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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7929–7945
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Curvature effects on optical emission of flexible organic light-emitting diodes

Ariel Epstein, Nir Tessler, and Pinchas D. Einziger  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7929-7945 (2012)
http://dx.doi.org/10.1364/OE.20.007929


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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

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

12. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

14

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

]. Furthermore, incorporating Deybe’s approximation [15

15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions : with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1970).

] 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

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]

], 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

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]

], 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 εn, μn and σn, respectively, for the nth layer. Furthermore, the nth and (n + 1)th layers are separated by the cylindrical shell ρ = an for n > 0 and ρ = an+1 for n < 0, and we define a0 = ρ′, aN+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 = aM, which is infinite for a PPOLED, and decreases as the FOLED is bent. Note that ε−1 = ε1, μ−1 = μ1 and σ−1 = σ1. 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 eI0 and mI0, 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 ejωt. The wave number and wave impedance of the nth layer are given as kn = ω{μnεn [1 − jσn/ (ωεn)]}½ = (ω/c)(nnjκn) and Zn = (μn/ {εn [1 − jσn/ (ωεn)]})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{kn} ≤ 0, leading to ℑ{Zn} ≥ 0. The 2D source vector is ρ⃗′ = (ρ′,φ′) and the observation point is denoted by ρ⃗ = (ρ).

Fig. 1 Two-dimensional configuration for the curved FOLED model. The source location is denoted by ρ⃗′ (red) and the observation point by ρ⃗ (blue). Only outbound azimuthal waves are taken into account by placing a ”perfect azimuthal absorber” at the azimuthal boundaries of the curved FOLED [13].

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

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

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

], 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

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]

]). 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. |knR| ≫ 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

Table 1. Wave Equation and Boundary Conditions for the 1D and 2D Cylindrical Green Functions

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In analogy to the planar model solution [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]

], 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 Hl,m(i) to indicate the Hankel function of the ith kind at the lth layer side of the mth interface, Hν(i)(klam), and a similar convention for the derivative, Hl,m(i)=Hν(i)(Ω)/Ω|Ω=klam; the normalized derivative is consequently defined as hl,m(i)=Hl,m(i)/Hl,m(i), and the order of the functions, ν, is usually omitted and can be inferred from the context.

Table 2. Recursive Relations of the Cylindrical 1D Green Function

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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

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]

], namely
Sρ(φφ)=limρ8PN+1dωp(ω)GN+1(ρ,ρ)[ρGN+1(ρ,ρ)]*
(5)
where mePn=|meI0|2kn(Zn)±1/16 is the power density radiated by a line source in unbounded homogeneous medium sharing the optical properties of the nth 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

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]

], 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

12. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

], pp. 710–712; [15

15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions : with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1970).

])
Hν(1)(Ω)[(2/π)2Ω2ν2]1/4{exp{j[(Ω2ν2)1/2ν(π2arcsinνΩ)]jπ4}ν<|Ω|exp{[(ν2Ω2)1/2νarccoshνΩ]jπ2}ν>|Ω|
(6)
and the approximated Hankel function of the second order is readily derived from Eq. (6) using the identity Hν(2)(Ω)=[Hν(1)(Ω*)]*. We define Ωl,m = klam and sinαl,m = νl,m, which in conjunction with the definitions of Table 2 yield the following approximation for the phase factors,
wl,m=(w^l,m)1{e2jΩl,m(cosαl,m+αl,msinαl,m)ν<|Ωl,m|ej(πνπ2)ν>|Ωl,m|
(7)
For the optical wavelengths considered, the condition |Ω| ∼ |knR| ≫ 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, Ωl,m ([12

12. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

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

], 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, ejν(φ φ′) and ejν(φ φ′). Considering these arguments, the saddle point condition for the multiple reflection term defined by a certain set of indices (lq, q, l′, l˜qp) in Eq. (4) reads [13

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

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

]
±(φφ)+n=1N+1(αn,nαn,n1)+2n=1N(αn,nαn,n1)(n+ln+p=2nl˜np)+2n=M1(αn,n+1αn,n)(l^n+1)+(l^(M+1)+1)(2α(M+1),Mπ)=0
(8)
As the azimuthal momentum values ν > |Ωl,m| constitute the evanescent spectrum of the problem’s Green function (complex αl,m), for far-field evaluation it is sufficient to consider solutions in the interval ν ∈ [0,Ωmin), where Ωmin = min {Ωn+1,n}. As was shown in detail by Einziger and Felsen [13

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

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

], the saddle point condition is a manifestation of the laws of geometrical ray-optics. In fact, Eq. (8) forms an ”angular conservation law”, as αn,n and αn,n−1 indicate the angles between the ray trajectory in the nth layer and the normal to the nth 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 (αn,nαn,n−1). In that sense, the saddle point value, νs (lq, q, l′, l˜qp), 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, (φφ′), are taken into consideration (See Fig. 4 of [18

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

]).

2.4. Closed-form solution for prototype device

Fig. 2 (a) Physical configuration of the prototype FOLED specified in Table 2.4. (b) For very large radius of curvature with respect to the device dimensions, the prototype PPOLED is received. A polar coordinate system (ρ̃,θ) is defined as in [16] (note the different origin).

Table 3. Geometrical and Electrical Properties of a Prototype FOLED, Corresponding to Fig. 1*

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We observe three important typical features of such a device. First, as knR ≫ 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 ≫ (dn+1dn), for |n| ≤ 1; this can be written alternatively as Δan+1,n = (an+1an) ≪ an, |n| ≤ 1. Thus we may approximate these layers’ contribution to the saddle point condition (Eq. (8)) by
αn+1,n+1αn+1,nΔan+1,nantanαn+1,n
(11)
which can be neglected, as normally Ωn+1,n ≫ Ωmin > 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, (−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.

Utilizing these features of the prototype FOLED yields the following reduced form of the saddle point condition,
|φφ|+(α4,4α4,3)+[2(l3+1)+1](α3,3α3,2)=0
(12)
where we recall that (l3 + 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. GN+1(ρ,ρ)GN+1±(ρ,ρ) for (φφ′) ≷ 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
G4(ρ,ρ)12πρρl1=1lmaxs1=0l1+1s2=0l2+1{ejν|φφ|2jk4cosα4,4tIS(ν)tDR(ν)KWM(l1,s1,s2)(ν)}ν=νs(l3)
(13)
where the azimuthal wavenumber at the saddle point, ν = νs (l3), 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 lmax = 3. Utilizing the definition of the optical path, Ln,m,l, of a ray propagating in the nth layer from ρ = am to ρ = al, namely,
knLn,m,l=Ωn,m(cosαn,m+αn,msinαn,m)Ωn,l(cosαn,l+αn,lsinαn,l)
(14)
the simplified Image-Source (IS) transmission factor reads
tIS(ν)=1Γ^1exp{2jk1L1,0,1}=l=01KIS(l)(ν)
(15)
where we define the IS interference cross-term as KIS(l)(ν)=(Γ^1)le2jlk1L1,0,1; the simplified Direct-Ray (DR) transmission factor is given by
tDR(ν)=n=24(1+Γn1)n=14cosαn,ncosαn,n1exp{jknLn,n,n1}
(16)
; and the simplified WM multiple reflection combination is defined as
KWM(l1,s1,s2)(ν)=1Ql3(ν)(l1+1s1)(l2+1s2)[Γ^1Γ1]s1[Γ^1(1Γ12)Γ2]s2[Γ^1(1Γ12)(1Γ22)Γ3]l3+1exp{2j[(l1+1)k1L1,1,1+(l2+1)k2L2,2,1+(l3+1)k3L3,3,2]}
(17)

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

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]

]; 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., (1Γ12)). 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

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]

]. These attenuation factors arise from the convolution of the emission pattern with the spectral distribution, assumed to be Gaussian with width of Δω, and are squared exponential in the ratio between the ray total optical path and the source ensemble coherence length, Lc = cω. In OLEDs, as was discussed in detail in [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]

], due to the typical small coherence length with respect to the substrate thickness (Lcd3), 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
Sρ(|φφ|)P4πρl1=13s1=0l1+1s2=0l2+1l1¯=13s1¯=0l1¯+1l=01l¯=01{|tDR(ν)|2k4ρcosα4,4KIS(l)(ν)KWM(l1,s1,s2)(ν)[KIS(l¯)(ν)KWM(l1¯,s1¯,s2¯)(ν)]*exp{12(k4Lc)2[2(ll¯){k1L1,0,1}+2(l1l1¯){k1L1,1,1}+2(l2l2¯){k2L2,2,1}]2}}ν=νs(l3=l3¯)
(18)
where it follows from the latter discussion regarding the spectral distribution effect that only terms with l3=l3¯ should be taken into account, which implies s2¯=l3l1¯+s1¯.

3. Results and discussion

Emission patterns for the prototype FOLED, Sρ (|φφ′|), 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, Sρ̃ (θ), calculated according to the formulation in [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]

]. 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

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]

]. As customary, the emission patterns are normalized according to the maximal value.

Fig. 3 Emission patterns of (a) the prototype PPOLED and (b)–(g) the prototype FOLED with radii of curvature varying from (b) R = 2000μm to (g) R = 50μm. The emission patterns are plotted for two source-cathode separations of z′ = 20nm (red dashed line) and z′ = 140nm (blue solid line). For the z′ = 140nm case, the local maximum angle is denoted in the legend of each subplot. (h) Illustration of the main curvature effects, emphasizing the difference between the planar (dotted lines and captions) and the curved (solid lines and captions) devices. The red lines demonstrate the behavior of a ray which departs the anode/substrate interface in an angle below the critical angle for the substrate/air interface, θc, and the blue lines follow a ray with an angle of departure above θc. Normals to the interfaces at the point of incidence are denoted by thin dotted lines. The geometrical ray-optics interpretation indicates that the angle of incidence, and hence the local reflection coefficients, at the substrate/air interface are always smaller for the curved FOLED than for the PPOLED.

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 50μm < d3 (Fig. 3(g)) to 2000μm ≫ d3 (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 Rd3 we may utilize Eq. (11) for n = 3, reducing the saddle point condition of Eq. (12) to its PPOLED limit, ρsinα4,4 = a3 sin (α4,4 + 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

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]

]. 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 (Rd3), 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, |φφ′|max, in which this extremum is achieved is observed. This effect is reversed as we cross from Fig. 3(e) (R = 200μm) to Fig. 3(f) (R = 100μm), 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 = 200μm (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

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]

]. 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 |φφ′| which yields the same substrate/air reflection coefficient, Γ3, as the one received when the PPOLED observation angle is, i.e. we require that Γ3PPOLED(θ)=Γ3FOLED(|φφ|). This implies cosα4,3 = cosθ [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]

], and consequently ν = k4a3 sinθ. Substituting this into Eq. (12) yields for a far field observation point
|φφ|=θ+arcsin(k4a3k3a2sinθ)arcsin(k4k3sinθ)
(19)
which implies that |φφ′| > 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. |1+Γ3PPOLED(θ=|φφ|)|<|1+Γ3FOLED(|φφ|)|. 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 = 200μm.

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, θ3 = α3,2, 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 α3,3. 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, θ3. According to the law of sines, α3,3 < θ3, 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
|φφ|view(R)=π/2+arcsin(k4a3k3a2)arcsin(k4k3)
(20)
and for a given set of materials, this expression can be maximized by choosing an optimal radius of curvature
a3a2=k3k4Roptk4k3k4d3|φφ|view(Ropt)=πarcsin(k4k3)
(21)
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, |φφ′| > 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)).

For the prototype device, Eq. (21) yields Ropt ≈ 175μm, and indeed when we cross this value, going from R = 200μm (Fig. 3(e)) to R = 100μm (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 μm, relatively close to Ropt, we find that |φφ′|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

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

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

23

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]

]. 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

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

]. 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

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

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]

]; 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

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]

]
ηesc,3DPPOLED=1cosθc=11(k4k3)2,ηesc,2DPPOLED=2θcπ=2πarcsin(k4k3)
(22)
where we considered the substrate/air reflection coefficient to be the limiting factor of the DR transmission, thus sinθc = k4/k3.

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 + Γ3|, the ray should incident the substrate/air interface with an angle α3,3 which satisfies the TIR condition, namely (α3,3)c = θc, as for the planar case; by definition, this implies ν = k4a3 sin (π/2). However, the angle in which the same critical ray departs the anode/substrate interface, (α3,2)c, and which determines the escape cone or triangle, may be much larger, and is given by assigning ν = k4a3 into the suitable definition
sinαc=sin(α3,2)c=a3a2sinθc=k4a3k3a2
(23)
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 (α3,3)c = θc departed from the anode/substrate boundary at a larger angle (α3,2)c > θc. 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,
ηesc,3DFOLED(R)=1cosαc=11(k4a3k3a2)2,ηesc,2DFOLED(R)=2αcπ=2πarcsin(k4a3k3a2)
(24)
which, as expected, respectively converge to Eq. (22) in the planar limit Rd3. 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 ηesc,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 Ropt 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 ηesc,2DFOLED(R)/ηesc,2DPPOLED.
Fig. 4 Outcoupling efficiency enhancement (with respect to the corresponding PPOLED) as a function of radius of curvature for the prototype FOLED, with emission zones located at z′ = 20nm (red circles) or z′ = 140nm (blue × symbols), along with the 2D substrate escape efficiency enhancement (green solid line, calculated according to Eqs. (24) and (22)).

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

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

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]

]. 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

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 17, 7562–7570 (2009). [CrossRef] [PubMed]

] or microlens array [27

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 18, 3238–3243 (2010). [CrossRef] [PubMed]

] to reduce incident angle, low-index thin-film to mediate the refractive mismatch [22

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]

], or low-index grid to reduce probability of supercritical incidence [28

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]

]. 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 d3. 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]

], 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.

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 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]

3.

N. Tessler, G. J. Denton, and R. H. Friend, “Lasing from conjugated-polymer microcavities,” Nature 382, 695–697 (1996). [CrossRef]

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 459, 234–238 (2009). [CrossRef] [PubMed]

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 17, 10136–10143 (2009). [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]

7.

C.-J. Chiang, C. Winscom, S. Bull, and A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. 10, 1268–1274 (2009). [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]

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]

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. 98, 151904 (2011). [CrossRef]

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 5, 753–757 (2011). [CrossRef]

12.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

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]

14.

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

15.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions : with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1970).

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]

18.

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

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. 11, 1039–1046 (2010). [CrossRef]

21.

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

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]

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]

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 17, 7562–7570 (2009). [CrossRef] [PubMed]

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 18, 3238–3243 (2010). [CrossRef] [PubMed]

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]

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]

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

  1. G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, 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, M. E. Thompson, “Vacuum-deposited, nonpolymeric flexible organic light-emitting devices,” Opt. Lett. 22, 172–174 (1997). [CrossRef] [PubMed]
  3. N. Tessler, G. J. Denton, R. H. Friend, “Lasing from conjugated-polymer microcavities,” Nature 382, 695–697 (1996). [CrossRef]
  4. D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009). [CrossRef] [PubMed]
  5. 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]
  6. 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]
  7. C.-J. Chiang, C. Winscom, S. Bull, A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. 10, 1268–1274 (2009). [CrossRef]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  12. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).
  13. P. Einziger, L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. 31, 863–870 (1983). [CrossRef]
  14. W. Wasylkiwskyj, “Diffraction by a concave perfectly conducting circular cylinder,” IEEE Trans. Antennas Propag. 23, 480–492 (1975). [CrossRef]
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions : with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1970).
  16. A. Epstein, N. Tessler, 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, 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]
  18. P. Einziger, L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. 31, 870–884 (1983). [CrossRef]
  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, A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010). [CrossRef]
  21. E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006). [CrossRef]
  22. 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]
  23. 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]
  24. 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]
  25. 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]
  26. 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]
  27. 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]
  28. 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]
  29. 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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