# Playing Ping Pong with Light:
Directional Emission of White Light

Heribert Wankerl  University of Regensburg, 93053 Regensburg, Germany  OSRAM
Opto Semiconductors GmbH, 93055 Regensburg, Germany  heribert.wankerl@ams-
osram.com  Christopher Wiesmann  OSRAM Opto Semiconductors GmbH, 93055
Regensburg, Germany  Laura Kreiner  OSRAM Opto Semiconductors GmbH, 93055
Regensburg, Germany  Rainer Butendeich  OSRAM Opto Semiconductors GmbH, 93055
Regensburg, Germany  Alexander Luce  OSRAM Opto Semiconductors GmbH, 93055
Regensburg, Germany  Max Planck Institute for the Science of Light, 91058
Erlangen, Germany  Friedrich Alexander University Erlangen Nuremberg, 91054
Erlangen, Germany  Sandra Sobczyk  OSRAM Opto Semiconductors GmbH, 93055
Regensburg, Germany  Maike Lorena Stern  OSRAM Opto Semiconductors GmbH, 93055
Regensburg, Germany  Elmar Wolfgang Lang  University of Regensburg, 93053
Regensburg, Germany

###### Abstract

Over the last decades, light-emitting diodes (LED) have replaced common light
bulbs in almost every application, from flashlights in smartphones to
automotive headlights. Illuminating nightly streets requires LEDs to emit a
light spectrum that is perceived as pure white by the human eye. The power
associated with such a white light spectrum is not only distributed over the
contributing wavelengths but also over the angles of vision. For many
applications, the usable light rays are required to exit the LED in forward
direction, namely under small angles to the perpendicular. In this work, we
demonstrate that a specifically designed multi-layer thin film on top of a
white LED increases the power of pure white light emitted in forward
direction. Therefore, the deduced multi-objective optimization problem is
reformulated via a real-valued physics-guided objective function that
represents the hierarchical structure of our engineering problem. Variants of
Bayesian optimization are employed to maximize this non-deterministic
objective function based on ray tracing simulations. Eventually, the
investigation of optical properties of suitable multi-layer thin films allowed
to identify the mechanism behind the increased directionality of white light:
angle and wavelength selective filtering causes the multi-layer thin film to
play ping pong with rays of light.

###### keywords:

Bayesian Optimization and Physics-guided Machine Learning and Ray tracing and
White Light Directionality and White Light-Emitting Diodes

##  1 Introduction

(a) Conceptual LED setup

(b) Color space according to the Commission internationale de l’éclairage
(CIE-color space adopted from Ding et al. [1])

Figure 1: An LED (a) consists of a chip, a conversion system and a multi-layer
thin film that features ten layers. The latter focuses the emitted light
spectrum into a $\pm 25^{\circ}$-angle cone. The spectrum is composited of
light rays (arrows) of different wavelengths. Such a spectrum can be
associated with a point $\mathbf{c}=\left(C_{x},C_{y}\right)$ in the color
space (b). In this work, the Euclidean distance between two color points is
denoted as the color point deviation $d$.

For many recent applications in our everyday lives, light-emitting diodes
(LED) are required to emit a light spectrum that features specific optical
properties. Such a light spectrum explains how the (relative) power provided
by an LED is distributed over its contributing wavelengths and occurring
emergent angles. In this work, we elaborate on how to shift the light
distribution of white LEDs towards forward directions using a multi-layer thin
film (MLTF). As illustrated in figure 1 $(a)$, the term white LED (package)
refers to a horizontal stack of a semi-conductor chip, the conversion system
and an optional multi-layer thin film. Further on, the common LED structure
without an MLTF is referred to as the reference design. By multiplication with
a color matching function[2, 3], each light spectrum can be characterized by
the so-called color point, a two-dimensional vector in the color space of
figure 1 $(b)$, which describes the color of a spectrum[1]. In the common case
of a white LED without a multi-layer thin film (MLTF), the chip emits blue
light that passes through the ensuing conversion system[4]. Here, two
conversion materials, compounded with weight percentages
$\mathbf{w}=(w_{1},w_{2})$, convert a portion of the blue light to green and
red light resulting in white light. The amount of conversion materials
determines the degree of conversion and is adapted such that the resulting
light spectrum corresponds to the application-specific white target color
point C. Another important characteristic of an LED is the power of its
radiated spectrum. In radiometry, this power is commonly referred to as
radiant flux. Hence, both the color point $\mathbf{c}^{\alpha}(\mathbf{w})$
and the forward power $P^{\alpha}(\mathbf{w})$ featured by an LED in a
$\pm\alpha$-angle cone are computed based on the anisotropic radiated light
spectrum. Unfortunately, such emitted spectra of LED packages are unknown a
priori for varying conversion materials and/or MLTFs. Thus, the deduction of
spectrum-related measures of a particular LED like its power or color point
would require expensive physical experiments; namely, to physically fabricate
and evaluate the spectra of such LED packages. An elegant and cheaper
alternative is to statistically trace a bunch of rays sampled from the light
distribution of the light injecting LED chip, until they exit the LED package
and form the emitted spectrum, or vanish due to non-radiative thermal losses.
Thereby, the behavior of each ray is governed by geometrical optics[5]. In
this framework, only the spectrum of the LED chip needs to be known and thus
is measured once. The spectrum of the LED chip in turn is assumed to remain
stable over changes of both, the conversion system and the MLTF. Such still
time-consuming and noisy ray tracing simulations are based on calibrated
optical models of LEDs [6, 7, 8, 9, 10] and allow an estimation of their
spectra. Notably, ray tracing simulations are non-deterministic and can be
considered Monte-Carlo simulations [9]: The random variables of interest
(power and color point) cannot be computed analytically and thus need to be
estimated via numerous realizations and processing (tracing) of observable
random variables (rays that exit the chip). Such tray tracing simulations
replace expensive experiments and allow to conduct extensive optimization of
LEDs.

Since the aforementioned so-called color point optimization problem based on
$\mathbf{w}$ is empirically unambiguous and convex, it is not possible to
further increase the power $P^{\alpha}(\mathbf{w})$ if

$\displaystyle\mathbf{w}=\text{argmin}_{\boldsymbol{\omega}}\\{d\left(\mathbf{C},\textbf{c}^{\alpha}\left(\boldsymbol{\omega}\right)\right)\\}$

holds based on the Euclidean distance $d\equiv\|\cdot\|_{2}$. The proposed
idea of this manuscript is to modify the anisotropic light spectrum so as to
focus more power $P^{\alpha}(\mathbf{w},\mathbf{t})$ into the forward angle
cone compared to the reference design by using an MLTF — parameterized by
$\mathbf{t}$ — on top of the conversion system; thus, increasing the
directionality of white light. In this work, directional emission or
directionality of white light refers to the power that is emitted in a
particular (solid) angle of interest induced by emitted rays of light
accumulated in these directions. The corresponding inverse design problem
features $T+2$ parameters that describe the modified LED: In addition to the
weight percentages of the conversion materials, the layer thicknesses
$\mathbf{t}=\left(t_{1},...,t_{T}\right)$ of $T$ alternating layers of
titanium dioxide ($\text{TiO}_{2}$) and silicon dioxide ($\text{SiO}_{2}$) can
be adapted. Notably, increasing the usable power may compete with achieving
the application-specific white target color point. For instance, due to the
Stokes shift [11], designing an MLTF that raises the ratio of blue light in
forward direction, obviously increases the usable power but will no longer
retain the desired color point. Namely, it will render the LED’s light to
appear bluish and no longer convenient for a white light application. In other
words, the MLTF is required to not only increase the number of rays that
emerge in forward direction but also achieve a particular ratio between blue,
green and red rays. The aforementioned challenges result in a multi-objective
optimization problem: improving the directionality of white light while
preserving the color point associated to the spectrum that exits the LED. More
precisely, we aim to increase $P^{\alpha}(\mathbf{w},\mathbf{t})$ while
keeping the Euclidean distance $d^{\alpha}(\mathbf{w},\mathbf{t})\equiv
d(\mathbf{c}^{\alpha}(\mathbf{w},\mathbf{t}),\mathbf{C})$ low. To summarize,
the contribution of this work is three-fold:

  * •

Bayesian optimization is used to adapt the layer thicknesses of an MLTF based
on ray tracing simulations of white LEDs

  * •

MLTFs are investigated with regard to their general ability to increase
directionality of white light emission of LEDs

  * •

The effect that explains the directionality increase of white LEDs in physical
terms is identified

##  2 State of the art

Although the directional emission of white light is crucial for many
applications like head lamps of cars, to the best of our knowledge, none of
the aforementioned points were investigated in previous work. Traditionally,
the directionality of incoherent light sources like LEDs is increased via
optical devices such as reflectors or lenses [12, 13], which shape the beam
profile of a light source [14]. Notably, such optics are often bulky and
therefore cause optical loss or limit the design. On the nanoscale, meta-[15,
16, 17] and microlenses[18, 19] have been introduced to manipulate light by
microscopic structures. In addition, scientific work has been conducted
towards the development of incoherent and directional light sources based on
periodic or random gratings like dielectric[20, 21] or plasmonic[22]
structures. Moreover, extensive studies regarding metasurfaces [23, 24] have
been conducted to increase the directional outcoupling of emitters. However,
all of these systems are challenging to fabricate in mass-production or apply
to coherent light sources only. Notably, they do not affect the light source
itself but rather shape the light distribution over angle that exits a light
source like an LED package. In contrast, the proposed epitaxial deposition of
an additional MLTF on top of the conversion system of an LED is
straightforward from an engineering viewpoint, allows compact integration in
existing LED packages, implies only low additional expenditure and is directly
applicable for mass-production of optical semiconductors. The closest
investigation to ours may be the study conducted by Yi Zheng and Matthew
Stough [25]: They proposed to use an MLTF as a wavelength selective filter
between the LED chip and the conversion system to increase the global efficacy
of white LEDs. This filter allows the blue light coming from the chip to pass
while simultaneously reflecting the green and red light emitted by the
conversion system. Thereby, reabsorbtion effects of non-blue photons in the
chip are suppressed. In other words, the MLTF enforces the green and red light
to exit the package rather than irradiating the chip and thus increases the
overall extraction efficiency. Such an MLTF is included into the optical LED
model used in this work and further considered as integral component of the
LED chip. In contrast, the contribution of our work is to improve the
directional emission into an angle cone rather than increasing the overall
outcoupling efficiency. Therefore, we demonstrate that an MLTF, acting as an
angle and wavelength selective filter between the conversion system and the
ambient air, can increase the directional emission of white light. Thereby,
the MLTF helps to shape the angular and spectral light distribution directly
through ray ping pong during its generation process rather than to collimate
the outcoupled light.

Designing MLTFs that feature a particular target reflectivity or transitivity
over angle of incidence and wavelength is a common engineering challenge and
therefore many optimization methods have been developed: Some of them are
based on gradients [26] or biological inspirations [27, 28]. Recently, even
some approaches including neural networks [29, 30, 31] or reinforcement
learning [32, 33] have been implemented to efficiently scan the search space
for suitable MLTF designs. These techniques rely on fast-to-evaluate
computations regarding the transfer matrix method[34, 35] (TMM) that allow to
compute the optical characteristics111e.g. reflectivity or transmittivity over
wavelength and angle of incidence of an MLTF. To measure222e.g. with a notion
of reconstruction error how close a particular MLTF’s optical characteristic
is to an optimal one takes not even a second in total. However, in contrast
with applications like anti-reflection coatings[28], a specific optimal
optical characteristic that causes an MLTF to increase the directionality of
white light is not known a priori. To circumvent this lack of information, we
conduct noisy ray tracing simulations in order to optimize the power and color
point in forward direction regarding the layer thicknesses of an MLTF. As a
side benefit, the optical characteristic of MLTFs is implicitly optimized
towards the a priori unknown optimal one and can be investigated further a
posteriori. Notably, these noisy simulations take about $4\text{ [min]}$ for a
given MLTF, which is $420$ times longer compared to computing the optical
characteristics of the MLTF itself based on TMM. Thus, ray tracing simulations
are relatively expensive-to-evaluate and render most of the aforementioned
data-hungry TMM-based optimization methods of MLTFs impractical. Therefore, we
propose to adapt the individual MLTF layer thicknesses in order to maximize
$P^{\alpha}(\mathbf{w},\mathbf{t})$ while minimizing
$d^{\alpha}(\mathbf{w},\mathbf{t})$ via a variant of Bayesian
optimization[36]. Although Bayesian optimization has shown satisfying results
on many mathematical test functions as well as expensive-to-evaluate real-
world problems regarding engineering, physical and chemical sciences [37, 38,
39], to the best of our knowledge no application towards ray tracing
simulations is reported in scientific literature. After optimizing an MLTF,
its optical characteristics, like transmittivity, allow to physically deduce
target behaviors that an MLTF needs to fulfill in order to further increase
the directionality of white light. Hereby, an effect called ray ping pong is
identified to be responsible for the improved directional emission of white
light. Related effects like photon recycling [40] were studied in previous
work for solar GaAs cells [41, 42, 43], perovskite LEDs [44] or thin films
themselves [45]. In our work, depending on their wavelength and incident
angle, the rays are considered to be partially trapped in the LED package by
an MLTF, enforcing interactions between rays and LED package materials like
scattering, (re-)absorbtion, (re-)emission or non-radiative effects. The MLTF
can statistically steer the degree of such interactions for particular parts
of the original spectrum emitted by the LED package. Thus, the MLTF has a
direct impact on the emitted spectrum and needs to balance the radiated
spectrum not only to appear as white in forward direction, but also to
suppress radiation in non-forward direction. In other words, the MLTF is found
to play angle and wavelength selective ping pong with the rays of light to
achieve an equilibirium of emitted rays, which is of advantage regarding
directionality while still holding the color point.

Figure 2: (a) Illustration of a multi-layer thin film consisting of two
dielectric materials ($\text{TiO}_{2}$ in light grey and $\text{SiO}_{2}$ in
dark grey). (b) The heatmap of the Euclidean color point deviation over weight
percentages of two conversion materials for the multi-layer thin film in (a).
The green arrow indicates the possible path of a local optimizer initialized
near the origin of the search space to the optimal converter weight
percentages.

1:Initialize $\mathcal{D}$, $\mathcal{W}$, $0<\alpha<90$, and set a stopping
criterion, e.g. a timeout

2:while stopping criterion not fulfilled do

3: Suggest (next) $\mathbf{t}$ based on $\mathcal{D}$ and TS-SOO [36]

4: Solve
$\mathbf{w}=\text{argmin}_{\boldsymbol{\omega}}\\{d^{\alpha}\left(\boldsymbol{\omega},\mathbf{t}\right)\\}$
based on Downhill-simplex algorithm

5: Add aquired data point to data set
$\mathcal{D}\leftarrow\mathcal{D}\cup\\{\left(\mathbf{t},f^{\alpha}(\mathbf{t}\right)\\}$
and track $\mathcal{W}\leftarrow\mathcal{W}\cup\\{\mathbf{w}\\}$

6:end while

Algorithm 1 Hierarchical optimization approach: Pseudo-code that illustrates
the proceeding during optimization. $\mathcal{D}$ denotes the acquired data
set of parameters and observations, $\alpha$ denotes the opening angle of the
forward cone, and the timeout criterion defines how long the optimization
endures. Optionally, the weight percentages are stored in $\mathcal{W}$ for
traceability.

##  3 Hierarchical optimization as a product of weights

Since hitting a suitable color point of an LED is mandatory for production, we
reformulate the problem introduced in section 1 as a hierarchical optimization
task

$\displaystyle\max_{\mathbf{t}}\big{\\{}P^{\alpha}\left(\mathbf{w},\mathbf{t}\right)|\mathbf{w}=\text{argmin}_{\boldsymbol{\omega}}\\{d^{\alpha}\left(\boldsymbol{\omega},\mathbf{t}\right)\\}\big{\\}}.$
(1)

Therefore, according to domain expert knowledge, we assume that the color
point optimization with respect to the conversion materials remains convex for
a fixed particular MLTF, if initialized near the origin (see figure 2).
Equation (1) allows to separate the two sets of parameters: The convex color
point optimization problem based on the weight percentages $\mathbf{w}$ of
conversion materials, and the non-convex power optimization problem including
the layer thicknesses $\mathbf{t}$. In this work, a variant of active learning
called Bayesian optimization is applied to maximize the power $P^{\alpha}$ by
adapting the thicknesses of the layers of an MLTF. Eventually, a variation of
the Downhill-simplex[46] optimizer is used to customize the conversion system
for each particular MLTF in order to retain the desired color point before
evaluating the power. To account for this hierarchical structure, we implement
a weighted power as physics-guided real-valued objective function

$\displaystyle
f^{\alpha}(\mathbf{t})=P^{\alpha}\left(\mathbf{w},\mathbf{t}\right)\cdot
W\left(d^{\alpha}\left(\mathbf{w},\mathbf{t}\right)\right)$
$\displaystyle\text{, where
}\mathbf{w}=\text{argmin}_{\boldsymbol{\omega}}\\{d^{\alpha}\left(\boldsymbol{\omega},\mathbf{t}\right)\\}$
(2) $\displaystyle\text{ and
}W\left(d^{\alpha}\left(\mathbf{w},\mathbf{t}\right)\right)=\text{exp}\left(a\cdot[d^{\alpha}\left(\mathbf{w},\mathbf{t}\right)]^{b}\right),$
(3)

which is maximized via Thompson-sampling single-objective Bayesian
optimization[36] (TS-SOO) in this work. Here, $\mathbf{w}$ is a solution to
the nested color point optimization in problem (1). As explained, the weight
percentage parameters $\mathbf{w}$ are not tuneable by the Bayesian optimizer
directly. However, for given thicknesses $\mathbf{t}$ of an MLTF, the color
point deviation $d^{\alpha}(\mathbf{w},\mathbf{t})$ may be bounded from below
with regard to the weight percentages $\mathbf{w}$. This means that the target
color point is not reachable for a given MLTF. In such cases, the comparison
of power values for different MLTFs at different color points becomes invalid
due to the Stokes shift. Here, $W(\cdot)$ allows to guide the course of
optimization towards suitable MLTFs: The weighting punishes excessive
deviations from the target color point by decreasing the objective, although
the power of an MLTF may be high. In practice, more weighting functions may be
introduced to account for various conflicting or competing effects during LED
development.

We compared the TS-SOO to the Thompson-sampling efficient multi-objective
optimization[36] (TS-EMO). Here, a high forward power $P^{\alpha}$ and a low
color point deviation $d^{\alpha}$ are considered as (competing) real-valued
objectives. Namely, TS-EMO directly searches for joint parameter vectors
$\left(\mathbf{w},\mathbf{t}\right)$ to entry-wise maximize

$\displaystyle\mathbb{R}^{2}\times\mathbb{R}^{T}$
$\displaystyle\rightarrow\mathbb{R}^{2}$ (4)
$\displaystyle\left(\mathbf{w},\mathbf{t}\right)$
$\displaystyle\mapsto\left(P^{\alpha}\left(\mathbf{w},\mathbf{t}\right),-d^{\alpha}\left(\mathbf{w},\mathbf{t}\right)\right),$

where the aforementioned Stokes shift renders the involved objectives to be
competitive in nature. The minus sign of the second entry of equation (4)
reflects the intention to minimize the color point deviation. Note that the
hierarchical structure of the engineering optimization problem is not longer
represented in the equation.

Figure 3: (left) The ray tracing for the reference design based on different
numbers of traced rays ($10\cdot 10^{3}$, $50\cdot 10^{3}$ and $100\cdot
10^{3}$) is started $15$ times. The conversion material weight percentages
remained unchanged. As expected, the corresponding boxplots indicate that the
noise level decreases with an increasing number of rays traced. The horizontal
dashed line indicates the reference power obtained with $10^{6}$ rays. This
noise effect holds for the estimated color point $\left(C_{x},C_{y}\right)$,
too. (right) The weighting function (3) for $a=-634914.5425$ and $b=3.3900$.
The dark lines emphasize the condition $W(0.01)=0.9$.

##  4 Implementation

For applications like headlights, we set $\alpha=25^{\circ}$ and set
$d^{\alpha}(\mathbf{w},\mathbf{t})=0.005$ as a preliminary upper bound of the
color point deviation from the required target color point
$\mathbf{C}=\left(\nicefrac{{1}}{{3}},\nicefrac{{1}}{{3}}\right)$ of a white
LED. This enables us to universally solve for the parameters
$\left(a,b\right)$ such that the empirically derived conditions

$\displaystyle W(0.005)$ $\displaystyle=0.99$ (5) $\displaystyle W(0.010)$
$\displaystyle=0.90$

hold, yielding $a=-634914.5425$ and $b=3.3900$. The resulting weighting
function is illustrated in figure 3 (a). The reference design refers to the
special case $t_{1}=...=t_{T}=0.0$. In general, we propose to formulate
hierarchical, competing objectives as a product of weights and a central
figure of merit, e.g. the power. Conceptually, this enforces strong
(AND-)conditions for all constraints explained by the weights. In other words,
a high figure of merit is only valuable if all constraints are fulfilled. On
the other hand, this formulation preserves continuity of the objective
function which makes it approximable with Gaussian processes [47]. As the
black-box function (2) is based on a ray tracing simulation of $25\cdot
10^{4}$ rays that takes several minutes ($\approx 4\text{ [min]}$) to be
evaluated and features non-linearity and non-convexity, the maximization is
conducted via an active learning appoach, the aforementioned TS-SOO. This
variant of Bayesian optimization is employed to optimize expensive-to-evaluate
engineering and chemical problems [37, 38, 39]. Moreover, as illustrated in
figure 3 (b), the evaluation of the objective function provides noisy samples
due to the conducted ray tracing. As explained in algorithm 1, in each
optimization iteration $n$ a thickness vector $\mathbf{t}^{n}$ is suggested
using TS-SOO. For this MLTF, a variant of the Downhill-simplex algorithm
solves the convex color point optimization nested in statement (1), yielding
$\mathbf{w}^{n}$. The acquired data point
$\left(\mathbf{t}^{n},f^{\alpha}\left(\mathbf{t}^{n}\right)\right)$ is added
to the data set $\mathcal{D}$. This data set is used to update the global
surrogate model — implemented as a Gaussian process — based on which the next
thickness vector $\mathbf{t}^{n+1}$ is derived until a predefined stopping
criterion is fulfilled, e.g. a timeout or a maximum number $N\geq n$ of
iterations. Basically, TS-EMO follows the same optimization routine, but
directly suggests joint parameter vectors
$\left(\mathbf{w}^{n},\mathbf{t}^{n}\right)$ to solve the multi-objective
optimization problem (4). For TS-SOO as well as TS-EMO, we set $T=12$ and
allow the thicknesses to vary between $10\text{ [nm]}$ and $200\text{ [nm]}$
for each layer. Moreover, both conversion material weight percentages are
adoptable between $0\text{ [wt\%}]$ and $25\text{ [wt\%}]$. After the
objective function (2) or (4) was optimized via TS-SOO or TS-EMO, a naive
Downhill-simplex algorithm [46] is conducted, respectively. Via not more than
$25$ optimizer steps the thickness parameters and conversion material weight
percentages are jointly fine-tuned. Each of these steps takes $8-9\text{
[min]}$ and is based on $5\cdot 10^{5}$ traced rays per simulation to evaluate
the respective objective function. During the local refinement, the acceptable
upper bound for the color point deviation in equation (5) is narrowed down
from $0.005$ to $0.002$, which is practically required for most applications.

Figure 4: Color point deviation $d^{\alpha}$ over power $P^{\alpha}$ after
joint (left, TS-EMO), hierarchical (middle, TS-SOO) and local (right, Downhill
simplex) optimization for $\alpha=25^{\circ}$. The horizontal black lines
denote the color point deviation of $0.005$ points, where
$W\left(0.005\right)=0.99$ holds. The vertical black lines denote the power of
the reference design without multi-layer thin film. For each sample in these
plots, the color map reflects the objective function values obtained by
equation (2).

##  5 Results

In this section, we present the results of our investigations. First, we
demonstrate that MLTFs can increase the directionality of white LEDs, which
may seem counterintuitive beforehand. Second, we give an explanation of the
underlying physical effects and discuss how we can make the latter observable
in the spectra. The results are summarized in table 5. Here, we report the
MLTFs suggested by TS-SOO and TS-EMO that provided the highest power in $\pm
25^{\circ}$, while exhibiting a color point deviation lower than $0.002$
points — which is an acceptable deviation for the most consumer products in
practice. As mentioned, a joint local optimization of both, thicknesses and
conversion material weight percentages is conducted after the global Bayesian
optimization. Unsurprisingly, the total power of the white LED decreases for
all considered MLTFs due to absorption losses. However, more power is
available at particular forward angles of interest. This directionality
increase is of value for applications like automotive headlamps or projection,
where non-forward light does not contribute.

Figure 5: (left) Common angular and spectral distribution of relative power of
a white LED without a multi-layer thin film. (right) Spectrum of a white LED
with a multi-layer thin film that contains $28.9\%$ more relative power
compared to the reference design in forward direction. In each case, the blue
chip wavelength ($440\text{ [nm]}$) and the green and red wavelengths
($555\text{ [nm]}$) and $600\text{ [nm]}$) of the conversion materials are
indicated by vertical dashed lines. The horizontal grey line indicates the
$\pm 25^{\circ}$-forward direction. The color map for the heat maps coincides
with figure 6 (right).

$\alpha$ $[^{\circ}]$ $25$ $45$ $90$ $P^{\alpha}\text{ [W]}$ 25cm Reference
TS-SOO TS-EMO 20cm$0.201$ $\mathbf{0.259}$ $0.210$ 20cm$0.559$ $0.640$ $0.569$
20cm$1.052$ $0.985$ $0.978$ $d^{\alpha}$ $[1]$ 20cm Reference TS-SOO TS-EMO
20cm$0.0015$ $\mathbf{0.0007}$ $0.0025$ 20cm$0.0071$ $0.0230$ $0.0262$
20cm$0.0146$ $0.0275$ $0.0281$

Table 1: Optimization results regarding color point deviation $d^{\alpha}$,
and power $P^{\alpha}$ of algorithm 1, and TS-EMO for $\alpha=25^{\circ}$, and
after local refinement. The designs featuring a multi-layer thin film are
compared with the reference design, where only a color point optimization is
conducted. In addition to the direct optimization objectives regarding $\pm
25^{\circ}$, we also report the color point deviations and energies for $\pm
45^{\circ}$ and $\pm 90^{\circ}$ observed for the respective optimized MTLFs.

###  5.1 Increase of directionality

The reference design provides
$P^{\alpha}\left(\mathbf{w},\mathbf{t}=\mathbf{0}\right)=0.201\text{ [W]}$
after adopting the weight percentages $\mathbf{w}$, while the deviation
between the LED’s and the target color point is given by
$d^{\alpha}\left(\mathbf{w},\mathbf{t}=\mathbf{0}\right)=0.00151<0.002$. After
optimizing an MLTF’s thickness vector $\mathbf{t}$ using algorithm 1, the
forward power of the LED is increased by $25.4\%$ to
$P^{\alpha}\left(\mathbf{w},\mathbf{t}\neq\mathbf{0}\right)=0.252\text{ [W]}$.
The Euclidean color point deviation for this design is given by
$0.0014<0.002$. Notably, using TS-EMO to solve the corresponding original
multi-objective optimization problem (4) achieved only almost $5\%$ more
forward power while keeping the color point deviation below the required value
of $0.002$. This is not surprising, as TS-EMO is not supposed to be used for
more than eight parameters[36]. As illustrated in the very right plot in
figure 4, the local refinement increased the power in forward direction by an
additional $3.5\%$ relative to the reference design. Thus, yielding
$0.259\text{ [W]}$ or $28.9\%$ more light compared to the reference design,
while maintaining an acceptable color point deviation of $0.0007<0.002$.
Notably, starting a local refinement with the joint parameters provided by the
TS-EMO implementation, either the color point deviation could not be reduced
under $0.002$ or the forward power increase remained below $10.0\%$. Due to
the high computational effort associated with ray tracing simulations, we
restarted each optimization procedure—based on TS-SOO and TS-EMO— only three
times for $48\text{ [hours]}$ each. The reported data corresponds to the best
run regarding the objective function for TS-SOO and TS-EMO, respectively.
Because the relative and absolute trends of the results of these experiments
appeared to be consistent and showed no unexpected anomalies, they were not
evaluated in detail. The Pareto fronts illustrated in figure 4 plot the
(forward) power against the color point deviation for both, joint optimization
using TS-EMO and hierarchical optimization using TS-SOO. As the time horizont
was fixed, the different numbers of samples are explained by the
implementation of the joint ($719$ samples) and hierarchical ($143$ samples)
optimization approach: For each thickness vector $\mathbf{t}$ suggested by TS-
SOO, the solving of the nested color point optimization (2) based on ray
tracing simulations takes about $20\text{ [min]}$. Contrariwise, the
evaluation of a joint parameter vector $\left(\mathbf{t},\mathbf{w}\right)$
requires only one ray tracing simulation and thus takes about $4\text{
[min]}$. The comparison between the left and middle Pareto fronts in figure 4
indicate that TS-SOO circumvents MLTFs that lead to high color point
deviations, as those are punished via multiplicative weights (3). In contrast,
TS-EMO is not implicitly informed about the engineering structure of the
problem via the objective function (4). Namely, an increase in forward power
at the cost of color point deviation is of no value for specific LED
applications. Therefore, most MLTFs suggested by TS-EMO indeed achieve the
same or even higher forward power values compared to TS-SOO, but bring along
an unacceptable color point deviation significantly above $0.005$.

Figure 6: (left) Transmission of a multi-layer thin film that exhibits
$28.9\%$ more light compared to the reference design in forward direction.
(right) The hypothetical, effective spectrum of light, which exits the
conversion system and irradiates the multi-layer thin film. In both cases, the
vertical dashed lines indicate the wavelengths contributing to the LED
spectrum and the horizontal lines denote the separation between forward and
non-forward direction.

###  5.2 Ping pong with light rays

The results of the previous section allow to deduce an explanation of the
mechanism of white light directionality using an MLTF. We can deduce a
hypothetical effective spectrum which irradiates the MLTF. Therefore, we
conduct pixel-wise division of the observed spectrum in figure 5 (b) through
the spectral and directional transmission of the optimized MLTF in figure 6
(a). This hypothetical spectrum is illustrated in figure 6 (b) and indicates
how much relative power is required to impinge on the MLTF from the conversion
system for each angle and wavelength, in order to explain the physically
detected spectrum that irradiates the ambient air after traversing the MLTF.
The transmission describes the probability of a light ray to pass the MLTF,
depending on its wavelength and angle of incidence. It was computed by a
parallelized version of the TMM package[35] provided by Luce et al.[34]. Here,
the light-injecting substrate (conversion system) is represented as silicone
($\text{SiO}_{2}$) of infinite thickness. Air of infinite thickness defines
the ambient environment. Aside from the relative power peak around the chip
wavelength of $440\text{ [nm]}$, another two contributions of relative power
at $555\text{ [nm]}$ and $600\text{ [nm]}$ appear as a single broad peak in
the spectra of figure 5. These green and red wavelength contributions
correspond to the conversion material emissions, respectively. As expected,
the transmission of the MLTF for all of these wavelengths is low for
inconvenient, large beam angles. Thus, non-forward light rays are trapped in
the LED package until physical interactions change their directional
properties such that they are likely to escape or the light rays vanish
optically due to non-radiative thermal effects. We suppose that this
phenomenon causes the directionality enhancement and refer to it as ray ping
pong. Here, the MLTF on top of the conversion system directly influences the
emitted spectrum over angle and wavelength of the LED. The MLTF does not only
function as an angle selective filter that reflects non-forward light rays
back into the LED package, but also balances the statistics of emitted blue,
green and red rays in order to appear as white light of a specific color. In
other words, the MLTF exploits the process of ray ping pong to reach a
statistical equilibrium of emitted rays that is advantageous regarding
directionality and color point of the outcoupled light. Notably, the
characteristic transmission pattern of an MLTF over wavelength and angle of
incidence (see figure 6) not only depends on the layer thicknesses, but also
the constituent materials. Studying table 5 implies that the MLTF decreases
the total efficacy of a white LED corresponding to $\alpha=90^{\circ}$: Rays
are trapped in the LED package and are thus more likely to further interact
with the LED materials instead of escaping into the ambient air. Such
interactions may include photon recycling or scattering, but also non-
radiative effects like thermal losses of rays absorbed by the conversion
materials. As mentioned, for applications like head lamps only power emitted
in forward direction $\left(\alpha\ll 90^{\circ}\right)$ is usable. In this
case, any increase in directionality obviously outweighs a (moderate) drop of
global efficacy.

###  5.3 Limits of directionality increase

Figure 7: The normalized intensity of an LED with and without a multi-layer
thin film (MLTF) is denoted on the left axis. The sinus-weighted integral over
the intensity difference (grey area) — the so-called cumulated intensity
difference — on the right axis indicates that a directionality increase can be
achieved up to $58.4^{\circ}$. However, the peak of directionality increase is
observed between $35^{\circ}$ and $40^{\circ}$.

The previous section allows us to understand the constraints of our approach.
Although no explicit optimizations for angles different from the application-
specific $25^{\circ}$ were conducted for this work, our investigations
regarding the cumulated intensity difference of figure 7 indicate that an
increase in forward power is possible up to almost $60^{\circ}$. Namely, for
angles $\alpha>58.4^{\circ}$, the MLTF would need to re-direct light rays of
unsuitable angles between $\alpha$ and $90^{\circ}$. Due to the Lambertian
radiation characteristics of LEDs, the quantity of such rays may be too low to
outweigh non-radiative effects like thermal losses, which are statistically
enforced by the MLTF for each ray. Thus, the deposition of an MLTF may not be
suitable for applications that leverage light with a broad range of incidence
angles, here larger than $58.4^{\circ}$.

##  6 Conclusions

Increasing the (forward) power of a white LED while still maintaining its
color point is a hierarchical multi-objective optimization problem: The Stokes
shift renders these objectives to be competitive by nature, because a naive
increase in power may turn the emitted light bluish instead of pure white.
Since the color point is a strict requirement for many applications, a higher
forward power of an LED at the cost of color changes is utterly undesired. In
this work, we enter into the competition between color point and power with a
Bayesian optimization approach that optimizes a physics-guided, weighted
objective function. Hereby, weights continuously implement hard constraints
and thus preserve approximability via Gaussian processes. The reported results
indicate that multi-layer thin films on top of white LEDs can increase the
light directionality. The epitaxial deposition of such multi-layer thin films
is only implying low additional expenditure and directly applicable for mass-
production of many optical semiconductors. Our analyses reveal that a
carefully designed multi-layer thin film functions as an angle and wavelength
sensitive filter: The filter statistically balances emitted rays of different
wavelengths to meet the color point. In addition, it traps rays that would
exit the LED at large angles in order to implicitly enforce their forward
(re-)emission. To summarize, the proposed objective function guides the
optimization towards a multi-layer thin film that leverages statistical ray
ping pong to enforce favorable properties of the spectrum radiated by the LED.
Thus, we shine a light on the previously enigmatic effect causing the
counterintuitive increase of white light directionality using a multi-layer
thin film.

## Material, Data, and Code Availability

The optical models and related (commercial) software that support the findings
of this study are available from OSRAM Opto Semiconductors GmbH but
restrictions apply to the availability of these items, which were used under
license for the current study, and so are not publicly available. The
generated and analysed data as well as any code-related information is
included in this manuscript or already published by third parties.

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## Acknowledgements

The authors would like to thank the engineers and scientists, especially
Daniel Grünbaum, at OSRAM Opto Semiconductors GmbH in Regensburg for their
exhaustive assistance and for providing the required optical models and
computational resources.

## Author contributions statement

H.W. developed the physics-guided objective function, wrote and ran the code,
devised the experiments and edited the manuscript. C.W. and L.K. interpreted
the acquired data in terms of the ray ping pong and contributed application-
specific knowledge about white light. R.B. analysed the results and advised
how to avoid the Stokes shift. A.L. provided code to evaluate and discuss
multi-layer thin films. S.S. provided the calibrated optical model of a white
LED to conduct ray tracing simulations. M.L.S and E.W.L made substantial
conceptional and editorial contributions to this manuscript. All authors
reviewed the manuscript and played a round of ray ping pong.