proposal a firm theoretical foundation by grounding it in an
independent information-theoretic principle of efficiency, the Information
Bottleneck (IB) principle (?, ?); they also showed that color naming systems
across languages are efficient in the IB sense, that optimally IB-efficient
systems resemble those found in human languages, and that the IB principle
accounts for important aspects of the data that had eluded earlier
explanations. Subsequent work has shown that container naming (?, ?),
grammatical categories of number, tense, and evidentiality (?, ?), and person
systems (?, ?) are also efficient in the IB sense.

In a commentary on this line of research, ? (?) asked how semantic systems
evolve to become efficient, and suggested that an important role may be played
by iterated learning (IL; e.g. ?, ?). In IL, a cultural convention is learned
by one generation of agents, who then provide training data from which the
next generation learns, and so on. The convention changes as it passes through
generations, yielding a cultural evolutionary process. The idea that such a
process could eventually lead to efficient semantic systems has since been
explored and broadly supported. ? (?) showed that chains of human learners who
were originally given a randomly generated color category system eventually
produced systems that were similar to those of the World Color Survey (WCS; ?,
?), a large dataset of color naming systems from 110 unwritten languages.
Although this study did not explicitly address efficiency, ? (?) drew that
link explicitly: they reanalzyed the data of ? (?) and showed that the color
naming systems produced by IL not only became more similar to those of human
languages – they also became more informative; the same paper also presented
analogous findings for semantic systems of spatial relations. In response, ?
(?) argued, on the basis of a Bayesian model of IL and experiments with human
participants, that learning actually contributes simplicity rather than
informativeness. Overall, there is support for the idea that IL can lead to
efficient semantic systems, with continuing debate over how and why. There are
also recent proposals that non-iterated learning – e.g. in the context of a
dyad of communicating agents (e.g. ?, ?, ?, ?), or in a single agent without
communication (e.g. ?, ?, ?) – can explain efficient color naming systems.
These recent contributions build on an important line of earlier work using
agent-based simulations cast as evolutionary models, without explicitly
addressing efficiency (e.g. ?, ?, ?, ?, ?, ?).

Several of these prior studies have engaged efficiency in the IB sense, and
two are of particular relevance to our own work. ? (?) showed that a dyad of
neural network agents, trained to discriminate colors via communication,
eventually arrived at color naming systems that were highly efficient in the
IB sense. However, these systems did not always resemble those of human
languages: their categories “depart to some extent from those typically
defined by human color naming” (?, ?, p. 11 of SI). ? (?) explored a similar
color communication game, and found that their neural agents gravitated to
color naming systems that are both essentially optimally efficient in the IB
sense, and similar to human color naming systems from the WCS. They achieved
this by optimizing an objective function that is based on the IB objective. To
our knowledge, earlier work leaves open whether both high IB efficiency and
similarity to human languages can be achieved by other means. We explore that
question here.

In what follows, we first demonstrate that there exist many possible color
naming systems that are highly efficient in the IB sense, but do not closely
resemble human systems. The existence of such efficient-yet-not-human-like
systems is not surprising given that IB is a non-convex optimization problem
(?, ?, ?), but it may be helpful in understanding how ? (?) achieved high IB
efficiency with systems that deviate from human ones. We then show that IL,
instantiated in communicating neural networks, gravitates toward efficiency
and, within the class of efficient systems, gravitates more toward human color
naming systems than toward others. Finally, we show that iterated learning
alone, and communication alone, do not yield that outcome as clearly. We
conclude that iterated learning and communication jointly provide a plausible
explanation of how human color naming systems become efficient.

## Not all efficient systems are human-like

Figure 1: Top: Color naming stimulus grid. Bottom: 9 color naming systems
displayed relative to this grid. The left column contains color naming systems
from 3 languages in the WCS (from top to bottom: Bete, Colorado, Dyimini).
Colored regions indicate category extensions, and the color code used for each
category is the mean of that category in CIELAB color space. The named color
categories are distributions, and for each category we highlight the level
sets between $0.75-1.0$ (unfaded area) and $0.3-0.75$ (faded area). The middle
and right columns contain randomly-generated systems of complexity comparable
to that of the WCS system in the same row. The middle column shows random
systems that are similar to the WCS system in the same row. The right column
shows random systems that are dissimilar to the WCS system in the same row; at
the same time, there is no other WCS system that is more similar to this
random system.

We considered a class of artificial color naming systems related to one
considered by ? (?). In the class we consider, each named category $w$ is
modeled as a spherical Gaussian-shaped kernel with mean (prototype) $x_{w}$ in
3-dimensional CIELAB color space, such that the distribution over words $w$
given a color chip $c$ is:

$\displaystyle S(w|c)\propto e^{-\eta||x_{c}-x_{w}||_{2}^{2}}$ (1)

where $\eta>0$ is a parameter controlling the precision of the Gaussian
kernel. We then generated artificial color category systems with $K=3\dots 10$
categories each, by first sampling $\eta$ randomly from a uniform distribution
over the interval $[0.001,0.005]$ and then sampling the prototype $x_{w}$ of
each category $w$ randomly, without replacement, from a uniform distribution
over the cells of the color naming grid shown at the top of Figure 1. In
analyzing these systems, we drew on the following three quantities from the IB
framework as presented by ? (?): the complexity of a category system, gNID (a
measure of dissimilarity between two category systems), and $\epsilon$ (a
measure of inefficiency, or deviation from the theoretical limit of
efficiency). We noted that the range of complexity (in the IB sense) for
systems in the World Color Survey (WCS) was $[0.84,2.65]$, and also noted that
our random model sometimes generated systems outside this range; we only
considered artificial systems with complexity within this range, and generated
$100$ such systems for each $K$; we refer to these systems as RM, for random
model.

The lower panels of Figure 1 show that some of these RM systems are similar
to, and others quite dissimilar to, natural systems in the WCS. In each row,
the rightmost system, which is dissimilar to the WCS system in that row, is
nonetheless more similar to that WCS system than to any other WCS system,
meaning that it is dissimilar to all WCS systems. Thus, there exist RM systems
that are quite dissimilar to naturally occurring systems. To quantify this
pattern, we separated the RM systems into two groups, based on whether their
gNID to the closest WCS system exceeded a threshold. We set this threshold to
the smallest gNID between systems in the left (WCS) and right (RM dissimilar)
columns of Figure 1, which is $0.29$. We then grouped all RM systems with gNID
to the closest WCS system below this threshold into one group,
$\text{RM}_{\text{s}}$ (for similar to WCS), and the other RM systems into
another group, $\text{RM}_{\text{d}}$ (for dissimilar to WCS). $38\%$ of the
RM systems fell in $\text{RM}_{\text{d}}$ and they spanned the complexity
range $[0.86,2.26]$. Thus, a substantial proportion of the RM systems are at
least as dissimilar to WCS systems as are those in the right column of Figure
1.

Figure 2: Efficiency of color naming, following ?, ?. The dashed line is the
IB theoretical limit of efficiency for color naming, indicating the greatest
possible accuracy for each level of complexity. The color naming systems of
the WCS are shown in orange, replicating the findings of ?, ?. Our RM systems
are shown in blue. It can be seen that the RM systems are often closer to the
IB curve than the WCS systems are. The inset shows the 9 color systems of
Figure 1, with the dissimilar random systems shown as +.

Figure 2 shows the results of an IB efficiency analysis of the WCS systems
(replicating ?, ?, and assuming their least-informative prior), and also of
our RM systems. It can be seen that all RM systems are highly efficient in the
IB sense – i.e. they are close to the IB curve that defines the theoretical
limit of efficiency in this domain. Mann-Whitney $U$ tests revealed (1) that
the RM systems tend to exhibit greater efficiency (lower inefficiency
$\epsilon$) than do the WCS systems in the same complexity range ($P\ll.001$),
and (2) that the $\text{RM}_{\text{d}}$ systems, which are dissimilar to WCS
systems, are also more efficient than WCS systems ($P\ll.001$, one-sided), and
slightly to marginally more efficient than $\text{RM}_{\text{s}}$ systems
($P=.019$ one-sided; Bonferroni corrections do not change the qualitative
outcome). These findings suggest that there is a substantial number of color
naming systems that are dissimilar to those of human languages, yet more
efficient than them. This in turn may help to make sense of ?’s (?) finding
that their evolutionary process yielded systems that were highly efficient but
not particularly similar to human ones: our analysis illustrates that there
are many such systems. Given this, we sought an evolutionary process that
would yield both efficiency in the IB sense, and similarity to human systems,
without specifying IB optimization as a part of that process (cf. ?, ?).

## Iterated learning and communication

As noted above, iterated learning (IL; e.g. ?, ?, ?) is a cultural
evolutionary process in which a cultural convention is learned first by one
generation of agents, who then pass that convention on to another generation,
and so on — and the convention changes during inter-generational transmission.
Some of the work we have reviewed above addresses IL (e.g. ?, ?, ?, ?).
However other work we have reviewed instead addresses cultural evolution
through communication within a single generation (e.g. ?, ?, ?, ?). We wished
to explore the roles of both IL and communication, and so we adopted an
approach that involves both, in a way that allows the role of each to be
highlighted. Specifically, we adopted the recently proposed _neural iterated
learning_ (NIL) algorithm (?, ?). In the NIL algorithm, artificial agents are
implemented as neural networks that communicate with each other within a
generation, and cultural convention (in our case, a color naming system)
evolves both from within-generation communication and from inter-generational
transmission, as the convention is iteratively passed down through generations
of artificial agents, with each new generation learning from the previous
one.111NIL, or neural iterated learning, is therefore not an entirely
informative name for this process, as it does not explicitly label the
important element of within-generation communication.

In the NIL algorithm, each generation $t$ (for time step) consists of two
artificial agents, a speaker $S_{t}$ and a listener $L_{t}$. The NIL algorithm
operates in three phases. (1) In the first phase, the _learning phase_ , both
agents are exposed to the naming convention of the previous generation. This
is done by first training the speaker $S_{t}$, using cross-entropy loss, on
color-name pairs generated by the speaker of the previous generation. The
listener $L_{t}$ is then trained via reinforcement learning in a few rounds of
a signaling game while keeping $S_{t}$ fixed: that is, the speaker learns from
the previous generation, and the listener then learns from the speaker. We had
the agents play the signaling game used by ?, ?, in which the speaker is given
a color chip $c$, sampled from a prior distribution over color chips, and
produces a category name describing that color. The listener then attempts to
identify the speaker’s intended color based on the name produced, by selecting
a color chip $\hat{c}$ from among those of the naming grid shown in Figure 1.
A reward is given to the listener depending on how perceptually similar the
selected chip is to the original color. (2) In the second phase, the
_interaction phase_ , the agents play the same signaling game but this time
both agents receive a joint reward and update their parameters during
communicative interactions. (3) In the third phase, the _transmission phase_ ,
color-name pairs are generated by sampling colors from the prior distribution
and obtaining names for them from the speaker $S_{t}$. These color-name pairs
are then passed on to the next generation of agents. In all three phases,
color chips are sampled according to the least-informative prior of ? (?). We
represent both the speaker and listener as neural networks with one hidden
layer consisting of $25$ units with a sigmoidal activation function.
Individual colors are represented in 3-dimensional CIELAB space when supplied
as input to the speaker, and category names as one-hot encoded vectors. For
the reinforcement learning parts of NIL we use the classical algorithm
REINFORCE (?, ?). For the transmission phase we sample $300$ color-name pairs,
out of the $330$ chips in the entire stimulus set; this ensures that the new
generation will have seen examples from most of color space but it is
impossible for them to have seen all color-name pairs. To optimize the neural
networks, we use the optimizer Adam (?, ?), both in the learning and
interaction phase, with learning rate $0.005$ and batch size $50$. For each
phase in the NIL algorithm we take $1000$ gradient steps. We stop the NIL
algorithm once the maximum difference in IB complexity and accuracy over the
ten latest generations is smaller than $0.1$ bit, i.e. when the last ten
generations are all within a small region of the IB plane. Algorithm 1
presents a schematic overview of the NIL algorithm, and ? (?) present a
detailed description. The hyperparameters were tuned empirically by studying
which parameters yielded the highest reward in a small set of experiments. We
found very little difference between different sizes of the network.

1: Initialize $D_{1}$ uniformly at random

2: for $t=1...$ do

3:  _Learning Phase_

4: Randomly initialize $S_{t}$ and $L_{t}$.

5: Train $S_{t}$ on $D_{t}$ using stochastic gradient descent and cross-
entropy loss.

6: Play signaling game between $S_{t}$ and $L_{t}$ and update parameters of
only $L_{t}$ using the rewards.

7:  _Interaction Phase_

8: Play signaling game between $S_{t}$ and $L_{t}$ and update parameters of
both agents using the rewards.

9:  _Transmission Phase_

10: Create transmission dataset $D_{t+1}$ consisting of color-name pairs,
$(c,w)$ by sampling colors from the prior $p(c)$ and providing them as input
to $S_{t}$.

11: end for

Algorithm 1 Neural Iterated Learning

Figure 3: Efficiency of the (top) IL+C, (middle) IL, and (c) C evolved color
naming systems, in each case compared with the natural systems of the WCS. The
black triangle indicates the end state of one run, shown in the inset color
map.

For each vocabulary size $K=3\dots 10$ and $K=100$ we ran $100$ independent
instances of the NIL algorithm. For each instance, we considered the color
naming system of the last speaker to be the result of that instance — we call
these systems IL+C, as they are the result of iterated learning plus
communication, and we evaluated the IL+C systems in the IB framework. As can
be seen in Figure 3 (top panel), the IL+C systems are highly efficient in the
IB sense: they lie near the theoretical efficiency limit (median inefficiency
$\epsilon=0.07$), and they are no less efficient than the random RM systems we
considered above (median inefficiency $\epsilon=0.09$), which in turn are more
efficient than the human systems of the WCS (see above). Thus, iterated
learning plus communication as formalized in the NIL algorithm leads to
semantic systems that are efficient in the IB sense. This is not entirely
surprising: the reward during the signaling game favors informativeness
(higher reward for similar colors, following ?, ?), and it has been argued
that learning favors simplicity (e.g. ?, ?). Interestingly, all the resulting
systems lie within the complexity range of the WCS systems even though NIL
could theoretically produce much more complex systems, especially when
$K=100$.

? (?) showed that chains of iterated human learners tended to gravitate toward
color naming systems that were similar to those of the WCS, and we wished to
know whether the same was true of computational agents in the NIL framework.
For each IL+C system, we determined the dissimilarity (gNID) between that
system and the most similar (lowest gNID) WCS system. We also determined the
analogous quantity (dissimilarity to the most similar WCS system) for each
random RM system. Figure 4 shows that IL+C systems tend to be similar to WCS
systems to a greater extent than RM systems do, and this was confirmed by a
one-sided Mann-Whitney $U$ test $(P\ll.001)$.

Figure 4: Distribution of dissimilarity to WCS systems (minimum gNID to any
WCS system), shown for IL+C and RM systems. The RM systems include both
$\text{RM}_{\text{s}}$ and $\text{RM}_{\text{d}}$.

Thus, the NIL process tends to gravitate toward human (WCS) systems to a
greater extent than a random but efficient baseline, RM.

We also asked whether NIL would transform efficient systems that were
dissimilar to those of the WCS (namely those of $\text{RM}_{\text{d}}$) into
comparably efficient systems that were more similar to the WCS. To test this,
we initialized the NIL algorithm with a system sampled from
$\text{RM}_{\text{d}}$, ran the NIL algorithm, and compared the initial system
to the one that resulted from NIL. Figure 5 illustrates the beginning and end
points of this process for a small set of systems, and shows that NIL
transforms systems that are efficient but unlike the WCS into systems that are
similar to particular WCS systems.

Figure 5: NIL transforms efficient color naming systems to become more similar
to the WCS. In each row, the left column shows an $\text{RM}_{\text{d}}$
system that was used to initialize NIL, the middle column shows the result of
running NIL from that initialization state, and the right column shows a WCS
system (from top to bottom: Bete, Colorado, Dyimini) that is similar to the
NIL result.

Figure 6 shows the same general pattern but aggregated over all
$\text{RM}_{\text{d}}$ systems. For each NIL chain initialized with an
$\text{RM}_{\text{d}}$ system, we measured the dissimilarity (gNID) of that
initialized system to the most similar WCS system, and the gNID of the end
result of NIL to its most similar WCS system. It can be seen that NIL
transforms $\text{RM}_{\text{d}}$ systems into systems that are more similar
to the human systems of the WCS. The mean gNID to WCS was $0.38$ before NIL
and $0.25$ after, and the reduction in dissimilarity to WCS after applying NIL
was significant (one-sided (paired) Wilcoxon signed-rank test, $n=302$,
$T=1113$, $P\ll.001$). The median inefficiency of $\text{RM}_{\text{d}}$ is
$\epsilon=0.09$ and the median inefficiency of the results of NIL is slightly
lower at $\epsilon=0.07$, meaning that NIL made the already-efficient
$\text{RM}_{\text{d}}$ systems slightly more efficient (one-sided (paired)
Wilcoxon signed-rank test, $n=302$, $T=7716$, $P\ll.001$). Thus, NIL moves
already-efficient systems closer to the attested systems of the WCS, while
maintaining and even slightly improving efficiency. Finally, it is noteworthy
that NIL with 3 terms converges to a system that is similar to a 3-term WCS
system (see the top row of Figure 5), because 3-term systems are the one case
in which IB optimal systems qualitatively diverge from human data (?, ?, p.
7941). Thus, this is a case in which NIL appears to provide a better
qualitative fit to the data than IB does (see also ?, ?, ?).

Figure 6: NIL transforms efficient $\text{RM}_{\text{d}}$ color naming systems
to become more similar to the WCS. The difference score is dissimilarity to
WCS (minimum gNID to any WCS system) before NIL, minus the same quantity after
NIL. A higher value indicates that NIL has moved the systems closer to the
WCS. There are no values below 0, meaning that NIL never caused a system to
become less similar to the WCS.

## Other possible evolutionary processes

So far, we have seen evidence that the NIL algorithm may provide a plausible
model of the cultural evolutionary process by which human color naming systems
become efficient. We have referred to the result of the full NIL algorithm as
IL+C systems, because these systems result from both iterated learning (IL)
and communication (C). This raises the question whether iterating learning
alone, or communication alone, would yield comparable results.

To find out, we ran two variants of the NIL algorithm. One variant included
only iterated learning but no communication (i.e. lines 6-8 of Algorithm 1
were omitted). The other variant included communication but no iterated
learning (i.e. there was only one pass through the main loop, which stopped at
line 9); this is exactly the experiment that was performed by ? (?). All other
aspects of the algorithm were unchanged. We refer to the results of the
iterated-learning-only algorithm as IL (for iterated learning), and the
results of the communication-only algorithm as C (for communication).

Comparison of the three panels of Figure 3 reveals that there are qualitative
differences in the profiles of the systems produced by the 3 variants of the
NIL algorithm (IL+C, IL, and C). We have already seen that IL+C systems (top
panel) are both efficient and similar to human systems; we also note that they
lie within roughly the same complexity range as the human systems of the WCS.
In contrast, the IL systems (middle panel) skew toward lower complexity than
is seen in human systems, and in fact about $6\%$ of the IL systems lie at the
degenerate point $(0,0)$ in the IB plane, at which there is a single category
covering the entire color domain. This skew toward simplicity is compatible
with ?’s (?) claim that iterated learning provides a bias toward simplicity.
At the same time, the IL systems are not only simple but also quite efficient
(i.e. informative for their level of complexity), which is in turn compatible
with ?’s (?) claim that iterated learning provides some bias toward
informativeness. Finally, the C systems (bottom panel) show the opposite
pattern: a bias toward higher informativeness, at the price of higher
complexity, extending well above the complexity range observed in the human
systems of the WCS. Taken together, these results suggest that iterated
learning alone over-emphasizes simplicity, communication alone over-emphasizes
informativeness, and iterated learning with communication provides a balance
between the two that aligns reasonably well with what is observed in human
color naming systems. We found that IL+C systems are slightly more efficient
(mean $\epsilon=0.07\pm 0.02$) than IL (mean $\epsilon=0.15\pm 0.08$) or C
(mean $\epsilon=0.11\pm 0.04$) systems, where the $\pm$ indicates plus or
minus one standard deviation. IL+C systems were also closer to the most
similar WCS system (mean gNID = $0.21\pm 0.05$) than were IL (mean gNID =
$0.57\pm 0.17$) or C (mean gNID = $0.27\pm 0.07$) systems. Overall, these
results suggest that iterated learning plus communication is a more plausible
model of the cultural evolutionary process that leads to efficient human color
naming systems than is either iterated learning alone, or communication alone,
as these ideas are formalized in the NIL algorithm.

## Discussion

We have shown (1) that there exists a reasonably sized class of color naming
systems that are highly efficient in the IB sense but dissimilar from human
systems; (2) that iterated learning plus communication, as captured in the NIL
algorithm, leads to color naming systems that are both efficient in the IB
sense and similar to human systems, and (3) that iterated learning alone, and
communication alone, do not yield that result as clearly. These findings help
to answer some questions, and also open up others.

As we have noted, the existence of highly efficient systems that do not align
with human ones is not in itself surprising. IB is a non-convex optimization
problem (?, ?, ?), so multiple optima and near-optima are to be expected.
However we feel that our identification of such systems may nonetheless be
helpful, because it highlights just how many such systems exist, and just how
dissimilar from human systems they sometimes are — which helps to make sense
of ?’s (?) finding that simulations of cultural evolution can lead to color
naming systems that exhibit high IB efficiency but deviate to some extent from
human systems. This in turn highlights the importance of identifying cultural
evolutionary processes that avoid these local near-optima and instead converge
toward systems we find in human languages.

We have argued that iterated learning plus communication, as cast in the NIL
algorithm, is such a process, and that it provides a better account than
either iterated learning alone, or communication alone. This idea, and our
findings supporting it, may help to resolve a question in the literature. As
we have noted, ? (?) argued that iterated learning alone can lead to
informative semantic systems, whereas ? (?) argued that iterated learning
provides a bias for simplicity, and communication provides a bias for
informativeness (see also ?, ? for a similar argument concerning linguistic
form). Our finding that both forces are needed to account for the data aligns
with ?’s (?) claim. However our finding that learning alone also converges to
efficient systems – although to overly simple ones – helps to make sense of
?’s (?) findings.

It is natural to think of NIL, or any such process of cultural evolution, as a
means by which the abstract computational goal of optimal efficiency might be
approximated — and for the most part, that seems an accurate and useful way to
frame the matter. The optimally efficient color naming systems on the IB curve
closely resemble those in human languages (?, ?), and the IL+C systems are
likewise highly efficient and similar to those in human languages. However,
there is an important exception to this pattern. As noted above, in the case
of 3-term systems, the IB optimal system qualitatively differs from the color
naming patterns found in the WCS (?, ?, p. 7941), whereas IL+C systems
qualitatively match them (see e.g. the top row of Figure 5, middle and right
panels). Thus, in this one case, it appears that human languages do not attain
the optimal solution or something similar to it, and instead attain a somewhat
different near-optimal solution that is apparently more easily reached by a
process of cultural evolution – a possibility anticipated by ? (?, p. 1054).

A major question left open by our findings is exactly why we obtain the
results we do. NIL is just one possible evolutionary process, and we have seen
that that process accounts for existing data reasonably well. It makes sense
intuitively that NIL strikes a balance between the simplicity bias of iterated
learning and the informativeness bias of communication (?, ?, ?) — but what is
still missing is a finer-grained sense for exactly which features of this
detailed process are critical, vs. replaceable by others, and what the broader
class of such processes is that would account well for the data (e.g. ?, ?). A
related direction for future research concerns the fact that the evolutionary
process we have explored is somewhat abstract and idealized, in that agents
communicate with little context or pragmatic inference. Actual linguistic
communication is highly context-dependent, and supported by rich pragmatic
inference — it seems important to understand whether our results would still
hold in a more realistic and richer environment for learning and interaction.
Finally, we have focused here on the domain of color, but the ideas we have
pursued are not specific to color, so another open question is the extent to
which our results generalize to other semantic domains.

## Acknowledgments

We thank Noga Zaslavsky and 3 anonymous reviewers for helpful comments on an
earlier version of this paper. Any errors are our own. Author contributions:
EC, DD, and TR designed the research; EC performed the research; EC analyzed
the data; and EC, DD, and TR wrote the paper. EC was funded by Chalmers AI
Research (CHAIR) and the Sweden-America Foundation (SweAm). Computing
resources used for the experiments were provided by the Swedish National
Infrastructure for Computing (SNIC).

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11institutetext:  1: SRON Netherlands Institute for Space Research,
Niels Bohrweg 4, 2333 CA Leiden, The Netherlands
11email<EMAIL_ADDRESS>

2: Department of Microelectronics, Delft University of Technology,
Mekelweg 4, 2628 CD Delft, The Netherlands

3: Else Kooi Laboratory, Delft University of Technology,
Feldmannweg 17, 2628 CT Delft, The Netherlands

4: Kavli Institute of Nanoscience, Delft University of Technology,
Lorentzweg 1, 2628 CJ Delft, The Netherlands

# Model and Measurements of an Optical Stack for Broadband Visible to Near-
Infrared Absorption in TiN MKIDs

K. Kouwenhoven1,2 I. Elwakil1 J. van Wingerden3 V. Murugesan1 D.J. Thoen2,4
J.J.A. Baselmans1,2 P.J. de Visser1

###### Abstract

Typical materials for optical Microwave Kinetic Inductance Detetectors (MKIDs)
are metals with a natural absorption of $\sim$30-50% in the visible and near-
infrared. To reach high absorption efficiencies (90-100%) the KID must be
embedded in an optical stack. We show an optical stack design for a 60 nm TiN
film. The optical stack is modeled as sections of transmission lines, where
the parameters for each section are related to the optical properties of each
layer. We derive the complex permittivity of the TiN film from a spectral
ellipsometry measurement. The designed optical stack is optimised for
broadband absorption and consists of, from top (illumination side) to bottom:
85 nm $\mathrm{SiO_{2}}$, 60 nm TiN, 23 nm of $\mathrm{SiO_{2}}$, and a 100 nm
thick Al mirror. We show the modeled absorption and reflection of this stack,
which has $>$80% absorption from 400 nm to 1550 nm and near-unity absorption
for 500 nm to 800 nm. We measure transmission and reflection of this stack
with a commercial spectrophotometer. The results are in good agreement with
the model.

###### Keywords:

Microwave Kinetic Inductance Detectors, TiN, Optical Stack, Broadband
Absorption

††journal: Journal of Low Temperature Physics

##  1 Introduction

There is a wide variation in the superconductors used for Microwave Kinetic
Inductance Detectors (MKIDs) Mazin2020 . Some common materials are Hafnium
(Hf), Titanium nitride (TiN), Platinum Silicide (PtSi), and alpha- or beta-
phase Tantalum (Ta). Although the properties of these metals differ, they have
one thing in common. They suffer from low absorption efficiency in the visible
to near-IR wavelength range. For $\beta$-Ta, TiN, PtSi, and Hf absorption is
limited to around 50% in the visible and 30% in the IR range, as illustrated
by Fig. 1 and presented in Coiffard2020 . To increase the power absorbed in
the MKID, we embed it in an optical stack consisting of a mirror (cavity) and
one or multiple matching layers. With this approach, unity absorption can be
achieved Rosenberg2004 ; Lita2008 ; Dai2019 . However, the bandwidth over
which high absorption is achieved is usually limited. Here we show the design
of an optical stack for a 60 nm thick TiN layer, based on a transmission line
model, that achieves absorption $>$ 80% over a band from 400 nm to 1550 nm,
with near-unity transmission from 500 nm to 800 nm. We verify the model with a
reflection and transmission measurement using a commercial spectrophotometer.

Figure 1: Absorbed power under normal incidence for two high resistivity
superconductors, TiN (blue) and beta phase tantalum ($\mathrm{\beta}$-Ta)
(red). The plotted absorption is calculated from the measured optical
constants of both films. Both films are 60 nm thick and deposited on a 350
$\mathrm{\mu}$m polished c-plane sapphire substrate. TiN: $T_{c}$ = 3.45 K,
$R_{s}$ = 237.2 $\mathrm{\Omega}$. $\beta$-Ta: $T_{c}$ = 0.99 K, $R_{s}$ =
38.8 $\mathrm{\Omega}$. (Color figure online.)

##  2 Model

The optical stack consists of the superconducting MKID layer (TiN), a
superconducting Al mirror, and two dielectric layers, as illustrated in Fig.
2. The 100 nm thick mirror is deposited on top of the sapphire substrate and
has to be made from a superconducting material to maintain the high quality
factor of the MKIDs Dai2019 . The basic version contains two dielectric
layers. One layer realizes the spacing between the MKID layer and the mirror.
The other layer, deposited on top of the MKID layer, is used to match the
impedance of the MKID plus mirror to the impedance of free-space $\eta_{0}$.
The number of layers on top of the MKID can be increased to realize either
multi-layer interference or tapered impedance matching structures.

The deposited dielectrics will increase the dielectric loss and Two-Level
System (TLS) noise levels in the MKID, especially in regions with a high
E-field density. For this reason, the optical stack should be limited to the
inductor area.

Figure 2: Illustration of the proposed optical stack. The MKID layer (red) is
backed by a 100 nm thick superconducting Al mirror (black). The stack contains
two dielectric layers (blue), one to create the back-short spacing between the
MKID and the superconducting mirror and a second to reduce reflection from the
MKID layer. The number of dielectric layers deposited on top of the MKID can
be increased to improve the stacks performance when necessary. To limit the
added dielectric loss and Two Level System (TLS) noise from the deposited
dielectrics, the capacitor should not be embedded in the stack. In this case
the MKID is front-side (top) illuminated. (Color figure online.)

###  2.1 Transmission Line Representation

Figure 3: Transmission line representation of the optical stack. Each of the
layers, either dielectric or metal, is represented by a section of
transmission line with length equal to the height of the layer. The
transmission lines are characterized by a complex impedance $\eta_{c}$ and a
complex propagation constant $k_{c}$. (Color figure online.)

To model the expected absorption of the optical stack based on the optical
properties of each layer, we represent each layer with a section of
transmission line as illustrated in Fig. 3. The length $l$ of the transmission
line is the thickness of the respective layer, and the complex impedance
$\eta_{c}$ and propagation constant $k_{c}$ are a function of the layer’s
optical properties. We characterize each layer by its relative permittivity
$\tilde{\epsilon_{r}}(\omega)=\epsilon^{\prime}_{r}(\omega)-i\epsilon^{\prime\prime}_{r}(\omega)$,
where $\omega$ indicates the frequency dependency of the relative
permittivity. For ease of readability, we will omit the frequency dependency
of $\tilde{\epsilon_{r}}(\omega)$ in the following equations. The transmission
line parameters are given by Pozar2011

$k_{c}=\beta_{d}\sqrt{1-i\tau},$ (1)

$\eta_{c}=\eta_{d}\frac{1}{\sqrt{1-i\tau}},$ (2)

where $\beta_{d}$ and $\eta_{d}$ are the phase constant and impedance for a
plane-wave in a lossless dielectric
$\beta_{d}=\frac{2\pi}{\lambda_{0}}\sqrt{\epsilon^{\prime}_{r}}$ and
$\eta_{d}=\frac{\eta_{0}}{\sqrt{\epsilon^{\prime}_{r}}}$ with $\eta_{0}$ the
free space impedance and $\lambda_{0}$ the wavelength in free space. The
electric loss tangent $\tan\delta_{e}$ is given by

$\tau=\tan\delta_{e}=\frac{\epsilon^{\prime\prime}_{r}}{\epsilon^{\prime}_{r}}.$
(3)

For a perfect conductor $\tau=\infty$ and for a lossless dielectric $\tau=0$.
The ABCD matrix is a useful tool for the analysis of the full optical stack.
The ABCD matrix of a transmission line section is Pozar2011

$\begin{bmatrix}A&B\\\ C&D\\\
\end{bmatrix}=\begin{bmatrix}\cos(k_{c}l)&\eta_{c}i\sin(k_{c}l)\\\
\eta_{c}^{-1}i\sin(k_{c}l)&\cos(k_{c}l)\\\ \end{bmatrix}$ (4)

and the matrix of the full stack is then given by the cascade of the ABCD
matrices corresponding to the different layers Pozar2011

$\begin{bmatrix}A&B\\\ C&D\\\ \end{bmatrix}=\begin{bmatrix}A_{1}&B_{1}\\\
C_{1}&D_{1}\\\ \end{bmatrix}\begin{bmatrix}A_{2}&B_{2}\\\ C_{2}&D_{2}\\\
\end{bmatrix}\cdots\begin{bmatrix}A_{n}&B_{n}\\\ C_{n}&D_{n}\\\
\end{bmatrix}.$ (5)

The reflection ($\Gamma$) and transmission ($T$) coefficients of the stack are
obtained as Frickey1994

$\Gamma=\frac{AZ_{02}+B-CZ_{01}^{*}Z_{02}-DZ_{01}^{*}}{AZ_{02}+B+CZ_{01}Z_{02}+DZ_{01}}$
(6)

$T=\frac{2\sqrt{\operatorname{Re}(Z_{01})\operatorname{Re}(Z_{02})}}{AZ_{02}+B+CZ_{01}Z_{02}+DZ_{01}},$
(7)

where $Z_{01}$ and $Z_{02}$ are the terminations of the transmission line
Frickey1994  as given in Fig. 3. For the structure in Fig. 2, $Z_{01}$ is the
wave impedance of free space ($\eta_{0}\approx 120\pi$ $\mathrm{\Omega}$) and
$Z_{02}$ the wave impedance of the substrate
($\eta_{0}/\sqrt{\epsilon_{r,sub}}$). The absorbed power fraction of the
complete stack is then given by $A=1-|\Gamma|^{2}-|T|^{2}$.

###  2.2 Spectroscopic Ellipsometry

We characterize the superconducting films with a room temperature
spectroscopic ellipsometry measurement. We measure the amplitude component
$\psi$ and phase difference $\Delta$ at multiple angles $\Theta$ ranging from
55∘ to 80∘ in steps of 5∘ (typically centered around the Brewster’s angle). We
assume that the films are isotropic, homogeneous and that the measured
reflection is dominated by the first reflection at the air-TiN interface. In
this case, the film can be considered infinitely thick. Under these
assumptions, the complex permittivity of the film is retrieved as

$\tilde{\epsilon}(\omega)=\sin^{2}(\Theta)\left[1+\tan^{2}(\Theta)\left(\frac{1-\rho}{1+\rho}\right)^{2}\right]$
(8)

with $\rho=\tan{(\psi)}\exp{(i\Delta)}$ and
$\tilde{\epsilon_{r}}=\epsilon^{\prime}_{r}-i\epsilon^{\prime\prime}_{r}$. The
complex permittivity $\tilde{\epsilon_{r}}$ is related to the complex
refractive index as

$\tilde{\epsilon_{r}}=(n+ik)^{2}.$ (9)

The derived complex permittivity for a 60 nm thick TiN film, deposited on a
sapphire substrate, is given in Fig. 4. The TiN film has a critical
temperature ($T_{c}$) of 3.45 K and a sheet resistance ($R_{s}$) of 237.2
$\mathrm{\Omega}$.

Figure 4: Optical constants (n,k) and relative permittivity for TiN. These
values are derived from a multi-angle spectroscopic ellipsometry measurement
of a 60 nm thick TiN film deposited on a 350 $\mathrm{\mu}$m polished c-plane
sapphire substrate. Individual measurements points are plotted, showing a gap
around 1000 nm where the set-up switches between two different detectors.
Measured from 191.35 nm to 1688.0 nm in 710 non-equidistant measurement
points.

###  2.3 Optimization for Broadband Absorption

For high absorption at single wavelengths, the optimization is easy. The
thicknesses of both dielectrics, see Fig. 2, should be chosen as
$\lambda_{d}/4$ (quarter wavelength) with $\lambda_{d}$ the wavelength in the
dielectric. For absorption over a large wavelength range, however, the optimal
thicknesses can no longer be calculated analytically since the optical
constants of the metal layer vary strongly with wavelength, as indicated by
Fig. 4. Therefore we parameterize the optical stack with respect to the layer
thicknesses and calculate the achieved absorption for multiple thickness
combinations. We optimize for maximum average absorption over a defined
wavelength range, in this case, 400 nm to 1550 nm. The optimal layer
combination for a 60 nm TiN film consists of, from top (illumination side) to
bottom: 85 nm $\mathrm{SiO_{2}}$, 60 nm of unpatterned TiN, 23 nm of
$\mathrm{SiO_{2}}$, and a 100 nm thick Al mirror. $\mathrm{SiO_{2}}$ is chosen
since it is transparent (lossless) for visible light and has a relative
permittivity close to the desired value for a $\lambda/4$ matching layer,
$\epsilon_{r,\lambda/4}=\sqrt{\epsilon_{r,1}\epsilon_{r,2}}$ with
$\epsilon_{r,1}$ and $\epsilon_{r,2}$ the relative permittivity of the two
media surrounding the matching layer. The material properties for
$\mathrm{SiO_{2}}$ and Al are taken from Rakic1995  and Marcos2016 .

The calculated absorbed and reflected power fractions for this stack are given
in Fig. 5. The given absorption is for the entire structure and contains the
power absorbed by the Al mirror in addition to the power absorbed in the TiN
MKID. The transmission line model presented here can be used to extract the
power absorbed per layer, separating the power absorbed in the TiN layer from
the power lost in the dielectric and Al layers. The model shows that of the
power absorbed by the stack, 1-5% is absorbed in the Al mirror with a $\sim$1%
absorption in the two dielectric layers for shorter wavelengths, see Fig. 5.
The authors of Verma2020  quote an absorption of 3% in the metallic mirror at
1550 nm, similar to the results presented here. For a single wavelength
application an optimal stack can be analytically designed for 100% absorption.
However, the power absorbed in the actual detector is limited to $\sim$95-98%
by absorption in the dielectric and Al layers.

##  3 Measurement

Figure 5: Top: Model (blue) and measurement (red) of the absorption (solid)
and reflection (dashed), of a stack consisting of, from top (illumination
side) to bottom: 85 nm layer of $\mathrm{SiO_{2}}$, a 60 nm thick unpatterned
TiN film, a 23 nm $\mathrm{SiO_{2}}$ layer and a 100 nm thick Al mirror
deposited on a 350 $\mathrm{\mu}$m polished c-plane sapphire substrate.The
absorbed power is derived from the measured reflection and transmission. The
measurement is performed with a PerkinElmer Lambda 1050+ spectrophotometer
equipped with an integrating sphere. Bottom: Power lost per material defined
as the fraction of total absorbed power. The material properties for
$\mathrm{SiO_{2}}$ and Al are taken from Rakic1995  and Marcos2016 . (Color
figure online.)

We measure the reflectance and transmittance from 400 nm to 1550 nm with a
commercial spectrophotometer (PerkinElmer Lambda 1050+) equipped with an
integrating sphere (IS), with the sample mounted at the front (transmission)
or back (reflection) port of the IS. The measurement is self-calibrating
through spectralon (white disk) reference measurements. At around 900 nm, the
spectrophotometer switches gratings, which causes a calibration error
resulting in fluctuations around the 900 nm point. The measured reflection and
derived absorption are given in Fig. 5, which shows a good agreement between
the model and measurement. The transmitted power is omitted from this figure
since it is 0% over the entire wavelength range for both the model and
measurement due to the 100 nm thick Al mirror. The $SiO_{2}$ layers are
sputter deposited from a quartz target.

The disagreement between model and measurement can be explained by a
relatively small change in optical properties or a deviation in the layer
thicknesses. We did not measure the layer thicknesses of the fabricated
sample. These changes can explain the larger deviation at shorter wavelengths
in Fig. 5 as they can create destructive interference close to the wavelength
range of interest.

The reflection and transmission measurement is performed at room temperature
while the operating temperature of MKIDs is around 100 mK. The change in
temperature can have an effect on the optical constants of the metallic
layers. To study the temperature dependence of these films, the optical
constants of metallic layers can be modelled as a combination of a Drude term
and a set of Lorentz oscillators, as is done for TiN in Patsalas2001  and for
Al in Rakic1995 . The Drude term scales with the conductivity of the sample
$\sigma$ and is thus temperature dependent. We assume the interband
transitions, modelled with the Lorentz oscillator model, are temperature
independent. We measure the temperature dependency of $\sigma$ with a four-
probe DC structure during a cool-down as the Residual-Resistance Ratio (RRR).
For TiN, RRR is $\sim$1 and for Al it is $\sim$3, indicating that the optical
constants of TiN should be temperature independent. Furthermore, the analysis
in Rakic1995  and Patsalas2001  show that the optical constants of both the Al
and TiN films are dominated by the temperature independent interband
transitions.

##  4 Conclusion and Outlook

We have designed an optical stack consisting of, from top (illumination side)
to bottom: 85 nm $\mathrm{SiO_{2}}$, 60 nm TiN (MKID inductor), 23 nm
$\mathrm{SiO_{2}}$ and a 100 nm thick Al mirror optimized for broadband
absorption in the visible to near-infrared. As presented in Fig. 5, the
initial results are promising, showing near-unity absorption from 600 nm to
800 nm and overall high absorption in the 400 nm to 1550 nm band. For the next
iteration, we will measure both the layer thickness and optical constants of
each deposited layer, to get a better agreement between model and measurement.
The final goal is to measure the absorption efficiency of a MKID with its
inductor embedded in an optical stack. In addition to the analysis in this
paper, this requires an understanding of the effect of temperature on the
optical properties of each layer, as well as the effect of a patterned MKID
layer (inductor lines).

###### Acknowledgements.

We acknowledge Martijn Tijssen and Olindo Isabella for their support with the
ellipsometry and reflection/transmission measurements. This work is
financially supported by the Netherlands Organisation for Scientific Research
NWO (Projectruimte 680-91-127 and Veni Grant No. 639.041.750). JJAB was
supported by the European Research Council ERC (Consolidator Grant No. 648135
MOSAIC). The datasets generated during and/or analysed during the current
study are available in the ZENODO repository,
https://doi.org/10.5281/zenodo.5549994.

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