## Abstract

We present an analytical model for the optical emission of a two-dimensional source in a flexible organic light-emitting diode formation with arbitrary curvature. The formulation rigorously produces closed-form analytical expressions which clearly relate the emission pattern and the device configuration, in particular, the radius of curvature. We investigate the optical properties of a prototype model through the resultant expressions, revealing that the bending induces a dramatic enhancement of emission to large angles, allowing for large viewing angle and reduced total internal reflection losses. These effects, shown to arise from geometrical considerations, demonstrate the unique advantages which curved flexible devices offer with respect to their planar counterparts. To the best of our knowledge, this is the first time that a rigorous analytical investigation of the optical characteristics of these novel devices is conducted. The resultant analytical formulae provide a robust basis for future analysis, as well as a set of design rules for efficient device engineering.

© 2012 Optical Society of America

## 1. Introduction

Organic light-emitting diodes (OLEDs) have been intensively investigated for the past two decades as promising candidates for novel electrooptic applications, such as thin flexible displays, low-cost lasers and efficient lighting instruments [1–4]. While rigid plane-parallel OLEDs (PPOLEDs) have drawn most of the research attention in previous years, lately there has been a growing interest in the emerging field of flexible OLEDs (FOLEDs) and their exciting applications [5–11]. However, most of the literature in this area focuses on the mechanical aspects or fabrication issues related to the device flexibility [2, 7, 10], while there is almost no reference to the optical effects that are introduced by this new degree of freedom.

In this paper we present a rigorous electromagnetic (EM) canonical model for curved (concentric) stratified media with an emitter embedded in one of the internal layers, namely, analysis of a layered cylindrical FOLED configuration. Since, in practice, these devices are characterized by large radius of curvature with respect to wavelength, azimuthal periodic effects should be ignored; this is most efficiently achieved by utilizing Felsen’s ”perfect azimuthal absorber” concept [12–14]. Furthermore, incorporating Deybe’s approximation [15] enables us to formulate clear and compact expressions for the EM fields in each layer, establishing a complete analogy to the standard model of rigid PPOLEDs [16], which in this representation is simply the limit of infinite radius of curvature. This approach, applied to a realistic prototype model, yields closed-form analytical expressions for the FOLED emission pattern, which reflect the relations between the latter and the device dimensions, emitter location, choice of materials, and most importantly, radius of curvature. This allows us to explore the effects of different radii of curvature on FOLED emission, revealing the unique optical properties arising from the cylindrical geometry. For the sake of simplicity and clarity, we focus on a two-dimensional (2D) canonical configuration excited by impulsive (line) sources, instead of using the more realistic three-dimensional (3D) model. However, as has been shown before [17], the essence of the physical phenomena remains the same, and insight gained by the results can, in general, be applied to 3D devices as well.

## 2. Theory

#### 2.1. Formulation

We consider a 2D device with *M* + *N* + 2 concentric layers, with a line source embedded at a certain distance from the origin, *ρ*′, sandwiched between layers (−1) and (+1), as depicted in Fig. 1. The homogenous layer formed by combining layers (−1) and (+1), containing the line source, is termed the active layer. Each layer is characterized by its permittivity, permeability and conductivity, marked *ε** _{n}*,

*μ*

*and*

_{n}*σ*

*, respectively, for the*

_{n}*n*th layer. Furthermore, the

*n*th and (

*n*+ 1)th layers are separated by the cylindrical shell

*ρ*=

*a*

*for*

_{n}*n*> 0 and

*ρ*=

*a*

_{n}_{+1}for

*n*< 0, and we define

*a*

_{0}=

*ρ*′,

*a*

_{N}_{+1}=

*ρ*and

*a*

_{−(}

_{M}_{+1)}= 0. The radius of curvature of the device is defined by the radius of the innermost cylindrical shell,

*R*=

*a*

_{−}

*, which is infinite for a PPOLED, and decreases as the FOLED is bent. Note that*

_{M}*ε*

_{−1}=

*ε*

_{1},

*μ*

_{−1}=

*μ*

_{1}and

*σ*

_{−1}=

*σ*

_{1}. For the sake of completeness, we treat here both transverse electric (TE) and transverse magnetic (TM) modes, excited via an electric line source and a magnetic line source, having current magnitudes

^{e}*I*

_{0}and

^{m}*I*

_{0}, respectively. Throughout the paper, we use

*e*and

*m*superscripts or subscripts to denote electric or magnetic cases, respectively. Both sources are assumed to be time harmonic, with time dependence of

*e*

^{j}

^{ω}*. The wave number and wave impedance of the*

^{t}*n*th layer are given as

*k*

*=*

_{n}*ω*{

*μ*

_{n}*ε*

*[1 −*

_{n}*j*

*σ*

*/ (*

_{n}*ωε*

*)]}*

_{n}^{½}= (

*ω*

*/c*)(n

*−*

_{n}*j*

*κ*

*) and*

_{n}*Z*

*= (*

_{n}*μ*

*/ {*

_{n}*ε*

*[1 −*

_{n}*j*

*σ*

*/ (*

_{n}*ωε*

*)]})*

_{n}^{½}, where

*c*, n and

*κ*denote the velocity of light in vacuum, refractive index and extinction coefficient, respectively. To satisfy the radiation condition we require that the imaginary part of the wavenumber will be non-positive, i.e. ℑ{

*k*

*} ≤ 0, leading to ℑ{*

_{n}*Z*

*} ≥ 0. The 2D source vector is*

_{n}*ρ*⃗′ = (

*ρ*′,

*φ*′) and the observation point is denoted by

*ρ*⃗ = (

*ρ*

*,φ*).

As the schematics in Fig. 1 reveals, only a segment of the cylindrical layered media is of interest in our problem, as the bent FOLED does not form a closed cylinder. Therefore, following Einziger and Felsen in their work on 2D radomes [13,18], we utilize the ”perfect azimuthal absorber” concept to rigorously solve the Maxwell equations for the FOLED scenario in a natural manner, i.e., emphasizing the true physical phenomena and ignoring fictitious effects. In the frame of this methodology, a planar ”perfect absorber” is placed at *φ* = 0 and *φ* = *π*. The aim of this absorber is to establish the realistic scenario in which power which flows azimuthally from the source to the edges of the formation is ”lost” (much like waveguided energy does in the plane-parallel scenario [16]). We note that the perfect absorber placement is quite arbitrary, and is selected as to fit the model physical configuration. The absorber is ”perfect” in the sense that the azimuthal waves impinging upon it are absorbed completely without any spurious reflections. Hence, this method is applicable when the radius of curvature is large enough such that edge-diffraction effects may indeed be ignored, i.e. |*k*_{n}*R*| ≫ 1, and when there are small losses in the device such that the waves are sufficiently attenuated towards the device facets. Note that this model is probably the most elegant analytical tool to recover the limit of infinite radius of curvature, leading to the planar configuration solution.

#### 2.2. Power relations

The line source excitations in our problem are formally defined as

*G*(

*ρ*⃗,

*ρ*⃗′), as the spectral response to this impulsive excitation, related to the transverse components of the EM fields by

^{e}*J*

*=*

_{s}

^{e}*I*

_{0},

^{m}*M*

*=*

_{s}

^{m}*I*

_{0}, and

^{m}*J*

*=*

_{s}

^{e}*M*

*= 0;*

_{s}

^{e}*E*

*=*

_{t}*E*

*,*

_{z}

^{e}*H*

*= −*

_{t}*H*

*,*

_{φ}

^{m}*E*

*=*

_{t}*E*

*, and*

_{φ}

^{m}*H*

*=*

_{t}*H*

*[19]. The source excitation can be decomposed to a set of azimuthal waves; consequently, the 2D cylindrical Green function can be expressed by the following spectral integral*

_{z}*g*(

*ρ*

*,*

*ρ*′) is termed the one-dimensional (1D) cylindrical Green function, and the parameter

*ν*is the order of the function, which serves as a normalized azimuthal (transverse) wavenumber, in analogy to the PPOLED case [16]. This form of the 2D Green function satisfies the ”perfect azimuthal absorber” conditions, letting only azimuthal waves propagating from the source azimuth towards the device edges to reside [13]. The wave equation and boundary conditions for the 1D and 2D Green functions are outlined in Table 1, where we used the definition of the impedance ratio, ${}_{m}^{e}{\gamma}_{n}={\left({Z}_{n+1}/{Z}_{n}\right)}^{\pm 1}$ [19].

In analogy to the planar model solution [16], we express the 1D cylindrical Green function in the various layers using recursive relations, as summarized in Table 2. For clarity we use the notation
${H}_{l,m}^{(i)}$ to indicate the Hankel function of the *i*th kind at the *l*th layer side of the *m*th interface,
${H}_{\nu}^{(i)}({k}_{l}{a}_{m})$, and a similar convention for the derivative,
${H}_{l,m}^{(i{)}^{\prime}}={\partial {H}_{\nu}^{(i)}(\mathrm{\Omega})/\partial \mathrm{\Omega}|}_{\mathrm{\Omega}={k}_{l}{a}_{m}}$; the normalized derivative is consequently defined as
${h}_{l,m}^{(i)}={H}_{l,m}^{(i{)}^{\prime}}/{H}_{l,m}^{(i)}$, and the order of the functions, *ν*, is usually omitted and can be inferred from the context.

In the spirit of previous work [16], we express the cylindrical 1D Green function in the observation region, *n* = *N* + 1, for *N,M* ≥ 1, as a sum of multiple reflections between the layer boundaries,

*K*) and backward (

*K̂*) multiple reflection parameters, which take into account all the possible combinations of cylindrical wave trajectories within the device for the considered azimuthal order

*ν*, are defined in Table 2. The summation limits vary for each combination, and are given by ${l}_{n}={\sum}_{k=1}^{n-1}({p}_{k}-{s}_{k})+{l}_{1}$, ${\widehat{l}}_{n}={\sum}_{k=n+1}^{-1}({\widehat{p}}_{k}-{\widehat{s}}_{k})+{\widehat{l}}_{-1}$,

*l̂*

_{−1}=

*l*′ +

*l*

_{1}and ${\tilde{l}}_{q}^{n}={\sum}_{k=n}^{q-1}\left({\tilde{p}}_{k}^{n}-{\tilde{s}}_{k}^{n}\right)+{\tilde{l}}_{n}^{n}$.

Finally, we utilize Eqs. (2)–(4) to formulate the variation of the far-field radiated power density with azimuth, for a given spectral distribution of sources *p* (*ω*) [16], namely

*n*th cylindrical shell.

#### 2.3. Debye approximation and saddle point evaluation

In order to analytically resolve the integrals of Eq. (5) we make use of the Debye approximation of the Hankel functions [13], which is applicable whenever the argument, denoted here generally by Ω, is large (|Ω| ≫ 1) and significantly different from the order, *ν*. Under these conditions, the Hankel function of the first kind is given by ([12], pp. 710–712; [15])

*=*

_{l,m}*k*

_{l}*a*

*and sin*

_{m}*α*

*=*

_{l,m}*ν*/Ω

*, which in conjunction with the definitions of Table 2 yield the following approximation for the phase factors,*

_{l,m}*k*

_{n}*R*| ≫ 1 is easily met in realistic devices, hence the Debye approximation conditions are satisfied.

Phase factors as the ones listed in Eq. (7) are suitable for the execution of the steepest descent path method for saddle point evaluation, as they contain a large argument, Ω* _{l,m}* ([12], pp. 382–391). In contrast with the saddle point evaluation for the plane-parallel structure [16], the saddle points for the various combinations of multiple reflections may differ significantly; thus, in general, we cannot neglect the contribution of the internal layer trajectories to the saddle point condition. Moreover, in order to fit the integrals of Eq. (5) to the standard saddle point evaluation format, we treat separately the two azimuthal harmonics which form the cosine factor, namely,

*e*

^{j}

^{ν}^{(}

^{φ}^{−}

^{φ}^{′)}and

*e*

^{−}

^{j}

^{ν}^{(}

^{φ}^{−}

^{φ}^{′)}. Considering these arguments, the saddle point condition for the multiple reflection term defined by a certain set of indices (

*l*

*,*

_{q}*l̂*

*,*

_{q}*l*′, ${\tilde{l}}_{q}^{p}$) in Eq. (4) reads [13, 18]

*ν*> |Ω

*| constitute the evanescent spectrum of the problem’s Green function (complex*

_{l,m}*α*

*), for far-field evaluation it is sufficient to consider solutions in the interval*

_{l,m}*ν*∈ [0,Ω

*), where Ω*

_{min}_{min}= min {Ω

_{n}_{+1}

*}. As was shown in detail by Einziger and Felsen [13, 18], the saddle point condition is a manifestation of the laws of geometrical ray-optics. In fact, Eq. (8) forms an ”angular conservation law”, as*

_{,n}*α*

*and*

_{n,n}*α*

_{n,n}_{−1}indicate the angles between the ray trajectory in the

*n*th layer and the normal to the

*n*th or (

*n*− 1)th interfaces, respectively. This points out a major difference from the planar scenario, responsible for most of the effects discussed further on, as due to the curved geometry, a ray meets the two boundaries of the same layer at different angles of incidence (

*α*

*≠*

_{n,n}*α*

_{n,n}_{−1}). In that sense, the saddle point value,

*ν*

*(*

_{s}*l*

*,*

_{q}*l̂*

*,*

_{q}*l*′, ${\tilde{l}}_{q}^{p}$), is the one that ensures, through the law of sines, that from all the possible rays which depart from the source and satisfy the laws of geometrical optics, only the ones which reach the observation point, i.e. cover the angular aperture between source and observation points, (

*φ*−

*φ*′), are taken into consideration (See Fig. 4 of [18]).

The asymptotic evaluation of the 2D Green function is therefore given by

#### 2.4. Closed-form solution for prototype device

In order to investigate the curvature effects on the optical emission of FOLEDs, we apply the general formalism developed in previous subsections to a prototype device. A literature survey reveals several strategies for FOLED design, differing mainly by the choice of anode and substrate materials [1, 2, 6–9]. In order to remain compatible with the prototype device analyzed for the planar scenario [16], we consider a device similar to [8], which is fabricated on a poly(ethylene terephthalate) (PET) substrate, whose refractive index is similar to that of glass, and an ITO anode, as specified in Table 2.4. The elementary device, depicted in Fig. 2(a), corresponds to Fig. 1 (setting *M* = 1 and *N* = 3) with an electric line source excitation located at (*ρ*′,
${\phi}^{\prime}=\frac{\pi}{2}$), radiates typically at *λ* ≈ 600nm. We demonstrate the implications of our analysis using the 2D TE source, as the interference processes attributed to parallel electric sources have been found to be the dominant contribution to typical OLED radiation [17, 20]. Note that the layers are assumed to conserve their thickness under bending, thus the device geometry is defined by the interface distances *d** _{n}* from the cathode/organic boundary, where the curvature effect is introduced by

*a*

*=*

_{n}*R*+

*d*

*; consequently we define the source-cathode separation as*

_{n}*z*′ =

*ρ*′ −

*R*, in analogy to the planar case. The corresponding PPOLED prototype, achieved by taking the limit

*R*→ ∞, is depicted in Fig. 2(b), along with the suitable polar coordinate system (

*ρ*̃

*,*

*θ*).

We observe three important typical features of such a device. First, as *k*_{n}*R* ≫ 1 (Subsection 2.3), the thin-film characteristics of the device imply that the radius of curvature is much larger than the thickness of the anode and the active layer, i.e. *R* ≫ (*d*_{n}_{+1} − *d** _{n}*), for |

*n*| ≤ 1; this can be written alternatively as Δ

*a*

_{n}_{+1}

*= (*

_{,n}*a*

_{n}_{+1}−

*a*

*) ≪*

_{n}*a*

*, |*

_{n}*n*| ≤ 1. Thus we may approximate these layers’ contribution to the saddle point condition (Eq. (8)) by

_{n}_{+1}

*≫ Ω*

_{,n}_{min}>

*ν*for |

*n*| ≤ 1 due to the typical refractive indices. In other words, rays crossing these thin-film layers do not vary significantly their angle of incidence between the two boundaries of the layer, therefore their contribution to the ”angular conservation law” enforced by the saddle point condition, is negligible. This leaves only contributions of thick layers with respect to the radius of curvature, namely, the substrate.

The second property which allows simplification of Eq. (8) is the metallic cathode occupying the innermost cylinder, *n* = −2. At optical wavelengths, the metal has high losses, which are expressed as a very large imaginary part of *α*_{−2,−1}; thus, the factor of this term must be zero in order to allow for a real solution of the saddle point condition. This forces the number of ray crossings from one side of the cylindrical structure to the other side through the innermost cylinder, (*l̂*_{−2} + 1), to be zero, which physically means that rays which penetrate the metallic cathode suffer from a rapid decay, therefore do not contribute to the far-field emission pattern.

The third observation is related to the magnitude of the reflection coefficients of the various interfaces. Considering the refractive index differences between the device layers (Table 2.4) we notice the formation of a weak-microcavity (WM), as for most ray trajectories the dominant reflections originate in the subtrate/air or organic/cathode interfaces [16]. This allows us to reduce substantially the number of multiple reflection terms taken into account in the series representation of the 1D Green function (Eq. (4)) and consequently simplify the saddle point condition.

Utilizing these features of the prototype FOLED yields the following reduced form of the saddle point condition,

*l*

_{3}+ 1) is a non-negative integer which signifies the number of multiple reflections of rays passing back and forth through the substrate. Returning to the geometrical ray-optics interpretation, the reduced saddle point condition states that the significant effects on the ray trajectory arise from its propagation in the relatively thick substrate, including multiple reflections, and the refraction from substrate to air. Note that as we removed the possibility to cross the innermost metallic shell, only one of the cosine harmonics has a saddle point in the valid solution domain, i.e. ${G}_{N+1}\left(\overrightarrow{\rho},{\overrightarrow{\rho}}^{\prime}\right)\sim {G}_{N+1}^{\pm}\left(\overrightarrow{\rho},{\overrightarrow{\rho}}^{\prime}\right)$ for (

*φ*−

*φ*′) ≷ 0, and there remains only one applicable saddle point condition for each observation angle, Eq. (12). Hence, the EM far-fields for the prototype device can be derived from a simplified version of the 2D Green function, given by

*ν*=

*ν*

*(*

_{s}*l*

_{3}), is a real solution for Eq. (12), i.e. it defines the appropriate angles of incidence as to satisfy the ”angular conservation law”; for the prototype device it is sufficient to take into account multiple reflections in the WM up to

*l*

*= 3. Utilizing the definition of the optical path,*

_{max}*L*

*, of a ray propagating in the*

_{n,m,l}*n*th layer from

*ρ*=

*a*

*to*

_{m}*ρ*=

*a*

*, namely,*

_{l}The physical interpretation of the expressions in Eqs. (13)–(17) is simple: the response of the device to an excitation in an internal layer consists of a geometrical factor, which is multiplied by transmission factors which take into account the interference between the source and the image induced by the reflecting cathode (IS), the transmission through the shell boundaries from the source to the observation point (DR), and a series of multiple reflections between the cathode/organic and substrate/air interfaces (WM). This forms a direct analogy to the Green functions derived for the planar prototype device [16]; the main differences are the requirement to recalculate the saddle point for multiple reflection combinations that differ by the number of times they cross the substrate, and the necessity to include in the calculation also transmission factors through the internal layers (e.g., $\left(1-{\mathrm{\Gamma}}_{1}^{2}\right)$). These requirements somewhat complicate the formal expressions, however they are necessary if an exact solution for the full range of radii of curvature is desirable.

As a final step, we incorporate the spectral distribution of the source (exciton) ensemble into our formalism, a step which is essential to produce emission patterns consistent with experimental results. This translates into multiplication of each multiple reflection combination cross-term, formed by the product of the series representation of the 2D Green function (Eq. (13)) and its derivative (Eq. (5)), with the suitable spectral broadening attenuation factor [16]. These attenuation factors arise from the convolution of the emission pattern with the spectral distribution, assumed to be Gaussian with width of Δ*ω*, and are squared exponential in the ratio between the ray total optical path and the source ensemble coherence length, *L** _{c}* =

*c*/Δ

*ω*. In OLEDs, as was discussed in detail in [16], due to the typical small coherence length with respect to the substrate thickness (

*L*

*≪*

_{c}*d*

_{3}), emission pattern features induced by interference of rays multiply reflected from the substrate/air interface are dramatically averaged when the spectral distribution is taken into account. The practical meaning is that cross-terms which contain interference phase accumulated about passage in the substrate can be completely omitted from the emission pattern calculation.

Finally, substituting Eq. (13) into Eq. (5), the emission pattern of the prototype FOLED reads

## 3. Results and discussion

Emission patterns for the prototype FOLED, *S** _{ρ}* (|

*φ*−

*φ*′|), with decreasing radii of curvature are plotted in Figs. 3(b)–3(g), where Fig. 3(a) presents the emission pattern for the corresponding PPOLED,

*S*

_{ρ}_{̃}(

*θ*), calculated according to the formulation in [16]. Each plot contains emission patterns for two emitter separations from the cathode, namely,

*z*′ = 20nm (red dashed line) and

*z*′ = 140nm (blue solid line). These two emission zone locations differ by their emission pattern characteristics due to different IS interference, the former yielding a quasi-Lambertian pattern, while the latter contains a distinct local maximum at an angle of 58°, for the PPOLED case [16]. As customary, the emission patterns are normalized according to the maximal value.

Before relating to the unique phenomena arising from the introduction of curvature, it is worth noting that a quick review of Fig. 3 confirms that as the radius of curvature increases from 50*μ*m < *d*_{3} (Fig. 3(g)) to 2000*μ*m ≫ *d*_{3} (Fig. 3(b)), the FOLED shell interfaces approach plane-parallelism and the emission patterns converge to the PPOLED emission patterns (Fig. 3(a)). This convergence can be validated rigorously using our formulation: for *R* ≫ *d*_{3} we may utilize Eq. (11) for *n* = 3, reducing the saddle point condition of Eq. (12) to its PPOLED limit, *ρ*sin*α*_{4,4} = *a*_{3} sin (*α*_{4,4} + *|φ* − *φ*′|), which can be solved analytically for *ν*. Introducing the resultant saddle point into the expressions listed in Table 2 yields exactly the closed-form analytical expressions obtained for the PPOLED case in [16]. The importance of carrying this validation procedure is that it indicates that the curvature effects become apparent only when the radius of curvature becomes comparable with the substrate thickness (*R* ∼ *d*_{3}), therefore translates into an important design rule for FOLEDs. It is worth noting that the PPOLED limit is reached in such a straightforward manner only due to the utilization of Felsen’s ”perfect azimuthal absorber” in our formulation, emphasizing once more the elegance of this method.

#### 3.1. Emission pattern ”stretching” and back-illumination

The first observation we make of the results presented in Fig. 3 is that as the radius of curvature decreases, it seems that the emission pattern undergoes ”stretching” towards large angles. For instance, if we follow the local maximum for the *z*′ = 140nm emission zone, a pronounced increase of the angle, |*φ* − *φ*′|_{max}, in which this extremum is achieved is observed. This effect is reversed as we cross from Fig. 3(e) (*R* = 200*μ**m*) to Fig. 3(f) (*R* = 100*μ**m*), and a ”compression” is observed when the radii of curvature further decrease. Note that this effect involves not only the extrema angle variation, but also the corresponding expansion (or shrinkage) of the range of angles to which substantial power is emitted; at some instances this effect even allows power emission towards observation angles larger then *π*/2 (e.g. *R* = 200*μ**m* (Fig. 3(e))). This phenomenon of back-illumination (BI) is unique to the curved formation, and cannot be observed for PPOLEDs.

The origin of the ”stretching” effect can be identified using our analytical formulation. As demonstrated by Eq. (13), the angular distribution of the emitted power is shaped by three physical processes: the IS interference, the DR transmission and the WM multiple reflections. From these three, the dominant contributions to the main emission pattern features come from the IS interference and the DR transmission from substrate to air [17]. Analyzing these contributions in the light of geometrical optics reveals that under bending the reflecting cathode becomes a (non-ideal) convex mirror; however, due to the small source-cathode separation with respect to the mirror focus, *z*′ ≪ *R*/2, the IS interference is almost unaffected by the curvature. This indicates that the curvature affects mostly the DR transmission factor, which is dominated by the substrate/air interface reflection.

To demonstrate this, we calculate the FOLED observation angle |*φ* − *φ*′| which yields the same substrate/air reflection coefficient, Γ_{3}, as the one received when the PPOLED observation angle is, i.e. we require that
${\mathrm{\Gamma}}_{3}^{\text{PPOLED}}(\theta )={\mathrm{\Gamma}}_{3}^{\text{FOLED}}\left(\left|\phi -{\phi}^{\prime}\right|\right)$. This implies cos*α*_{4,3} = cos*θ* [16], and consequently *ν* = *k*_{4}*a*_{3} sin*θ*. Substituting this into Eq. (12) yields for a far field observation point

*φ*−

*φ*′| >

*θ*. In other words, the same substrate/air reflection losses are obtained at larger observation angles in FOLEDs. From a slightly different point of view, the last result means that for a given observation angle, the transmission through the substrate/air interface would be larger for a curved FOLED than for a PPOLED, i.e. $\left|1+{\mathrm{\Gamma}}_{3}^{\text{PPOLED}}\left(\theta =\left|\phi -{\phi}^{\prime}\right|\right)\right|<\left|1+{\mathrm{\Gamma}}_{3}^{\text{FOLED}}\left(\left|\phi -{\phi}^{\prime}\right|\right)\right|$. This indicates that application of bending enhances the transmission of power emitted by the source towards large angles, or alternatively, that the DR transmission factor, which acts as an angular low-pass filter, has a wider angular bandwidth. Considering the relative insensitivity of the IS interference to curvatures, the emission pattern of the curved FOLED, which is a product of the DR and IS contributions, appears to be a ”stretched” form of the corresponding PPOLED pattern, in the sense that the IS interference features which appear at larger angles become more pronounced due to the wider angular bandwidth. The practical meaning of this observation is that the emission direction can be modulated by simply increasing or decreasing the radius of curvature of the FOLED, as demonstrated in Figs. 3(b)–3(g). As the information in the subplot legends indicates, the extremum angles increase from 58.7° for the planar case up to 83.6° for

*R*= 200

*μ*

*m*.

The geometrical ray-optics interpretation of this phenomenon is presented in Fig. 3(h), where the red solid line refers to a ray departing from the anode/substrate interface at a certain angle, *θ*_{3} = *α*_{3,2}, simultaneously for the planar and curved devices. The substrate/air interface is met first for the curved device, and it is indicated by a refraction of the red solid line with an incidence angle of *α*_{3,3}. In the planar case, however, the ray continues further to meet the substrate/air interface, as the dotted red line demonstrates, and the incidence angle remains the same, *θ*_{3}. According to the law of sines, *α*_{3,3} < *θ*_{3}, therefore the reflection coefficient for the curved device will also be smaller, which means an enhancement of the emission to large angles in FOLEDs with respect to PPOLEDs, in consistency with the above discussion.

The extent of emission pattern expansion can also be extracted from our analysis. More specifically, assigning *θ* = *π*/2 to Eq. (19) yields the FOLED angle which corresponds to the PPOLED critical angle, i.e. defines the maximal possible viewing angle of the FOLED, for a given radius of curvature

*φ*−

*φ*′| >

*π*/2. It is not guaranteed that significant power will be back-illuminated; this is mainly dependent on the IS interference, which predominantly determines the extrema angles (e.g., see differences between blue solid plots and red dashed plots in Figs. 3(a)–3(g)).

Further reduction of the radius of curvature below *R*_{opt} yields *k*_{4}*a*_{3} ≥ *k*_{3}*a*_{2}, and the second term of Eq. (20) becomes invalid. In this scenario, the radius of curvature is so small that the limiting reflection coefficient is no longer related to the substrate/air interface, as all possible rays incident upon it with angles below the critical angle (no total internal reflection (TIR) at the substrate/air interface). Instead, the reflections from the anode/susbtrate interface, Γ_{2}, become the dominant factor of the DR transmission. Interestingly, it can be shown that the same *R*_{opt} of Eq. (21) maximizes the viewing angle for both cases *k*_{4}*a*_{3} ≷ *k*_{3}*a*_{2}. Therefore for a given set of materials there is an optimal radius of curvature, *R*_{opt}, which maximizes the viewing angle.

For the prototype device, Eq. (21) yields *R*_{opt} ≈ 175*μ**m*, and indeed when we cross this value, going from *R* = 200*μ**m* (Fig. 3(e)) to *R* = 100*μ**m* (Fig. 3(f)) we observe the reversal of the ”stretching” effect. Moreover, the maximal viewing angle for a given radius of curvature may be extracted by drawing a tangent to the corresponding emission pattern as it approaches its zero. For *R* = 200 *μ**m*, relatively close to *R*_{opt}, we find that |*φ* − *φ*′|_{view} ≈ 120° (Fig. 3(e)). This is in consistency with Eq. (20), which analytically predicts this value to be 123.3°. This forms another efficient design rule.

#### 3.2. Total internal reflections and light escape cone

The second phenomenon that we observe is related to the first, as the same physical mechanism rules them both. As the radius of curvature decreases, an enhancement of the side lobe emission with respect to the forward-illumination (FI, |*φ* − *φ*′| = 0) is observed (Figs. 3(a)–3(g)), particularly pronounced for the *z*′ = 140nm case. As the IS interference is hardly affected by the curvature, and at |*φ* − *φ*′| = 0 so is the DR transmission, the FI should hold a constant level for all plots of Figs. 3(a)–3(g). This implies that a dramatic enhancement in the total device emission is achieved upon application of bending, due to the enhanced emission to large angles.

In order to formally assess this enhancement we utilize the concepts of ”escape cone” and ”escape efficiency”, which are commonly used for analyzing the TIR losses in LEDs [21–23]. In the frame of the ”escape” approximation, one refers to the source as omni-directional and considers rays which incident the various interfaces with angles below the critical angle as fully-transmitted and rays which incident one of the interfaces with an angle above the critical angle as totally-reflected. In 3D configurations, a cone is formed by the critical angle, and it is termed the ”escape cone”; in 2D configurations, such as the one considered herein, this cone is reduced to an ”escape triangle”. Consequently, the ”escape efficiency” is defined as the power of the outcoupled emission divided by the total source power, under these assumptions [21]. This is, of course, an approximation, as it does not take into account the specific reflection coefficient for each individual ray, non-radiative processes, the fluorescence efficiency of the emitting specie, the angular distribution of the source emission, multiple reflections etc. [24, 25]; however, it gives a good qualitative analysis of the effect of the TIR on the extraction efficiency, using very intuitive and simple relations. For a bottom-emitting PPOLED the 3D and 2D substrate escape efficiencies are given by [22]

*θ*

*=*

_{c}*k*

_{4}/

*k*

_{3}.

On the other hand, when we consider the curved FOLED, the critical angle for the substrate/air interface is larger due to the curvature effect. In order to zero |1 + Γ_{3}|, the ray should incident the substrate/air interface with an angle *α*_{3,3} which satisfies the TIR condition, namely (*α*_{3,3})* _{c}* =

*θ*

*, as for the planar case; by definition, this implies*

_{c}*ν*=

*k*

_{4}

*a*

_{3}sin (

*π*/2). However, the angle in which the same critical ray departs the anode/substrate interface, (

*α*

_{3,2})

*, and which determines the escape cone or triangle, may be much larger, and is given by assigning*

_{c}*ν*=

*k*

_{4}

*a*

_{3}into the suitable definition

*α*

_{3,3})

*=*

_{c}*θ*

*departed from the anode/substrate boundary at a larger angle (*

_{c}*α*

_{3,2})

*>*

_{c}*θ*

*. The dotted blue line shows the trajectory of the same ray for a planar formation, indicating that the same ray that was outcoupled to air in the curved FOLED, undergoes TIR in the corresponding PPOLED, hence the enhanced escape efficiency. Consequently, we should generalize the escape efficiency expressions to account for the enhancement due to a finite radius of curvature, namely,*

_{c}*R*≫

*d*

_{3}. It should be noted that the generalized 3D escape efficiency of Eq. (24) is valid only when the bending is applied uniformly across three dimensions, i.e. for 3D spherical FOLEDs; for the 2D cylindrical FOLED considered herein, only

*η*

_{esc,2D}is applicable.

This is a very important result. It indicates that the escape efficiency (of substrate modes) can be increased by decreasing the radius of curvature. Moreover, it shows that for *R*_{opt} defined in Eq. (21) the escape efficiency is 1, i.e. none of the rays departing the anode/substrate interface incidents the substrate/air boundary at an angle larger than the critical angle.

A quantitative estimate of the potential improvement in FOLED performance due to this effect is
provided by Fig. 4, where we plotted the
outcoupling efficiency enhancement as a function of the radius of curvature, for
the two previously considered emission zone locations of the prototype FOLED,
*z*′ = 20nm and *z*′
= 140nm. The graph presents the outcoupling efficiencies of the bent
device normalized by the outcoupling efficiencies of the corresponding PPOLED,
both calculated by numerical integration over angle of the emission patterns of
Fig. 3, rescaled according to FI. For
comparison, we also plot our analytical closed-form estimate for this
enhancement, i.e. the escape efficiency enhancement, given by ${\eta}_{\text{esc},2\text{D}}^{\text{FOLED}}(R)/{\eta}_{\text{esc},2\text{D}}^{\text{PPOLED}}$.

Figure 4 shows that indeed, the outcoupling efficiency is enhanced as the radius of curvature decreases, reaching 1.3-fold or 4-fold improvement at *R* = *R*_{opt} when the emission zone is at *z*′ =20nm or *z*′ =140nm, respectively. The reason for this difference is that the IS interference of the *z*′ =20nm sources produces a quasi-Lambertian emission pattern, whereas the pattern produced by the *z*′ =140nm emitters contains significant side-lobes at large-angles (see Fig. 3(a)). Consequently, the latter suffers much more TIR losses than the former to begin with, and the outcoupling enhancement, which is related to the reduction of these losses, acts accordingly. Following the same argumentation, it is not surprising that the escape efficiency enhancement lies in between these two trends, as its derivation assumes an isotropic emitter. Therefore, it reaches an intermediate value of 2.3-fold enhancement, which also demonstrates the quality of this parameter as an approximate measure for the outcoupling efficiency.

The last discussion is of high importance as it addresses one of the most severe problems which to-date limits OLED outcoupling efficiency, namely the TIR at the substrate/air interface, which accounts for ∼25% of the losses [24, 25]. Over the years different methods have been proposed to overcome this inherent refractive index mismatch and extract these substrate modes, e.g. by using high-index substrate and a matching half-sphere [26] or microlens array [27] to reduce incident angle, low-index thin-film to mediate the refractive mismatch [22], or low-index grid to reduce probability of supercritical incidence [28]. However, all of these device modifications require unique processing methods or introduction of specialized materials, whereas the presented enhancement effect induced by the FOLED curvature is achieved without additional effort, and is inherent to the geometry formed once the device is bent, an action the device was mechanically and electrically designed to perform particularly well.

It is worth noting that significant enhancement of FOLED outcoupling efficiency can be achieved with much more moderate bending if a thicker substrate is used, as the curvature effects depend only on the ratio between *R* and *d*_{3}. Nevertheless, there have been demonstrations of flexible organic electronic devices which remained functional at radii of curvatures below 1mm [29], indicating the principled feasibility of the proposed enhancement concept.

## 4. Conclusion

We have presented a rigorous solution to the Maxwell equations for a 2D source embedded in a concentric cylindrical shell formation, and formulated closed-form analytical formulae for the FOLED emission pattern; utilization of the ”perfect azimuthal absorber” methodology enabled natural convergence of the formulation to the PPOLED limit for infinite radius of curvature. Based on the resultant expressions, a geometrical ray-optics interpretation was given, and two important phenomena induced by the device curvature were observed and discussed. The first is the ”stretching” of the emission pattern features towards larger observation angles, an effect which allows BI. The second phenomenon is the reduction of substrate/air TIR losses, an effect which enhances dramatically the outcoupling efficiency of substrate modes, which is one of the main barriers for improving OLED external efficiencies. A modified expression for the FOLED escape efficiency was formulated, clearly and intuitively indicating the curvature induced efficiency enhancement. For both effects an optimal radius of curvature was found analytically.

This is the first time, to the best of our knowledge, that such a formulation of the optical fields and emission of FOLEDs is presented, followed by an analytical investigation of the curvature effects. The presented work is especially important in the face of the novelty of the flexible devices, revealing unique phenomena, and offering a set of design rules which relate the curvature to optical properties, thus can be efficiently utilized by engineers for device optimization.

## Acknowledgments

A. E. gratefully acknowledges the support of The Clore Israel Foundation Scholars Programme.

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