Markov Chains for Modeling Complex Luminescence, Absorption,

and Scattering in Nanophotonic Systems

A. Ryan Kutayiah1, Smriti Kumar1, Rivi Ratnaweera1, Kenny Easwaran2, and MatthewSheldon1,3

1Department of Chemistry, Texas A&M University, College Station, TX, USA2Department of Philosophy, Texas A&M University, College Station, TX, USA

3Department of Material Science and Engineering, Texas A&M University, College Station, TX, USA

Abstract

We develop a method based on Markov chains to model fluorescence, absorption, and scattering innanophotonic systems. We show that the method reproduces Beer-Lambert’s Law and Kirchhoff’s Law,but also can be used to analyze deviations from these laws when some of their assumptions are violated.We show how to use the method to analyze a luminescent solar concentrator (LSC) based on semiconductornanocrystals.

Contents

1 Introduction 3

2 Methods 4

2.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Emission Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Absorption Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.3 Transmission Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.4 Self-absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.5 Transitions from Environment to Medium . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.6 Transitions to Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.7 Variable Quantum Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Results and Discussion 11

3.1 Deviations from Beer-Lambert’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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3.1.1 Gallium Arsenide Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.2 Colloidal Gold Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Kirchhoff’s Law and Violations of Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Light Trapping in Luminescent Solar Concentrators . . . . . . . . . . . . . . . . . . . . . . . 16

4 Conclusion 17

2

1 Introduction

Recent progress in the study of nanophotonics, en-compassing research topics such as metamaterials,plasmonics, and quantum confined matter has estab-lished that the optical interaction between light andmatter is broadly tailorable across the electromag-netic spectrum via precise control of sub-wavelengthscale optical structures [1, 2, 3, 4, 5]. Developmentsin radiation theory, aided by sophisticated computa-tional methods, have informed experimental studiesin nanophotonics, for example, as encapsulated in theinverse design approach [6] (and references therein).Complex features of the optical response of nanoscalestructures including scattering, absorption, emission,and nonlinear behavior are increasingly well under-stood [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

However, a significant outstanding challenge is thebehavior of macroscopic systems composed of largeassemblies of sub-wavelength nanophotonic elements.In such systems complex interactions between in-dividual elements on the nano- and mesoscale giverise to the macroscopic optical response. The chal-lenge is compounded when these systems are opticallypumped and driven far from equilibrium, so that sim-plifying assumptions commonly employed, such asKirchhoff’s law for equating absorptivity and emis-sivity, are not a priori guaranteed to be applicable[18, 19]. Further, standard analytical or computa-tional strategies, such as finite element methods, maybe difficult or intractable to extend beyond nanoscalevolumes or without simplifying boundary conditions[20].

This work is focused on demonstrating how a com-putational strategy based on statistical analysis usingMarkov chains provides a powerful tool for modelingcomplex interactions between assemblies of nanopho-tonic elements that give rise to a macroscopic op-tical response. These interactions are mathemati-cally described as pathways of radiative and non-radiative energy exchange within the optical systemand between itself and the environment, includinganisotropic absorption, photoluminescence, scatter-ing, and loss. We show how this approach gives deep

insight into the steady state behavior of the distri-bution of light and energy throughout the system, aswell as other important optical properties, such asthe angle-dependent emission or scattering from theassembly as a whole, even when the system is drivenout of equilibrium.

Markov chains have been used to model light-matter interactions before, particularly in the con-text of radiative transfer, for example, see [21, 22].Mathematically, Markov chains also share some sim-ilarities with the more commonly used computationalapproach of Monte Carlo ray tracing. While MonteCarlo ray tracing is popular, it is computationallytaxing and it has been shown that the use of Markovchains applied to the same problems is computation-ally more efficient [23, 24]. Monte Carlo methods ap-proximate a distribution by sampling many instancesfrom it, while the Markov Chain method allows us tocalculate the distribution directly.

Moreover, we show how modeling nanophotonic as-semblies using a Markov chain is also conceptuallyappealing. The strategy is based on identifying theprobabilities that connect the pathways of energy ex-change between the nanoscale elements of the system,for example the angle-dependent emission or absorp-tion profile of individual emitters. These descriptionsof the optical behavior of individual elements in theensemble can be obtained using other methods, forexample Mie theory [25] or extended Mie theory [26],finite element modeling [27] or finite-difference time-domain [28]. Once the probabilities are identified thesteady state behavior of the system is obtained sim-ply by solving one eigenvalue problem.

This work is organized as follows: Section 2 de-scribes the basics of a Markov chain for the kinds ofsystems we will consider. Section 3 fills in the de-tails of these Markov chains to analyze the opticalbehavior of three specific systems. The first two sys-tems provide an analysis of Beer-Lambert’s law andKirchhoff’s law, respectively. The third models thebehavior of a luminescent solar concentrator (LSC)comprised of semiconductor nanocrystals. Section 4concludes the paper.

3

2 Methods

The primary method used to model optical inter-actions in this work is a probabilistic model knownas a Markov chain. First, we give a mathematical in-troduction to discrete-time Markov chains, Sec. 2.1.Then in Sec. 2.2, we set up the specific Markov chainfor the optical systems we analyze.

2.1 Markov Chains

Using terms defined below, a discrete time, time-homogeneous Markov chain is an appropriate mod-eling device for a “memoryless” system in which thetransitions at any moment depend only on the cur-rent state of the system, and not how the systemhas been at various points in the past, or how longit has been operating. Under certain simple condi-tions, the probability distribution of the system con-verges to a unique stationary distribution, that canbe used to find the steady state of the system it rep-resents. Our introduction to Markov chains looselyfollows [29, 30, 31].

A Markov chain is formally defined as a stochasticprocess that obeys the Markov property. A stochasticprocess is a collection of random variables X = Xt :t ∈ T indexed by time. Here, we consider a discreteset of times T . This “time” variable merely countsthe number of elapsed transitions between states. Wedo not assume any correspondence of the number oftime steps to a precise measurement of physical timein an experiment. The point of the model will notbe to determine the precise time at which any eventoccurs, but rather the steady state distribution ofthe photons in the system. The development hereassumes that the lifetime of each state is roughlyequal, and that transition times are negligible, butthe model can be modified to accommodate knowndivergences from these assumptions.

The random variables Xt all take values in a fi-nite set S of possible states. Since the set of states isfinite, the probability distribution for a random vari-able Xt is determined by the probabilities of the indi-vidual states, P (Xt = i). For convenience, we denote

P (Xt = i) by d(t)i , and the vector of probabilities for

all states as ~d(t) = 〈d(t)i : i ∈ S〉.

If the variables are all independent of each other,then the joint distribution P ((X0 = i0)&(X1 =i1)&(X2 = i2)& . . . ) is just the product of the dis-

tributions of the individual variables d(0)i0d

(1)i1d

(2)i2. . . .

But a general stochastic process may have arbitrarydependencies among the variables, so that comput-ing this joint distribution would require informationabout the conditional probabilities of each variableon every combination of the earlier ones. However,if the variables obey the Markov property, then theseconditional probabilities are determined by the valueof the previous variable, and are independent of thevalues of all earlier variables. That is,

P (Xt = j|Xt−1 = i,Xt−2 = it−2, . . . , X0 = i0)

= P (Xt = j|Xt−1 = i). (1)

Furthermore, if the process is time-homogeneous,then this conditional probability is independent oft, and we can define

pji = P (Xt = j|Xt−1 = i). (2)

This pji, the transition probability, is the probabilityof the system transitioning from state i to state j.

By the Law of Total Probability,

P (Xt = j) =∑i∈S

P ((Xt = j)&(Xt−1 = i)) (3)

=∑i∈S

P (Xt = j|Xt−1 = i)P (Xt−1 = i).

(4)

ord

(t)j =

∑i∈S

pjid(t−1)i . (5)

If we think of ~d(t) as a column vector and let P bethe matrix with pji in the jth row and ith column,called the transition matrix, then this means

~d(t) = P ~d(t−1). (6)

Since the columns of P sum to 1 (as they representthe probabilities of all values of Xt conditional on a

4

particular value of Xt−1), it is said to be a stochasticmatrix.

A Markov chain is said to be reducible if there arestates i and j such that, if one of the variables evertakes value i, then no later variable ever takes thevalue j, and irreducible otherwise. It is said to beperiodic if there is some integer n > 1 such that Xt =Xt′ only if |t − t′| is a multiple of n, and aperiodicotherwise. All our Markov chains are irreducible andaperiodic.

Each vector ~d(t) represents the probability distribu-tion of the system at a time. If there is a distribution~d(s) with

P ~d(s) = ~d(s), (7)

then it is said to be a stationary distribution of thesystem. For an irreducible, aperiodic Markov chain,there is always a unique stationary distribution [29,30, 31], and furthermore, for any initial distribution~d(0), we have

limt→∞

P t~d(0) = ~d(s). (8)

Thus, for such a Markov chain, the stationary dis-tribution represents the steady state behavior of thesystem, and we can find the stationary distributionby solving Eq. (7). Note that this is an eigenvalueequation with an eigenvalue of 1.

2.2 Model

We first introduce a system with some simplifyingassumptions in order to establish a concrete connec-tion between the Markov chain formalism (Sec. 2.1)and fluorescence, scattering, and absorption. In thiswork, a system comprises a medium and an envi-ronment, which interact with each other via photonexchange. The environment sends photons into themedium at a constant rate and angular profile.

The medium could be a solid such as a semi-conducting slab (Sec. 3.1), a colloidal suspension ofnanoparticles (Sec. 3.1), or nanoparticles embeddedin a solid polymer matrix (Sec. 3.2 and 3.3). In thissection, the medium is a solid matrix of uniform re-fractive index nm that hosts uniformly distributed

fluorophores. The environment around the mediumhas refractive index ne.

The emission of the fluorophores is assumed to beazimuthally symmetric. Thus, we represent angle ofemission just by the zenith angle θ, and assume emis-sion into any angle is proportional to some functionf(θ).

We consider two kinds of fluorophores and labelthem as isotropic or dipolar. By an isotropic flu-orophore, we mean that the fluorophore emits ra-diation with equal probability in all directions; wealso assume that there is no anisotropy in its absorp-tion cross-section σ. Thus, for isotropic fluorophores,f(θ) = 1. In contrast, the dipolar fluorophores dis-play anisotropy in the emission and absorption-crosssection, specifically f(θ) = sin2 θ, see Ref. [5]; andσ(θ) ∝ sin2 θ, see Ref. [32]. Further, all of the dipolarfluorophores emit radiation with the dipole orientedalong the same axis, normal to the top surface of themedium. We will later draw a parallel between thisbehavior and ensembles of homeotropically alignednanorod-shaped semiconductor nanocrystals in LSC-like devices.

We also assume that the light emitted by eithertype of fluorophores is monochromatic. At first weassume light coming from the environment is thesame wavelength as the fluorophore emission, but inSec. 3.3 this assumption is relaxed. The host materialof the medium is non-absorbing, and the real part ofthe fluorophores’ refractive index is matched to thatof the host medium. We assume the fluorophores arefar enough apart so that near-field effects and reso-nant energy transfer can be neglected in the prob-abilistic description for energy transfer. Finally, weassume that the medium is of infinite extent in thex- and y- directions and has a finite thickness L inthe z-direction.

Now we are in a position to describe the state spaceS, random variables Xt with distributions ~d(t), andtransition matrix P, as they relate to this system.The Markov chain tracks the movement of an indi-vidual photon through the system. For a system withmultiple photons, the stationary distribution of eachphoton will be the same, and the observed distribu-

5

tion of photons in the steady state will be propor-tional to this distribution.

The state space S, is constructed by stratifying themedium into M equal thickness parallel slabs. Wecall these slabs layers. For a medium of thicknessL, a layer has thickness ∆z = L/M . Each layer istreated as a state representing a possible location forthe photon at a time. In all our simulations reportedhere, we use M = 1000. We found that increasingM to 10, 000 only made a difference to the results onthe order of 10−8.

Xt represents the “position” of a photon at a timet, where positions are layers. Note that we only trackthe position of a photon along the z-axis since Xt

takes values from S which is created by discretizingthe system along z. It does not account for the lo-cation of the photon in the x and y directions. Ourstandard assumption is that the systems are largeenough and homogeneous enough in these horizontaldimensions that we can treat them as infinite, thoughsee further discussion in Sec. 3.3.

The probability d(t)i = P (Xt = i), represents the

probability that a photon had its most recent absorp-tion event at layer i at time t, that is, the probabilitythat a photon is “located at” this layer. In describinga system with multiple photons, we will sometimes re-

fer to d(t)i as the population (or number of photons)

of layer i.

A transition probability pji of the system is iden-tified as the probability that a photon from layer iat time t − 1 moves to layer j at time t. To movefrom layer i to layer j, a photon must be emittedfrom layer i, transmitted through all intervening lay-ers, and absorbed at layer j. In this simplified modelthe probabilities of these three events depend onlyon the zenith angle at which the photon is emitted;thus, we calculate each of these three probabilities ateach angle, multiply them, and integrate over all pos-sible angles of travel between these two layers. (Asdiscussed in Sec. 2.2.3, these are the angles from 0 toπ/2.) Denoting the emission probability density asdp(e)(θ), the transmission probability as p(t)(θ, i, j)and the absorption probability as p(a)(θ), the transi-

tion probability pji can be written as

pji =

∫ π/2

0

dp(e)(θ)p(t)(θ, i, j)p(a)(θ). (9)

We describe dp(e)(θ) in greater detail in Sec. 2.2.1,p(a)(θ) in Sec. 2.2.2, and p(t)(θ, i, j) in Sec. 2.2.3. InSec. 2.2.4 we discuss the special features of the casewhere i = j. The transition probabilities and statesare pictorially represented by a transition diagram inFig. 1b. Importantly, since the emission probabili-ties are assumed to be proportional to f(θ), and thetransmission probability depends only on the pathtraversed from one layer to the next, these transi-tion probabilities do not depend on any aspect of thesystem other than what layer the photon was mostrecently absorbed by, so the system obeys the Markovcondition.

In the physical system we are modeling, photonsare sent into the system at a constant rate, moveaccording to the transition probabilities from layerto layer within the medium, and eventually escape.While it does have a unique steady state, such a sys-tem is not strictly a Markov process, because of theentry and escape of photons from the set of states.However, for any such system, there is a related fic-titious system with the same transition probabilitiesamong the states in the medium, but with an extradummy state. Photons entering the medium tran-sition from the dummy state to the medium, whilephotons escaping the medium transition from themedium to this dummy state. That is, in this fic-titious system the photons entering the medium arethe photons that just escaped. This fictitious sys-tem is a Markov process because the photon numberis conserved, and therefore has a unique, identifiablesteady state distribution.

This fictitious system does not behave like thephysical system outside the steady state; in the ficti-tious system the number of photons entering the sys-tem at a time t is the number that escaped at timet−1, while in the physical system it is constant. How-ever, the steady state distributions must be the same.Both systems have the same transition probabilitiesamong the states within the medium. Furthermore,

6

in the steady state of the physical system (assumingfor now no non-radiative loss, but see Sec. 2.2.7), thenumber of escaping photons will equal the numberof entering photons, just like in the fictitious system.Thus, our analysis focuses on finding the steady stateof this fictitious Markov process, which is also thesteady state of the physical system.

We refer to the dummy state as “the environment”.The entire set of states is S = 0, 1, . . . ,M, wherelayer 0 is the environment, while the remaining layers,1, . . . ,M , belong to the medium, see Fig. 1. Tran-sition probabilities from and to the “environment”layer are discussed in Secs. 2.2.5 and 2.2.6 respec-tively.

The steady state distributions ~d(s) are eigenvectors,and thus can be scaled by any constant. We will al-ways display steady state distributions scaled so thatthe population of the “environment” layer is 1. Thiscan be interpreted physically as treating all modeledsystems as having the same intensity of light incidenton them, and comparing the relative populations ofthe layers of the different systems under this condi-tion. The populations of the layers are sensitive tohow thinly the medium is sliced into these discretelayers, but the trend will be the same. Thus, thenormalized steady state distribution is given by

~d(s) →

⟨1,d

(s)1

d(s)0

, . . . ,d

(s)M

d(s)0

⟩. (10)

2.2.1 Emission Probability Density

We develop the emission probability density byfirst considering emission from a layer inside themedium. We will discuss emission from the environ-ment to the medium on a case-by-case basis as dif-ferent scenarios are considered in Secs. 3.1, 3.2, and3.3.

For a layer of the medium, the emission probabil-ity density dp(e)(θ) is defined as the probability thata photon is emitted at an angle whose zenith is in-finitesimally close to θ. If f(θ) is proportional to theprobability of emission in solid angle dΩ = sin θdθdφ,

then integrating over azimuth angle φ yields

dp(e)(θ) = 2πνf(θ) sin θdθ, (11)

where ν is a normalization constant to ensure thatthe total probability is 1. That is,

ν =1∫ π

02πf(θ) sin θdθ

. (12)

For isotropic fluorophores, f(θ) = 1 and ν = 1/4π.For dipolar fluorophores, f(θ) = sin2 θ and ν = 3/8π.

2.2.2 Absorption Probability

For a photon traveling at angle θ that starts outsidea layer and then reaches it, the probability of beingabsorbed by that layer can be determined from thewell-known Beer-Lambert law [33] (see Ref. [34] fora derivation of Beer-Lambert’s law based on stochas-tics). Recall that ∆z is the thickness of the layer,so the path length through the layer along this an-gle is ∆z/| cos θ|. Let α(θ) = ρσ(θ), where ρ is thefluorophore concentration and σ(θ) is the absorptioncross-section (which has angle dependence in the caseof anisotropic fluorophores). Then, the probabilityof transmitting through the layer is e−α(θ)∆z/| cos θ|

while the probability of being absorbed by the layeris given by

p(a)(θ) = 1− e−α(θ)∆z/| cos θ|. (13)

2.2.3 Transmission Probability

For a photon that has been emitted by one layer iat an angle θ, to find the probability of it being trans-mitted to another layer j, we calculate the probabilityof it not being absorbed along the path from layer i tolayer j. The length of the full path the photon travelsbefore reaching layer j depends on where precisely inlayer i the photon was emitted. We approximate byassuming that all emissions occur from the center ofthe layer, and divide the medium into a large numberof layers to ensure that this is a good approximation.

The path from layer i to layer j can be direct, orcan involve reflections. In all our models, the top

7

surface of layer 1 is an interface between the mediumand the environment, and reflection occurs if θ is out-side the escape cone; the escape cone is formed by thecritical angle θc which is determined from Snell’s law.In Sec. 3.1, the bottom surface of layer M is also aninterface between the medium and the environmentthat reflects at these angles, while in Sec. 3.2 and 3.3the bottom surface is an ideal mirror that reflects atany angle.

These reflections change the angle of travel fromθ to π − θ and vice versa. But in all our systems,dp(e)(θ) = dp(e)(π − θ) and p(a)(θ) = p(a)(π − θ).Thus, in these calculations, we will always treat θ asa number between 0 and π/2, understanding that itmight represent the angle from the vertical in eitherdirection, so that reflections do not change it.

Because direct paths and reflected paths are twoseparate ways for a photon emitted at angle θ fromlayer i to reach layer j, the transmission probabilityis the sum of the direct and reflection transmissionprobabilities

p(t)(θ, i, j) = p(t)D (θ, i, j) + p

(t)R (θ, i, j). (14)

Figure 2a depicts the paths associated with the directand reflection (one reflection) transmission probabili-ties. Figure 2b shows a probability tree for the trans-mission probability. The total transmission proba-bility Eq. (14) is what goes into the computing thetransition probability from layer i to j using Eq. (9).However, as we observe at the end of this section,much of the reflection term is negligible in cases wherethe medium is thick and highly absorbing.

Under the assumption that emission occurs at thecenter of layer i, the length of the direct path fromlayer i to layer j is (j − i− 1/2)∆z/ cos θ if i < j or(i− j − 1/2)∆z/ cos θ if i > j, see Fig. 1a. Thus thedirect transmission probability from layer i to layerj along an angle θ is

p(t)D (θ, i, j) = e−α(θ)(|i−j|−1/2)∆z/ cos θ. (15)

(Since θ is now assumed to be between 0 and π/2, weno longer need to take the absolute value of cos θ).

Calculating p(t)R (θ, i, j) involves considering several

cases, depending on whether a mirror is present on

(a) (b)

Figure 1: (a) Illustration showing the stratificationof the system into layers along with how distancesbetween the layers are computed. The purple arrowindicates the transition probability density from layeri to layer j at an angle θ: dp(e)(θ)p(t)(θ, i, j)p(a)(θ).The transition probability pji is obtained by integrat-ing the transition probability density over θ from 0to π/2. (b) A transition diagram for the system. Therectangles represent the states and the arrows repre-sent the transition probabilities.

the bottom layer, and whether θ is within the es-cape cone for medium-environment interfaces. Forangles within the escape cone θc, there is only onepossible reflected path, which occurs only if there isa mirror on the bottom layer, and the photon ini-tially travels downward from layer i and is reflectedback up to layer j. But for angles outside the escapecone, the light can initially travel either upwards ordownwards, and can reflect any number of times, re-gardless of whether the top and bottom are mirrorsor interfaces with the environment.

Thus, for θ < θc, the probability of transmission by

reflection, p(t)R is either 0 (if there is no mirror on the

bottom) or (in the presence of a mirror) can be calcu-lated by means of the path length of the downwardsreflection. This path passes throughM−i+1/2 layersas it moves downwards (from layer i to the mirror),and then M − j layers after reflection (from the mir-ror to the bottom of layer j). In this case, where the

8

(a) (b)

Figure 2: (a) Illustration showing transitions fromlayer i to j by a direct path (solid arrow) and byreflections (dashed arrows). (b) A probability treerepresenting the different paths a photon, first emit-ted into the lower hemisphere, can take depending onthe angle at which it is emitted in that hemisphere.

only reflection is a single reflection of a downwardspath,

p↓R(θ, i, j) = e−α(θ)∆z(2M−(i+j)+1/2)/ cos θ, (16)

But if θ ≥ θc, a photon can travel from i to j bymeans of any number of reflections of either an up-wards or downwards path. It is convenient to write

p(t)R as the sum of several terms: p↓OR , representing

transmission probability along any path that startsby moving downwards and has an odd number of re-flections; p↓ER , representing transmission probabilityalong any path that starts by moving downwards andhas an even number of reflections; p↑OR , representingtransmission probability along any path that startsby moving upwards and has an odd number of reflec-tions; and p↑ER , representing transmission probabilityalong any path that starts by moving upwards andhas an even number of reflections.

For each of these terms, there is a shortest path,with either one or two reflections. The longer paths ineach term differ in length by 2M∆z/ cos θ, since theycross the entire thickness of the medium twice to gettwo more reflections. Thus, the transmission proba-

bility along each path is e−2α(θ)M∆z/ cos θ times theprobability along the path with two fewer reflections.The sum of these transmission probabilities is thusa geometric series, whose value can be calculated bymultiplying the first term by 1/(1−e−2α(θ)M∆z/ cos θ).To find the first term in the series, we calculate theshortest path length starting with an emission in therelevant direction and either one or two reflections.The shortest path for a photon that starts downwardand has an odd number of reflections was given inequation (16), and the others are calculated similarly:

p↓OR (θ, i, j) =e−α(θ)∆z(2M−(i+j)+1/2)/ cos θ

1− e−2αM∆z/ cos θ, (17)

p↓ER (θ, i, j) =e−α(θ)∆z(2M+j−i−1/2)/ cos θ

1− e−2αM∆z/ cos θ, (18)

p↑OR (θ, i, j) =e−α(θ)∆z(i+j−3/2)/ cos θ

1− e−2α(θ)M∆z/ cos θ, (19)

p↑ER (θ, i, j) =e−α(θ)∆z(2M+i−j−1/2)/ cos θ

1− e−2α(θ)M∆z/ cos θ. (20)

We can input these expressions into Eq. (14) andsee the following. When there is no mirror at thebottom and θ < θc, then p(t)(θ, i, j) is just given bythe term in Eq. (15). When there is a mirror at thebottom and θ < θc, then p(t)(θ, i, j) is the sum of theterms in equations (15) and (16). When θ ≥ θc, andthere is either a mirror or an environmental interfaceat the bottom, then p(t)(θ, i, j) is the sum of the termsin Eqs. (15), (17), (18), (19), and (20).

Note that many of these terms depend one−α(θ)M∆z/ cos θ. This is the transmission probabil-ity of a photon going through the entire thickness ofthe medium at angle θ. When the medium is thickand highly absorbing at angle θ, this probability isvery close to 0. In this case it is a good approxi-mation to treat the denominator of the fraction as1 (that is, to take just the first term in these infi-nite series of reflections, rather than the sum of allterms). It is also a good approximation to ignorethe even terms entirely (since even two reflections in-volves passing completely through the thickness ofthe medium). Additionally, the odd reflection termscan largely be ignored except for the ↑ O term wheni and j are both small, or the ↓ O term when i and j

9

are both close to M .

2.2.4 Self-absorption

In Secs. 2.2.2 and 2.2.3, we made the assumptionthat i 6= j. This was essential because it allowedus to multiply a single absorption probability by thesum of the transmission probabilities along all paths.But when i = j, the probability of absorption onthe direct path is different from the probability ofabsorption on reflected paths. On the direct path, aphoton emitted from the center of the layer only hashalf of the thickness of the layer in which it can beabsorbed, instead of the full thickness of the layer,though it can be emitted in the upward or downwarddirection on this angle. But on the reflected paths,the photon can be absorbed anywhere along the fullthickness of the layer.

To find the probability of self-absorption on the di-

rect path, p(a)S (θ), we use the Beer-Lambert law for a

path length of (1/2)∆z/ cos θ. We double this prob-ability, to account for both upward and downwardpaths at angle θ:

p(a)S (θ) = 2(1− e−α(θ)∆z/2 cos θ). (21)

Thus, the transition matrix element pii is given by

pii =

∫ π/2

0

dp(e)(θ)(p(a)S (θ) + p

(t)R (θ, i, i)p(a)(θ)),

(22)

where p(a)(θ) and p(t)R (θ, i, i) are as described in

Secs. 2.2.2 and 2.2.3, respectively. (Recall that

p(t)R (θ, i, i) is 0 when θ < θc and there is no mirror

on the bottom; it is given by the term in Eq. (16)when θ < θc and there is a mirror on the bottom;and it is given by the sum of the terms in Eqs. (17),(18), (19), and (20) when θ ≥ θc.)

2.2.5 Transitions from Environment toMedium

In all systems we consider, photons enter the sys-tem from a source above layer 1, with some char-acteristic distribution for that system. Thus, we

model transitions from the “environment” layer tothe medium by

pj0 =

∫ θc

0

dp(e)env(θ)p

(t)(θ, 0, j)p(a)(θ). (23)

Although the integration goes up to the critical angle,the emission density may be zero for some range ofangles less than the critical angle. Specifically, if lightfrom the environment impinges on the medium at anangle θe then dp(e) is zero in the range θm < θ ≤ θc.Here θm = sin−1([ne/nm] sin(θe)). The amount of

light, or the emission density dp(e)env, from the environ-

ment will differ in the different systems we consider,and will thus be described in each of the relevant sec-tions. p(a)(θ) is the same as in Sec. 2.2.2.

However, p(t)(θ, 0, j) has some more significant dif-ferences from p(t)(θ, i, j). Light from the environmentis incident precisely at the border of the medium,rather than in the middle of a layer, so the distancetraversed is an integer number of layers, rather thana half integer. Furthermore, light from the environ-ment only moves in the downward direction.

Transmission probability from the environment tolayer j along the direct path is given by

p(t)D (θ, 0, j) = e−α(θ)j∆z/ cos θ. (24)

For θ within the escape cone, with no mirror onthe bottom, the direct term is the only term, so

p(t)(θ, 0, j) = p(t)D (θ, 0, j). For θ within the es-

cape cone, in cases with a mirror on the bottom,

p(t)(θ, 0, j) = p(t)D (θ, 0, j) + p↓R(θ, 0, j), where

p↓R(θ, 0, j) = e−α(θ)∆z(2M−j)/ cos θ, (25)

Light from outside the system only enters at an-gles within the escape cone, so we do not considertransmission probabilities at angles outside the es-cape cone.

2.2.6 Transitions to Environment

Photons that escape the system are said to transi-tion to the “environment” layer. While other transi-tions can happen at any angle from 0 to π/2, these

10

transitions can only happen at angles within the es-cape cone. Thus,

p0i =

∫ θc

0

dp(e)(θ)p(t)(θ, i, 0). (26)

dp(e) is just as in Sec. 2.2.1. We no longer need toaccount for absorption, because photons that reachthe interface with the environment within the escapecone automatically escape.

When there is no mirror on the bottom layer,there are two direct paths from layer i to the en-vironment — an upward and a downward path.Thus, p(t)(θ, i, 0) = p↑(θ, i, 0) + p↓(θ, i, 0), wherep↑(θ, i, 0) = e−α(θ)(i+1/2)∆z/ cos θ and p↓(θ, i, 0) =e−α(θ)(M−i+1/2)∆z/ cos θ.

When there is a mirror on the bottom layer, thereis a direct upwards path from layer i to the en-vironment, and a downward path with one reflec-tion. Thus, p(t)(θ, i, 0) = p↑(θ, i, 0) + p↓R(θ, i, 0),

where p↑(θ, i, 0) is as above, and p↓R(θ, i, 0) =e−α(θ)(2M−i+1/2)∆z/ cos θ.

Finally, we must consider transitions directly fromthe environment to itself.

p00 =

∫ θc

0

dp(e)env(θ)p

(t)(θ, 0, 0). (27)

As above, we only consider angles within the escapecone, and there is no need for an absorption term.

As in Sec. 2.2.5, dp(e)env(θ) depends on the system. As

mentioned in Sec. 2.2.5 dp(e) may be zero for someangles.

p(t)(θ, 0, 0) consists only of a single term, and is ei-ther e−α(θ)2M∆z/ cos θ or e−α(θ)M∆z/ cos θ, dependingon whether there is or is not a mirror at the bottomof the medium.

2.2.7 Variable Quantum Yield

So far, we have assumed that every photon that isabsorbed by a layer is re-emitted by that layer. Thatis, we have assumed there is no non-radiative loss,and in particular that the quantum yield of the flu-orophores ηQ = 1. However, when there are such

losses, these can be modeled as just another sortof transition from each layer of the medium to thedummy “environment” layer. We will consider non-radiative losses due to non-unity quantum yield ofthe fluorophores.

To transform the transition matrix P assumingunity quantum yield to the matrix P ′ accounting fornon-unity quantum yield, we need to slightly mod-ify the transitions. All transition probabilities due toemission of photons from the medium are scaled downby the quantum yield, while the remaining probabil-ity is added to the probability of transition to the“environment”. Transitions from the environmentare unchanged.

p′ji = ηQpji i, j 6= 0 (28)

p′0i = ηQp0i + (1− ηQ) i 6= 0 (29)

p′j0 = pj0 (30)

The transformation described by Eqs. (28), (29)and (30) preserves the stochastic nature of the tran-sition matrix and the total number of photons in thefictitious system will be conserved. Its steady statewill still correspond to the steady state of the phys-ical system, in which the sum of all losses (radiativeand non-radiative) is equal to the number of photonsentering the system.

3 Results and Discussion

3.1 Deviations from Beer-Lambert’sLaw

The well-known Beer-Lambert Law allows one toanalyze how a non-fluorescent and non-scatteringbody absorbs and transmits radiation. A standardexperimental geometry usually consists of a colli-mated light source incident on a sample with knownthickness, so the path length of the light as well asthe incident and transmitted intensity can be quan-tified. From this, one can determine the absorptioncoefficient of a sample, or if the absorption coefficientis known, one can predict the transmitted intensity.

11

However, when the sample is strongly fluorescentand/or scatters light at the same wavelength as itabsorbs or emits, it is not straightforward to analyzethe optical response. Photons can be absorbed, re-emitted, and/or scattered multiple times before exit-ing the material, and the overall behavior is not welldescribed by the Beer-Lambert Law. By explicitlyaccounting for these more complex interactions, ourmodel can predict the transmitted intensity; angle-dependent emission and scattering profile into theenvironment; and the light intensity, i.e. photon pop-ulation, as a function of position in the medium. Thelatter information can be used to calculate local vari-ations in the photo-induced chemical potential.

For analyzing these deviations from Beer-Lambert’s law we consider two systems: one isa slab of the photoluminescent semiconductingmaterial gallium arsenide (GaAs) and the other is acolloidal dispersion of gold nanoparticles with strongoptical scattering. In both systems, the mediumis sandwiched between two environment layers (nomirror at the bottom), see Fig. 3a and Fig. 4a.Monochromatic light from the environment impingesupon the media at normal incidence i.e. θe = 0.

Because light from the environment is received onlyat a single angle, the probability density is given by

the Dirac delta function dp(e)env(θ) = δ(θ). Thus, the

integrals in Secs. 2.2.5 and 2.2.6 can be simplified.Note that light from the environment only travels atangle θ = 0 and there is no mirror on the bottom,so the transmission probabilities only have a singleterm, yielding

pj0 = e−α(0)j∆zp(a)(0), (31)

andp00 = e−α(0)M∆z. (32)

3.1.1 Gallium Arsenide Slab

We model GaAs as an isotropic emitter. We takethe absorption coefficient and the refractive index ofGaAs at a wavelength corresponding to its bandgapenergy at room temperature, i.e. α = 14247 cm−1,and nm = 3.63. The GaAs slab is 10−3 cm (or

10 microns) thick, as such the entire slab absorbs99.99994% of light impinging upon it at normal inci-dence, assuming the light is monochromatic. Fig. 3bplots the steady state population as a function ofdepth for different cases that analyze how its behav-ior depends on changes in quantum yield ηQ with theincident light intensity held constant.

The case of ηQ = 0 corresponds to when Beer-Lambert’s law should be strictly valid (no emission,only absorption). This case is given by the lasttrace in Fig. 3b. The constant negative slope cor-responds to exponential decay, consistent with usualBeer-Lambert behavior.

(a) (b)

Figure 3: (a) The illustration depicts incoming lightin orange impinging at normal incidence. The lightemitted by the medium is in dark blue. The environ-ment is represented by light orange and the mediumby light blue. (b) This log-scaled plot shows thesteady state photon population as a function of depthfor various fluorescent quantum yields in a 10 µmthick slab of GaAs illuminated at its band gap en-ergy at normal incidence.

Deviations from the Beer-Lambert law occur whenemission is “turned on”, i.e. ηQ 6= 0. As ηQ in-creases, the steady state population of each layer in-creases monotonically. At any ηQ, population de-creases with depth — it is always higher towards thetop, where light is incident, and lower towards thebottom, where light is only emitted to the environ-ment. The decrease of population with depth is lesspronounced for higher ηQ, but even with ηQ = 1,the steady state distribution is not uniform. Most

12

significantly, the Markov model enables us to calcu-late the deviation from strict exponential decay forany ηQ > 0. Such deviations can be clearly seen inFig. 3. Upon close inspection, one observes a subtletilde-shaped features to plots in Fig. 3 especially for0.5 ≤ ηQ ≤ 0.99. These features are due to totalinternal reflections. The internally reflected light, re-flecting off the top and bottom interfaces has a higherprobability of being absorbed in the layers nearest tothe interface where the photons were reflected. Thiscauses an increase in the population near the top andbottom of the medium leading the to curvy featuresobserved in Fig. 3.

3.1.2 Colloidal Gold Nanoparticles

We consider a colloid composed of goldnanospheres in water. For this model, the “ab-sorption” and “fluorescence” terms developed aboveare interpreted to represent incoming and outgoinglight, respectively, that is scattered by the nanopar-ticles. As a simple approximation we assume thatthe gold spheres scatter light isotropically. A morerealistic model would involve Rayleigh-Gans orLorenz-Mie phase functions; the scattering phasefunction specifies the angular distribution of thescattered light [35]. Since the angular distributionof scattered light depends on the incident angle, aMarkov model for these systems would need to tracknot just the location of the most recent interaction,but also the incident angle, in its state space. Herewe use isotropic scattering as a first rough approx-imation that can be calculated with the simplerstate space. Models where the phase function isemployed for light scattering using Markov chainsare considered in Refs.[22, 23, 36].

For comparison with common experimental con-ditions, the number density of the gold particlesis 7.15 × 1015 m−3, the extinction cross section is2.93 × 10−15 m2. The thickness of the medium is1 cm and the refractive index is taken to be thatof water 1.33. For our analysis we consider behav-ior that results when the extinction cross-section σe,which is the sum of the absorption cross-section σand the scattering cross-section σs, remains fixed as

the absorption and scattering cross-section are var-ied. We keep track of only the variation of the scat-tering cross-section in terms of the extinction cross-section ω = σs/σe, where ω is called the scatteringalbedo. The scattering albedo is varied from zero toone. For a scattering albedo of zero the particles onlyabsorb light (no scattering), whereas for a scatteringalbedo of one, the particle only scatters light (no ab-sorption).

We assume that the particles and incident light sat-isfy the conditions required for elastic and isotropicscattering.

(a)

(b)

Figure 4: (a) Illustration of a cuvette filled with col-loidal gold nanoparticles. The orange cylinder rep-resents collimated light impinging on the cuvette atnormal incidence. (b) Plot of the light intensity as afunction of depth for various scattering albedos, ω.

In the case of zero scattering albedo, one sees fromFig. 4 that there is an exponential decay of pho-tons (the vertical axis is logarithmic), as expected.As the scattering albedo increases, the photons havea chance of being scattered. It is apparent fromFig. 4 that the attenuation increases as the scatteringalbedo decreases.

3.2 Kirchhoff’s Law and Violations ofDetailed Balance

In this section we reproduce Kirchhoff’s law of radi-ation, and analyze deviations from it. Kirchhoff’s lawof radiation states that the emissivity of a medium atany angle is equal to its absorptivity at that angle,

13

in thermal equilibrium. This is a stricter conditionthan a steady state, in which it is only required thatthe total flow of photons from the medium to the en-vironment is equal to the total flow of photons fromthe environment to the medium.

Kirchhoff’s law requires that in equilibrium thereis also a detailed balance, meaning that the flowof photons from the medium to the environmentat an angle is equal to the flow of photons fromthe environment to the medium at that same an-gle. We show that although this detailed balanceholds in equilibrium systems, and is very well ap-proximated in a non-equilibrium steady state systemwith isotropic fluorophores, it is clearly violated ina non-equilibrium steady state system with aligneddipolar fluorophores.

We consider two types of environmental radiationinteracting with a medium. The first is an isotropicradiation field characteristic of thermal equilibrium,e.g. the interior radiation field of a blackbody cav-ity. Under these conditions, light is incident on themedium with a Lambertian angular profile [37], with

dp(e)env(θ) =

2

sin2 θccos θ sin θdθ, (33)

for θ < θc. The second radiation source we consideris light received from the solar disk, i.e. sunlight, at

normal incidence to the medium. We use dp(e)env(θ) =

ν sin θ cos θdθ for θ ≤ 0.267 and dp(e)env(θ) = 0 for

θ > 0.267, where ν is a normalization constant equalto n2

m/(πn2e sin2(0.267)). Since our medium emits at

all angles, but absorbs only at angles correspondingto the solar disk, its steady state is not an equilib-rium.

We also consider two versions of the medium, onewith isotropic fluorophores and the other with dipo-lar fluorophores. The absorption, transmission, andemission probabilities for these are defined in Sec. 2.2.For the isotropic case, we consider a slab of GaAswith parameters (thickness, refractive index, absorp-tion coefficient and emission/absorption wavelengthof light) as described in Sec. 3.1.1. For the dipolarcase, we consider cadmium selenide/cadmium sulfide(CdSe/CdS) dot-in-rod crystals that are in a chlo-

roform solution, and are aligned with their long axisnormal to the surface of the medium. The concentra-tion of the CdSe/CdS dot-in-rod fluorophores is 10−6

molar concentration or molarity (M), extinction co-efficient of the fluorophores is 5× 106 cm−1M−1, thewavelength of emission/absorption is taken as 615nm. The refractive index of the medium is that ofchloroform at 615 nm, i.e. 1.445. The thickness ofthe medium is 2 cm.

In each of the four cases (Lambertian or solar in-cident radiation; isotropic or dipolar fluorophores)we always have a mirror at the bottom of layer M ,and assume the refractive index of the environmentis ne = 1.

To probe the angle distribution of photons enter-ing and leaving the system, we partition the range0 ≤ θ ≤ θc into a number bmax of bins, each of size∆θ = θc/bmax. We index these bins by a value branging from 1 to bmax. The bin b consists of anglesfrom (b − 1)∆θ to b∆θ. Note that for the equilib-rium case the light entering the medium, upon re-fraction, spans the range θ ≤ θc. However, for thenon-equilibrium case, the angular range of light en-tering the medium is less than θc. In particular,it spans the range θ ≤ sin−1([ne/nm] sin(0.267)).However, we still use the same bin range while keep-ing in mind that the absorption corresponding to binsin the range sin−1([ne/nm] sin(0.267)) < θ ≤ θc iszero.

We analyze transitions in greater detail now byidentifying the transition probabilities at angleswithin each bin. That is, we break up the defini-tions of the transition probabilities p0i and pj0 sothat instead of integrating over all angles, we inte-grate just over angles within the bin. We will havep0i =

∑bmax

b=1 p0i(b) and pj0 =∑bmax

b=1 pj0(b). Fori > 0, p0i(b) is the probability of a photon in layer ito leave the medium at an angle in bin b:

p0i(b) =

∫ b∆θ

(b−1)∆θ

dp(e)(θ)p(t)(θ, i, 0), (34)

as in Eq. (26). Similarly, for j > 0, pj0(b) is the prob-ability of a photon from the environment to reach

14

layer j at an angle in bin b:

pj0(b) =

∫ b∆θ

(b−1)∆θ

dp(e)env(θ)p

(t)(θ, 0, j)p(a)(θ), (35)

as in Eq. (23). p00(b) is the probability of light fromthe environment entering the medium, being reflectedoff the mirror on the bottom, and emerging againwithout being absorbed in the medium. That is:

p00(b) =

∫ b∆θ

(b−1)∆θ

dp(e)env(θ)p

(t)(θ, 0, 0), (36)

as in Eq. (27).

Once we have these detailed probabilities, we cananalyze the total absorption and emission of themedium within each of these bins by summing overall layers. In the steady state, the angle-dependentabsorption of the medium at angles in bin b is

a(b) =

M∑j=0

d(s)0 pj0(b), (37)

and the angle-dependent emission of the medium atangles in b is

e(b) =

M∑i=0

d(s)i p0i(b). (38)

Note that a(b) can be simplified to∫ b∆θ

(b−1)∆θdp

(e)env(θ),

because the sum of p(t)(θ, 0, j)p(a)(θ) is 1 (sinceall light must end up somewhere), though there isno comparable simplification of e(b). Because this

is the steady state, it is clear that∑bmax

b=1 a(b) =∑bmax

b=1 e(b). But the detailed balance predicted byKirchhoff’s law in equilibrium would require thata(b) = e(b) for each bin b.

Eqs. (37) and (38) are plotted for the isotropic flu-orophores and the dipole fluorophores in Figs. 5b and5d, respectively, for the case of an isotropic incidentradiation field (i.e. the equilibrium state). One seesthat the angle-dependent absorption and the angle-dependent emission are coincident which implies thatKirchhoff’s law is upheld.

(a) (b)

(c) (d)

Figure 5: Kirchhoff’s Law Analysis. The equilibriumcase. (a) and (c) illustrate the two types of mediaconsidered, one that emits isotropically (the greencircular object in (a)) and another that emits like adipole (the green two-lobed object in (c)). For both(a) and (c) the orange semicircle and orange struc-tures represent the incident light whereas the darkblue represents the light exiting the medium. (b)is a polar plot of the light entering (red trace) themedium and light leaving the medium (black trace)in the steady state for the isotropic emitting medium.(d) is the same as (b) except for the dipole emittingcase.

15

(a) (b)

(c) (d)

Figure 6: Kirchhoff’s Law Analysis. The nonequi-librium case. (a) and (c) illustrate the two types ofmedia considered, one that emits isotropically andanother that emits like a dipole. For both (a) and(c) the orange semicircle and orange structures rep-resent the incident light whereas the dark blue rep-resents the light exiting the medium. Light entersthe medium in the angular range 0 ≤ θ ≤ θm =sin−1([ne/nm] sin(0.267)). (b) is a polar plot of theangular distribution of light leaving the medium inthe steady state for the isotropic emitting medium.(d) is the same as (b) except for the dipole emit-ting case. Incident light, not represented in the plots,spans the solar angle 0 ≤ θ ≤ 0.267. One readily ob-serves that the isotropic emitters emit with a Lam-bertian radiation pattern unlike the dipolar emitters.

In contrast, the non-equilibrium case shows thatthe angular distributions of the incoming and out-going light does not match, i.e. a(b) 6= e(b). Thiscan be seen in Fig. 6 where the incoming light spans0 ≤ θ ≤ 0.267 (not pictured). Light exiting themedium for the isotropic case is shown in Fig. 6bwhile the exiting light for the aligned dipolar fluo-rophores it is shown in Fig. 6d. It is commonly as-sumed that most bulk materials fluoresce in a Lam-bertian manner, like equilibrium radiation fields. Wecan see that this happens when the system is com-posed of isotropic fluorophores, but not when it iscomposed of aligned dipolar ones, at least with thethickness of medium that we consider. Interestingly,the anisotropy of the emission in the dipolar case isdriven more by the lack of reabsorption along certainangles, rather than by the excess emission along otherangles.

3.3 Light Trapping in LuminescentSolar Concentrators

Luminescent solar concentrators (LSCs) have beenproposed as an alternative to conventional lens-basedoptical concentrators for improving the performanceand lowering the cost of high efficiency photovoltaic(PV) solar converters, see Ref. [17] and referencestherein. An LSC consists of fluorophores such as dyemolecules or semiconductor nanocrystals embeddedin a dielectric medium. Light is absorbed by the flu-orophores and re-emitted. Refractive index contrastbetween the medium and environment means thatsome re-emitted light is trapped by total internal re-flection into “waveguide modes”. PV elements arethen placed at the edges of the LSC, where light inthe waveguide modes is directed.

The performance of an LSC, and the consequentPV conversion efficiency, depends on the ratio oflight occupying waveguide modes compared to theincident sunlight. A useful proxy for this value isthe fraction of light each fluorophore emits into thewaveguide mode, ηtrap, which for isotropic emitters

is√

1− (ne/nm)2 or ∼ 74% for a typical organicpolymeric waveguide medium (nm/ne = 1.5) [38].

16

Anisotropic fluorophores, such as aligned semicon-ductor nanorods, have been proposed as fluorophoresbecause they emit preferentially into the waveguidemodes, and can thus provide a higher ηtrap. How-ever, a full analysis of LSC performance based on ei-ther fluoruphore should also account for the efficiencyof absorption of incident light, as well as further re-absorption and re-emission events. Monte Carlo sim-ulations of three-dimensional structures have beenused to account for these complexities, as well asscattering and non-unity quantum yield [39, 40, 41].Markov models of the sort described in this papercan make this fuller analysis more computationallytractable.

An LSC can be described with the same geometryanalyzed in Sec. 3.2, with a mirror on the bottom ofthe medium. We assume that the fluorophores aresuspended in a solid matrix of refractive index 1.49,while the refractive index of the environment is 1. Weconsider both isotropic fluorophores, as in Fig. 5a,and dipolar as in Fig. 5c. We assume diffuse incidentlight with a Lambertian profile. We do not track thewavelength of light specifically, but to account fora Stokes shift, we use an absorption coefficient forincident solar radiation of 5×108cm−1M−1, while weuse 5 × 105cm−1M−1 ( ∼ 1000x lower cross section)for light emitted from the fluorophores, in agreementwith common values for semiconductor nanocrystals[17].

We can use the previous analysis to find thesteady state population of excited fluorophores inthe medium,

∑Mi=1 di/d0. The trapping ratio of

re-emitted light is ηtrap =∫ π/2θc

dp(e)(θ) (which is√1− (ne/nm)2 for isotropic emitters). With quan-

tum yield ηQ, the net occupation of the waveguidemodes, which we will call the “occupation factor”, isgiven by the product of these three factors,

O = ηQηtrap

M∑i=1

did0. (39)

Because our model assumes an infinite horizontalextent of the medium, it does not directly track howmuch light reaches the photovoltaics at the edges. In-

stead it identifies how much light occupies the waveg-uide modes at any point in the medium, just as theanalysis based on ηtrap alone does. To fully trackthe three-dimensional motion of the photons from ini-tial incidence until PV conversion, the Markov modelwould require states representing differential volumeelements, rather than thin layers, as well as transitionprobabilities between them, but the process of findingthe stationary distribution would be the same.

Figure 7 plots the occupation factor with dipo-lar and isotropic fluorophores and variable quantumyield, as a function of the concentration of nanocrys-tals in the polymer matrix of the LSC, spanninga range commonly studied experimentally. Valuesgreater than 1 can be achieved because a single pho-ton may be absorbed and re-emitted several times.Note that dipolar fluorophores achieve higher occupa-tion factors when concentration is high, due in part tohigher ηtrap, while they have lower occupation factorat low concentrations due to absorbing less incidentlight than isotropic fluorophores.

Figure 7: Occupation of waveguide modes comparedto incident photons, as a function of concentration offluorophores in the LSC. Solid traces correspond todipolar fluorophores while dashed traces correspondto isotropic fluorophores.

4 Conclusion

We have demonstrated the use of a Markov modelto study the steady state behavior of several optical

17

systems. The method works by finding an eigenvectorof the matrix of transition probabilities for photonsmoving through the system. We have shown that thismethod yields the expected results in systems thatallow for an analytical solution (Beer-Lambert’s lawor Kirchhoff’s law). This model also gives a descrip-tion of steady states for which analytical solutions arenot available. We propose that this method could beparticularly useful for describing complex mesoscaleinteractions between nanophotonic elements, such asin an LSC.

The particular model we developed assumes trans-lational symmetry in two dimensions, and only tracksthe movement of photons in the third dimension.This model also assumes that the angle of emissionof a photon is independent of the angle of absorption,which makes less accurate predictions when appliedto analysis of scattering. However, more detailedMarkov models can track these additional variablesto give more precise results. Transition probabili-ties can also be modified to take into account addi-tional features of radiation, such as near-field inter-actions, wavelength dependence, and deviation fromperfect isotropic or sine-squared emission from thefluorophores.

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