## Curvature effects on optical emission of flexible organic light-emitting diodes |

Optics Express, Vol. 20, Issue 7, pp. 7929-7945 (2012)

http://dx.doi.org/10.1364/OE.20.007929

Acrobat PDF (2078 KB)

### 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 OSA

## 1. Introduction

14. W. Wasylkiwskyj, “Diffraction by a concave perfectly conducting circular cylinder,” IEEE Trans. Antennas Propag. **23**, 480–492 (1975). [CrossRef]

16. A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. **46**, 1388–1395 (2010). [CrossRef]

17. A. Epstein, N. Tessler, and P. D. Einziger, “Analytical extraction of the recombination zone location in organic light-emitting diodes from emission pattern extrema,” Opt. Lett. **35**, 3366–3368 (2010). [CrossRef] [PubMed]

## 2. Theory

### 2.1. Formulation

*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*).

13. P. Einziger and L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. **31**, 863–870 (1983). [CrossRef]

18. P. Einziger and L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. **31**, 870–884 (1983). [CrossRef]

*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

16. A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. **46**, 1388–1395 (2010). [CrossRef]

*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

16. A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. **46**, 1388–1395 (2010). [CrossRef]

*i*th kind at the

*l*th layer side of the

*m*th interface,

*ν*, is usually omitted and can be inferred from the context.

**46**, 1388–1395 (2010). [CrossRef]

*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

*l̂*

_{−1}=

*l*′ +

*l*

_{1}and

*p*(

*ω*) [16

**46**, 1388–1395 (2010). [CrossRef]

*n*th cylindrical shell.

### 2.3. Debye approximation and saddle point evaluation

13. P. Einziger and L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. **31**, 863–870 (1983). [CrossRef]

*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, For the optical wavelengths considered, the condition |Ω| ∼ |*

_{l,m}*k*

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

*([12], pp. 382–391). In contrast with the saddle point evaluation for the plane-parallel structure [16*

_{l,m}**46**, 1388–1395 (2010). [CrossRef]

*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*′,

13. P. Einziger and L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. **31**, 863–870 (1983). [CrossRef]

18. P. Einziger and L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. **31**, 870–884 (1983). [CrossRef]

*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*

_{,n}**31**, 863–870 (1983). [CrossRef]

18. P. Einziger and L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. **31**, 870–884 (1983). [CrossRef]

*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*′,

*φ*−

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

**31**, 870–884 (1983). [CrossRef]

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

1. G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature **357**, 477–479 (1992). [CrossRef]

2. G. Gu, P. E. Burrows, S. Venkatesh, S. R. Forrest, and M. E. Thompson, “Vacuum-deposited, nonpolymeric flexible organic light-emitting devices,” Opt. Lett. **22**, 172–174 (1997). [CrossRef] [PubMed]

6. T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. **8**, 494–499 (2009). [CrossRef] [PubMed]

9. B. Park and H. G. Jeon, “Spontaneous buckling in flexible organic light-emitting devices for enhanced light extraction,” Opt. Express **19**, A1117–A1125 (2011). [CrossRef] [PubMed]

**46**, 1388–1395 (2010). [CrossRef]

8. C.-J. Chiang, C. Winscom, and A. Monkman, “Electroluminescence characterization of FOLED devices under two type of external stresses caused by bending,” Org. Electron. **11**, 1870–1875 (2010). [CrossRef]

*M*= 1 and

*N*= 3) with an electric line source excitation located at (

*ρ*′,

*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

17. A. Epstein, N. Tessler, and P. D. Einziger, “Analytical extraction of the recombination zone location in organic light-emitting diodes from emission pattern extrema,” Opt. Lett. **35**, 3366–3368 (2010). [CrossRef] [PubMed]

20. M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. **11**, 1039–1046 (2010). [CrossRef]

*d*

*from the cathode/organic boundary, where the curvature effect is introduced by*

_{n}*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*).

*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*

*), for |*

_{n}*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 which can be neglected, as normally Ω

_{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.

*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.

*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.

*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, the simplified Image-Source (IS) transmission factor reads where we define the IS interference cross-term as*

_{l}**46**, 1388–1395 (2010). [CrossRef]

**46**, 1388–1395 (2010). [CrossRef]

*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

**46**, 1388–1395 (2010). [CrossRef]

*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.

## 3. Results and discussion

*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

**46**, 1388–1395 (2010). [CrossRef]

*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

**46**, 1388–1395 (2010). [CrossRef]

*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

**46**, 1388–1395 (2010). [CrossRef]

*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

*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.

17. A. Epstein, N. Tessler, and P. D. Einziger, “Analytical extraction of the recombination zone location in organic light-emitting diodes from emission pattern extrema,” Opt. Lett. **35**, 3366–3368 (2010). [CrossRef] [PubMed]

*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.

*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

*α*

_{4,3}= cos

*θ*[16

**46**, 1388–1395 (2010). [CrossRef]

*ν*=

*k*

_{4}

*a*

_{3}sin

*θ*. Substituting this into Eq. (12) yields for a far field observation point which implies that |

*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.

*R*= 200

*μ*

*m*.

*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.

*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 and for a given set of materials, this expression can be maximized by choosing an optimal radius of curvature Equations (20)–(21) indicate that BI can be achieved for FOLED, i.e. we can design our FOLED in such a way that a substantial amount of the emission will reach the back of the device, |

*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)).

*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

21. E. F. Schubert, *Light-Emitting Diodes* (Cambridge University Press, 2006). [CrossRef]

23. M. Ma, F. W. Mont, X. Yan, J. Cho, E. F. Schubert, G. B. Kim, and C. Sone, “Effects of the refractive index of the encapsulant on the light-extraction efficiency of light-emitting diodes,” Opt. Express **19**, A1135–A1140 (2011). [CrossRef] [PubMed]

21. E. F. Schubert, *Light-Emitting Diodes* (Cambridge University Press, 2006). [CrossRef]

24. S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. **104**, 123109 (2008). [CrossRef]

25. R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. **97**, 253305 (2010). [CrossRef]

22. T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. **13**, 1149–1152 (2001). [CrossRef]

*θ*

*=*

_{c}*k*

_{4}/

*k*

_{3}.

_{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 This is merely a manifestation of the law of sines, as demonstrated in Fig. 3(h), where the solid blue ray impinging upon the curved substrate/air interface at the critical angle (

*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, which, as expected, respectively converge to Eq. (22) in the planar limit*

_{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.

*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.

*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

24. S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. **104**, 123109 (2008). [CrossRef]

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22. T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. **13**, 1149–1152 (2001). [CrossRef]

28. Y. Sun and S. R. Forrest, “Enhanced light out-coupling of organic light-emitting devices using embedded low-index grids,” Nat. Photonics **2**, 483–487 (2008). [CrossRef]

*R*and

*d*

_{3}. Nevertheless, there have been demonstrations of flexible organic electronic devices which remained functional at radii of curvatures below 1mm [29

29. T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. **9**, 1015–1022 (2010). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature |

2. | G. Gu, P. E. Burrows, S. Venkatesh, S. R. Forrest, and M. E. Thompson, “Vacuum-deposited, nonpolymeric flexible organic light-emitting devices,” Opt. Lett. |

3. | N. Tessler, G. J. Denton, and R. H. Friend, “Lasing from conjugated-polymer microcavities,” Nature |

4. | D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature |

5. | B. Park, C. H. Park, M. Kim, and M. Han, “Polarized organic light-emitting device on a flexible giant birefringent optical reflecting polarizer substrate,” Opt. Express |

6. | T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. |

7. | C.-J. Chiang, C. Winscom, S. Bull, and A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. |

8. | C.-J. Chiang, C. Winscom, and A. Monkman, “Electroluminescence characterization of FOLED devices under two type of external stresses caused by bending,” Org. Electron. |

9. | B. Park and H. G. Jeon, “Spontaneous buckling in flexible organic light-emitting devices for enhanced light extraction,” Opt. Express |

10. | H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. |

11. | Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics |

12. | L. B. Felsen and N. Marcuvitz, |

13. | P. Einziger and L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. |

14. | W. Wasylkiwskyj, “Diffraction by a concave perfectly conducting circular cylinder,” IEEE Trans. Antennas Propag. |

15. | M. Abramowitz and I. A. Stegun, |

16. | A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. |

17. | A. Epstein, N. Tessler, and P. D. Einziger, “Analytical extraction of the recombination zone location in organic light-emitting diodes from emission pattern extrema,” Opt. Lett. |

18. | P. Einziger and L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. |

19. | A. Cherkassky, “Optimization of electromagnetic power absorption in a lossy circular cylinder,” Master’s thesis, Technion - Israel Institute of Technology (2006). |

20. | M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. |

21. | E. F. Schubert, |

22. | T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. |

23. | M. Ma, F. W. Mont, X. Yan, J. Cho, E. F. Schubert, G. B. Kim, and C. Sone, “Effects of the refractive index of the encapsulant on the light-extraction efficiency of light-emitting diodes,” Opt. Express |

24. | S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. |

25. | R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. |

26. | S. Mladenovski, K. Neyts, D. Pavicic, A. Werner, and C. Rothe, “Exceptionally efficient organic light emitting devices using high refractive index substrates,” Opt. Express |

27. | K.-Y. Chen, Y.-T. Chang, Y.-H. Ho, H.-Y. Lin, J.-H. Lee, and M.-K. Wei, “Emitter apodization dependent angular luminance enhancement of microlens-array film attached organic light-emitting devices,” Opt. Express |

28. | Y. Sun and S. R. Forrest, “Enhanced light out-coupling of organic light-emitting devices using embedded low-index grids,” Nat. Photonics |

29. | T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(230.3670) Optical devices : Light-emitting diodes

(250.3680) Optoelectronics : Light-emitting polymers

(310.6845) Thin films : Thin film devices and applications

**ToC Category:**

Optical Devices

**History**

Original Manuscript: December 22, 2011

Revised Manuscript: March 7, 2012

Manuscript Accepted: March 8, 2012

Published: March 21, 2012

**Citation**

Ariel Epstein, Nir Tessler, and Pinchas D. Einziger, "Curvature effects on optical emission of flexible organic light-emitting diodes," Opt. Express **20**, 7929-7945 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7929

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### References

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- B. Park, C. H. Park, M. Kim, M. Han, “Polarized organic light-emitting device on a flexible giant birefringent optical reflecting polarizer substrate,” Opt. Express 17, 10136–10143 (2009). [CrossRef] [PubMed]
- T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009). [CrossRef] [PubMed]
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- C.-J. Chiang, C. Winscom, A. Monkman, “Electroluminescence characterization of FOLED devices under two type of external stresses caused by bending,” Org. Electron. 11, 1870–1875 (2010). [CrossRef]
- B. Park, H. G. Jeon, “Spontaneous buckling in flexible organic light-emitting devices for enhanced light extraction,” Opt. Express 19, A1117–A1125 (2011). [CrossRef] [PubMed]
- H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011). [CrossRef]
- Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011). [CrossRef]
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- P. Einziger, L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. 31, 870–884 (1983). [CrossRef]
- A. Cherkassky, “Optimization of electromagnetic power absorption in a lossy circular cylinder,” Master’s thesis, Technion - Israel Institute of Technology (2006).
- M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010). [CrossRef]
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- T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. 13, 1149–1152 (2001). [CrossRef]
- M. Ma, F. W. Mont, X. Yan, J. Cho, E. F. Schubert, G. B. Kim, C. Sone, “Effects of the refractive index of the encapsulant on the light-extraction efficiency of light-emitting diodes,” Opt. Express 19, A1135–A1140 (2011). [CrossRef] [PubMed]
- S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104, 123109 (2008). [CrossRef]
- R. Meerheim, M. Furno, S. Hofmann, B. Lussem, K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97, 253305 (2010). [CrossRef]
- S. Mladenovski, K. Neyts, D. Pavicic, A. Werner, C. Rothe, “Exceptionally efficient organic light emitting devices using high refractive index substrates,” Opt. Express 17, 7562–7570 (2009). [CrossRef] [PubMed]
- K.-Y. Chen, Y.-T. Chang, Y.-H. Ho, H.-Y. Lin, J.-H. Lee, M.-K. Wei, “Emitter apodization dependent angular luminance enhancement of microlens-array film attached organic light-emitting devices,” Opt. Express 18, 3238–3243 (2010). [CrossRef] [PubMed]
- Y. Sun, S. R. Forrest, “Enhanced light out-coupling of organic light-emitting devices using embedded low-index grids,” Nat. Photonics 2, 483–487 (2008). [CrossRef]
- T. Sekitani, U. Zschieschang, H. Klauk, T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. 9, 1015–1022 (2010). [CrossRef] [PubMed]

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