## Abstract

In this paper, numerical algorithms for extraction of opto-electronic material and device parameters in organic light-emitting devices (OLEDs) are presented and tested for their practical use. Of particular interest is the extraction of the emission profile and the source spectrum. A linear and a nonlinear fitting method are presented and applied to emission spectra from OLEDs in order to determine the shape of the emission profile and source spectrum. The motivation of the work is that despite the existence of advanced numerical models for optical and electronic simulation of OLEDs, their practical use is limited if methods for the extraction of model parameters are not well established. Two fitting methods are presented and compared to each other and validated on the basis of consistency checks. Our investigations show the impact of the algorithms on the analysis of realistic OLED structures. It is shown that both fitting methods perform reasonably well, even if the emission spectra to be analyzed are noisy. In some cases the nonlinear method performs slightly better and can achieve a perfect resolution of the emission profile. However, the linear method provides the advantage that no assumption on the mathematical shape of the emission profile has to be made.

© 2010 OSA

## 1. Introduction

Organic light-emitting devices (OLEDs) are ideal light sources for future display applications, general lighting and chemical sensors [1] for their low power consumption and beneficial emission characteristics. The outcoupling efficiency of OLEDs is sensitive to the shape of the emission profile. Unfortunately, this shape is typically unknown and the analysis of light extraction is therefore often based on the assumption of a delta-shaped emission profile [2,3]. The emission profile is determined by the recombination profile and exciton diffusion mechanisms, as well as other electrical and device parameters [4–6]. Already in the early days, Tang used luminescent sensing layers to estimate the shape of the emission profile [7]. Recently, experimental investigations were conducted using a combinatorial method for device fabrication for comprehension of the emission profile and its relation to electrical parameters [8,9]. In recent years, models have been developed to accurately simulate the optical and electrical behavior of OLEDs [10–13]. With the help of optical simulation, Roberts illustrated how the shape and location of the emission profile is related to the outcoupling efficiency [14]. There have also been several reports, where simulation results were combined with a fitting method to extract the emission profile. Kuma et al. measured electroluminescence (EL) spectra, which were used for a least-square fit of simulated EL spectra as a function of the emission profile [15]. Generally, a superposition of simulated emission spectra originating from different positions of the emissive dipoles in the light-emitting layer has been used for the determination of the emission profile [12,15–17]. van Mensfoort et al. [17] have presented new aspects of parameter extraction and introduced an asymmetric analytical shape for the description of the emission profile in single-layer polymer LEDs, which goes to zero at the layer boundaries. They studied spectral and angular emission data and estimated the resolution of the method based on the condition number of a related matrix equation. However, we are not aware of any publication where different fitting methods and aspects of parameter extraction (i.e. a linear fitting method vs. a nonlinear fitting method, the incorporation of angular information, multi-emitter OLEDs, noise on the emission spectrum, etc.) have been summarized, evaluated with adequate examples and validated on the basis of consistency checks.

The objective of this study is thus to present and test numerical fitting algorithms for the extraction of the emission profile and source spectrum. This is achieved by an optical model, where a transfer-matrix theory approach for multi-layer systems is used in combination with a dipole emission model. The optical model is implemented in the semiconducting emissive thin film optics simulator (SETFOS) [18]. With SETFOS, we simulate the emission spectrum of an OLED based on an assumed emission profile together with a known source spectrum. The fitting methods are then applied to the calculated emission spectra in order to estimate the emission profile and source spectrum. The comparison between the obtained and assumed emission profile and source spectrum is an indication of how successfully the inverse problem can be solved.

This article is divided into two chapters. In Sec. 2, the mathematical background of the fitting methods is presented. In Sec. 3, we compare the results obtained by the algorithms with simulated emission spectra in order to deliver a consistency check.

## 2. Theoretical background

The theoretical background of the fitting methods studied in this article is introduced in this section. It is distinguished between a linear fitting method and a nonlinear one. The linear fitting method relates the measured emission spectrum of an OLED linearly to the unknown emission profile parameters. These parameters directly form the emission profile, whereas the nonlinear algorithm makes use of an analytical description of the emission profile with several parameters that are nonlinearly related to the resulting emission spectrum. Normally, not more than three parameters are necessary to determine the shape of the emission profile, which typically is assumed to be be gaussian or exponentially shaped [19]. An alternative, asymmetric analytical shape has been introduced by van Mensfoort et al. [17]. So the latter fitting method takes advantage of a reduced number of unknown parameters. Note that our classification of linear an a nonlinear fitting method is motivated by their mathematical (and not physical) nature. Both methods consider a superposition of emission spectra originating from distinct dipole locations.

#### 2.1. The linear least-square method

### 2.1.1. Mathematical formulation of the problem

The linear least-square algorithm is used to determine the shape of the emission profile, given a measured emission spectrum from an OLED. For simplicity the mathematical formulation for the least-square problem is derived for only one emitter. The emitter is characterized by the emission profile. For the moment we also assume that the emission spectrum is measured in normal direction and therefore the light is unpolarized. In a later section of this paper, this approach is extended further to multiple emitters described by several emission profiles and emission spectra measured for several angles *θ*. The extracted emission profile *P _{e}*(

*δ*) is discretized at

_{j}*N*relative positions

*δ*=

_{j}*d*/

_{j}*L*in the light-emitting layer, where

*d*is expressed as an absolute position and

_{j}*L*is the width of the layer. The emission spectrum is divided into

*M*wavelengths

*λ*(

_{i}*i*= 1...

*M*). The fitted emission spectrum

*I*(

_{f}*λ*) can be written as

_{i}*I*(

_{c}*λ*,

_{i}*δ*) is the emission intensity for the wavelength

_{j}*λ*and assuming a discrete emission profile (dirac function) at the relative position

_{i}*δ*in the layer. The emission intensity is given by

_{j}*I*(

*λ*,

_{i}*δ*) is the emission intensity for emissive dipoles with spectrally constant intensity.

_{j}*S*(

*λ*) is the source spectrum. Between the measured emission spectrum

_{i}*I*(

_{m}*λ*) and fitted emission spectrum

_{i}*I*(

_{f}*λ*), a residuum can be defined and written as

_{i}*M*>

*N*) and thus ill-posed. In matrix notation, the problem can be formulated as

*r*

_{1}=

*A*·

*x*

_{1}–

*b*

_{1}. The matrix A has the following structure

*b*

_{1}is a vector containing the measured emission spectrum

*I*(

_{m}*λ*) and the vector

_{i}*x*

_{1}corresponds to the a priori unknown emission profile

*P*(

_{e}*δ*). The term linear refers now to the linear combination between the matrix

_{j}*A*and the vector

*x*

_{1}of unknown weights. In every column of the matrix

*A*, an emission spectrum is calculated for a dirac shaped emission profile at the position

*δ*. The emission profile

_{j}*P*(

_{e}*δ*) at the relative position

_{j}*δ*is the weight of the corresponding spectrum, respectively the column. The mathematical task is to minimize the length of the vector ||

_{j}*r*

_{1}||.

### 2.1.2. Analyzing systems of multiple emitters and emission angles

The most general case of the emission spectrum is determined by the emission profile of multiple emitters *P _{e}*(

*δ*) and emission angles

^{k}*θ*. Given that the emission spectrum is measured at

_{l}*O*different angles (

*l*= 1...

*O*) and the OLED consists of

*Q*different emitters (

*k*= 1...

*Q*) in the same or in separate layers, the relation stated in Eq. (1), combined with the definition of the residuum in Eq. (4), can be extended to

*λ*. We assume a dirac shaped emission profile at the relative position ${\delta}_{j}^{k}$ for emitter

_{i}*k*and an emission angle of

*θ*. ${P}_{e}\left({\delta}_{j}^{k}\right)$ is the emission profile at relative position

_{l}*δ*for emitter

_{j}*k*. Equation (6) represents a system of linear equations ${r}_{2}^{s,p}\hspace{0.17em}=\hspace{0.17em}{A}^{s,p}\cdot {x}_{2}\hspace{0.17em}-\hspace{0.17em}{b}_{2}^{s,p}$, where the matrix

*A*contains the s-polarized and p-polarized emission spectra, the vector

^{s,p}*x*

_{2}contains the information of several emission profiles and the vector ${b}_{2}^{s,p}$ represents the measured emission spectrum. The mathematical task is again to minimize the length of the vector $\left|\right|{r}_{2}^{s,p}\left|\right|$.

### 2.1.3. Extraction of the emission profile and source spectrum

In the case of a single emitter, van Mensfoort et al. [17] presented a method to extract the source spectrum of the light-emitting material. The source spectrum can be obtained by replacing the emission intensity
${I}_{c}^{s,p}\hspace{0.17em}\left({\lambda}_{i},\hspace{0.17em}{\delta}_{j}^{k},\hspace{0.17em}{\theta}_{l}\right)$ by the emission intensity for emissive dipoles with spectrally constant intensity *I ^{s,p}*(

*λ*, ${\delta}_{j}^{k}$,

_{i}*θ*) in Eq. (6). We employ and evaluate this method both in our linear and the nonlinear fitting method.

_{l}#### 2.2. The nonlinear fitting method

### 2.2.1. Definition of the error function

The nonlinear method uses a multivariable and multitarget optimization algorithm based on a conjugated gradient method in order to minimize the residuum. The residuum between the measured emission intensity ${I}_{m}^{s,p}\hspace{0.17em}({\lambda}_{i},\hspace{0.17em}{\theta}_{l})$ and calculated emission intensity from the model ${I}_{c}^{s,p}\hspace{0.17em}({\lambda}_{i},\hspace{0.17em}x,\hspace{0.17em}{\theta}_{l})$ is

*x*is a set of unknown parameters for the analytical description of the emission profile. The calculation of the emission intensity is nonlinear in terms of

*x*. The nonlinear method is applied to minimize the following RMS error function

In case of a gaussian or exponential emission profile, the set of parameters consists of the location of the peak (*r*) and the width (*w*), written as *x* = (*r*, *w*). In case of the parameterization according to van Mensfoort et al. [17], the set of parameters is the location of the peak (*p*), the width (*w*) and the asymmetry (*a*), written as *x* = (*p*, *w*, *a*). The vector *x* can also include multiple emission profiles. For a multi-emitter OLED, where all the emission profiles are gaussian shaped, the set of parameters extends to *x* = (*r*_{1}, *w*_{1}, *r*_{2}, *w*_{2}, ...*r _{N}*,

*w*). For a single emitter in the nonlinear case, the source spectrum can be extracted analogously as stated in Sec. 2.1.3.

_{N}## 3. Results

In this chapter, we address the reliability and limitations of the linear and nonlinear fitting method after they were mathematically deduced and described in Sec. 2.1. and Sec. 2.2. First, we assume a given source spectrum from a light-emitting material. Then we also suppose an emission profile, stating where the dipoles are located in the device. In this paper we do not focus our discussion on exciton quenching effects and allow non-zero values for the emission profiles close to the electrodes. The effect of quenching and the use of parameter extraction algorithms in such circumstances is discussed in more detail elsewhere [20]. Further, the emission spectrum is generated by an optical dipole model described by Novotny [21] and available in the simulator SETFOS [18]. The calculated emission spectrum is used to solve the least-square problem in Eq. (6), respectively minimizing the error function *f*(*x*) as defined in Eq. (8). This allows the extraction of both, source spectrum and emission profile. The comparison of the extracted and assumed emission profile reveals the reliability of the presented algorithms. Throughout the paper, an open cavity and a cavity OLED are used for the consistency checks without loss of generality. The methods presented here may also be applied to small-molecule based OLEDs. Both OLEDs investigated here have a broad light-emitting polymer (LEP) layer of 120 *nm* in thickness. For the open cavity OLED, the light-emitting layer is embedded between a PEDOT:PSS anode and an aluminum cathode. The exact structure is depicted schematically in Fig. 1(a). With respect to an experimental setup, the diameter of the semi-sphere glass lens is at least an order of magnitude larger than the diameter of the OLED. In the case of the cavity OLED, the PEDOT:PSS and ITO layers are replaced by a 30 *nm* thick silver layer. The structure of the cavity OLED is illustrated in Fig. 1(b). Flämmich et al. [22] have recently elaborated on the orientation of the dipoles, which influences the outcoupling efficiency. For simplicity, we assume isotropically oriented dipoles here. However, we have verified that our nonlinear optimization algorithm can easily be applied to emission spectra to extract the orientation of the dipoles as well. In order to achieve an absolute quantity of the emission intensity and emission profile, the assumed current density in all considered devices is 10 *mA/cm*^{2}.

#### 3.1. Evaluation of the linear fitting method

### 3.1.1. Open cavity organic LEDs

Three different sets of emission spectra are calculated using three distinct gaussian emission profiles. The first profile has its peak at the relative position *r*_{a,1} = 0.3 and the width of the shape is *w _{a}*

_{,1}= 50

*nm*. The subscript

*a*stands for an assumed parameter of the emission profile. In addition, the parameters are numerated. The parameters of the second profile are

*r*

_{a,2}= 0.5 and

*w*

_{a,2}= 70

*nm*, respectively

*r*

_{a,3}= 0.9 and

*w*

_{a,3}= 70

*nm*for the third profile. Using the emission spectra in Fig. 2a as data and solving Eq. (6) leads to the extracted emission profiles plotted in Fig. 2(b). This plot shows the comparison between the extracted and assumed emission profiles. The extracted maxima have their peak at

*r*

_{e,1}= 0.33,

*r*

_{e,2}= 0.51, respectively at

*r*

_{e,3}= 0.67. The subscript

*e*describes an extracted parameter resulting from an emission spectrum fit. The peak position and width for the first two emission profiles is well represented. Only the fit results

*r*

_{e,3}= 0.67 deviates somewhat from the true position

*r*

_{a,3}= 0.9. Figure 2(a) shows that all three fitted emission spectra match the data very accurately. As we show further blow in Sec. 3.1.3, the agreement could be improved if the angular information would be incorporated in the analysis.

### 3.1.2. Cavity organic LEDs

Again three sets of emission spectra are calculated using the same three distinct gaussian emission profiles as in Sec. 3.1.1. Figure 3(a) shows the comparison between the emission spectra used for the linear fitting method and the fitted emission spectra.

Figure 3(a) illustrates the differences between the supposed shape and extracted shape of the emission profile. The extracted maxima obtained are *r*_{e,1} = 0.31, *r*_{e,2} = 0.51, respectively at *r*_{e,3} = 0.67. The fitted emission spectra match the calculated emission spectra very accurately. Compared to the open cavity OLED case, the extracted emission profiles have not changed considerably. In comparison the spectra in Fig. 2, the spectra show little dependence on the emission profile (see Fig. 3). Nevertheless, the extracted emission profiles [see Fig. 3(b)] reproduce the assumed ones very accurately. The agreement is similar as in the open cavity case [see Fig. 2(b)].

For illustrating how sensitive the emission spectrum is towards the position of the emitter, we include in our analysis a case where delta-shaped emission profiles are assumed at the following relative positions *r*_{e,1} = 0.3, *r*_{e,2} = 0.5 and *r*_{e,3} = 0.9. Figure 4(b) shows the reference delta-shaped emission profiles as vertical points. The lines stand for the extracted emission profiles from the calculated emission spectra in Fig. 4(a) depicted as points for the three assumed delta-shaped emission profiles. For comparison, Fig. 4(a) shows the fitted emission spectra as straight lines.

It can be seen from Fig. 4(a), that the absolute intensity emission value decreases, if the position of the assumed delta-shaped emission profile is shifted towards the cathode. Figure 4(b) indicates, that the extracted emission profiles are broadened with respect to the reference position. The peak position of the extracted emission profiles is accurately reproduced for the assumed emission profile at the relative position *r*_{a,1} = 0.3 and *r*_{a,2} = 0.5. In the third case where the assumed relative position is at *r*_{a,3} = 0.9, the estimated peak position is at the nearby cathode. This result is remarkable, given that we used a cavity OLED and perpendicular emission spectra only for this test. As the following section shows, the extracted emission profiles in general improves further, if addition angular information is considered.

### 3.1.3. Incorporation of angular information

This section the relevance of incorporating angular information in case of the linear algorithm. The experimental situation can be seen in the setup as depicted in Fig. 1, where the semi-sphere glass lens ensures that no refraction takes place at the glass-air interface. One emission profile is obtained from an emission spectrum calculated for an observation angle *θ* of 0 *deg*. The other is extracted from the s-polarized and p-polarized emission spectra for angles *θ* between 0 *deg* and 90 *deg*. One expects that the results extracted by the linear fit must improve when the angular information is taken into account. The device investigated is the open cavity OLED. It is assumed that the emission profile is gaussian distributed with *r _{a}* = 0.5 and

*w*= 50

_{a}*nm*.

Figure 5(a) shows the extracted emission profile for emission spectra calculated between an angle *θ* of 0 *deg* to 90 *deg* and an emission spectrum calculated with an angle *θ* of just 0 *deg*. The peaks found in both extracted emission profiles are at a similar relative position of *r _{e}* = 0.49, which is very close to the supposed one of

*r*= 0.5. As Fig. 5(a) shows, the fit where angular information is omitted results in a narrow shape of the emission profile. In the case where the angular information is taken into account, the resulting fit quality is improved. The RMS error between the assumed and extracted emission profile is reduced by a factor of 3 and the assumed emission profile can be reproduced very precisely. Figure 5(b) illustrates the radiance obtained by integration over the emission spectrum for each angle. It shows the difference between the radiance based on the emission spectrum calculated by the assumed emission profile and the one based on the emission spectrum resulting from the extracted emission profile. The radiances fit perfectly. In Fig. 5(c) the resulting emission spectrum calculated by the assumed emission profile is plotted. As a comparison, Fig. 5(d) illustrates the emission spectrum calculated by the extracted emission profile. The two angular emission spectra show hardly any difference.

_{a}### 3.1.4. Extraction of the source spectrum

The linear fitting method makes no assumption about the shape of the emission profile. In general, also no a priori statement about the source spectrum is necessary. The linear algorithm can calculate the source spectrum according to Sec. 2.1.3. based on the assumption that the dipoles emit light with spectrally constant intensity. For evaluation of this method, the emission intensity is calculated using an assumed source spectrum of the light-emitting material as plotted in Fig. 6(b). The device studied is again the open cavity OLED. The supposed emission profile is gaussian shaped with the peak at *r _{a}* = 0.4 and a width of

*w*= 40

_{a}*nm*. The shape is shown in Fig. 6(a). The linear extraction method is applied to the calculated emission spectrum. Figure 6(a) shows the comparison between the extracted emission profile, where the source spectrum is known and where it is left as another degree of freedom. In the latter case, we observe that the extracted source spectrum is extracted perfectly [see Fig. 6(b)], even though the emission profile is not reproduced very well [see Fig. 6(a)]. This suggests that it is a suitable method for estimating the source spectrum.

### 3.1.5. Multi-emitter organic LEDs

The extension to multiple emitters is demonstrated in the following example. Every emitter can be described by its representative emission profile. This allows to extract distinct emission zones of dye-doped or multi-color multi-layer OLEDs. The device used here is an open cavity OLED which has two emitters in the same layer. Such a situation is realized experimentally for an emissive host material and an emissive guest material. The emission profile of the first emitter is assumed to be gaussian shaped with the parameters *r*_{a,1} = 0.2 and *w*_{a,1} = 10 *nm*. The second emission profile is gaussian shaped with *r*_{a,2} = 0.6 and *w*_{a,2} = 30 *nm*. The emission spectrum used for the linear fitting method is shown in Fig. 7(a), where also the fitted emission spectrum is shown. In Fig. 7(b), the comparison between the different emission profiles is shown. The maxima of the extracted emission profiles are *r*_{e,1} = 0.20 for the first emitter, respectively *r*_{e,2} = 0.58 for the second emitter. One can see from Fig. 7(b) that the algorithm is able to reproduce the emission profiles of multi-emitter OLEDs.

### 3.1.6. Experimental noise on the emission spectrum

The last investigation of the linear fitting method deals with experimental noise on the emission spectrum. Normal distributed noise is added to a simulated emission spectrum as a function of the signal-to-noise ratio. The signal-to-noise ratio (*s*/*n*) is defined as

*max*(

*I*(

_{c}*λ*,

_{i}*θ*)) is the maximum emission intensity value and

_{l}*σ*is the standard deviation of the noise. The noise on the emission spectrum gives rise to a degradation of the extracted emission profile compared to the assumed as seen in Fig. 8(a). For a high signal-to-noise ratio (

*s*/

*n*= 100), the extracted emission profile is very close to the the assumed one. Only for poor signal-to-noise ratios (

*s*/

*n*≤ 40), the extracted emission profile diverges from the assumed one. This demonstrates that the fitting method is rather robust against experimental noise. The relative error between the extracted and the assumed emission profile is shown in Fig. 8(b) and approaches asymptotically a minimal error the less noise we add to the emission spectrum. As demonstrated in the investigations above where no noise is assumed on the emission spectra, there is always a small error between the extracted and assumed emission profile which is caused by the method of how we solve the least-square problem in Eq. (6).

#### 3.2. Evaluation of the nonlinear optimization method

### 3.2.1. Cavity organic LEDs

In Sec. 3.1.2, we showed the application of the linear method in the case of a cavity device. The linear method was not able to accurately reproduce the shape of emission profiles in some circumstances. This is true when light is emitted from dipoles which are located close to an electrode. The application of the nonlinear algorithm in this section will show if the results of the extracted emission profiles can be improved compared to the linear method, where the same cavity device is used. Also, the parameters of the three assumed emission profiles remain identical and again the analysis is performed without angular information.

Figure 9(b) illustrates the assumed and extracted emission profiles where the nonlinear method is used. In comparison, Fig. 9(a) shows the differences between the fitted and calculated emission spectra for all three tested parameter sets. The results clearly show that, unlike the linear method, the nonlinear algorithms can determine the assumed emission profile exactly. Hence, the differences in the emission spectra and emission profiles have vanished. However, one shall not neglect that the type of the expected emission profile must be known before an emission fit can be made.

In the case where delta-shaped emission profiles are assumed (see Fig. 4), we could reproduce the assumed relative positions *r*_{a,1} = 0.3, *r*_{a,2} = 0.5 and *r*_{a,3} = 0.9 very accurately using the nonlinear optimization method.

### 3.2.2. Incorporation of angular information

The application of the linear method in Sec. 3.1.3. showed an improved extracted emission profile compared to the assumed if we take angular information into account. This section will demonstrate the differences to the nonlinear method. The parameters of the emission profile as well as the device studied remain the same as in Sec. 3.1.3. The device is the open cavity device and a centrally located, gaussian shaped emission profile in the light-emitting polymer is assumed. The parameters are *r _{a}* = 0.5 for the peak position and

*w*= 50

_{a}*nm*for the width of the shape.

Figure 10(a) demonstrates the extracted emission profile compared to the assumed. The emission spectrum was calculated for angles *θ* between 0 *deg* and 90 *deg*. The location of the peak and width found during the optimization routine are *r _{e}* = 0.50 and

*w*= 50

_{e}*nm*. In Fig. 10(b), the radiance based on the fitted emission spectrum and the one based on the assumed emission spectrum is plotted. The nonlinear method showed better results than the linear method in comparison between Fig. 3 and Fig. 9. In this case, the results plotted in Fig. 10 are not improved with respect to the linear method presented in Fig. 5, except for the extraction of the emission profile, which is in this case perfectly represented. The comparison between the emission spectrum calculated from the assumed and extracted emission are illustrated in Fig. 10(c), respectively Fig. 10(d). The plots show very good agreement. Without angular information the emission profile extraction works well too (as exemplified above in Fig. 9).

### 3.2.3. Extraction of the source spectrum

If the source spectrum is not known, the nonlinear method can handle the extraction as well. To demonstrate this, the emission spectrum is calculated by an a priori known source spectrum and an assumed emission profile. This emission spectrum is used as a dataset to test the nonlinear algorithm. The error function *f* (*x*) between the dataset and the emission spectrum calculated by a source spectrum with spectrally constant intensity must be minimized. For the analysis, the open cavity OLED is used. The assumed emission profile is identical to the linear application test as described in Sec. 3.1.4. The assumed peak position is *r _{a}* = 0.4 and the width is

*w*= 40

_{a}*nm*.

Figure 11(a) shows the extracted emission profile. The obtained parameter set *x* for the emission profile by the nonlinear optimization is *r _{e}* = 0.39 for the peak location, respectively

*w*= 49

_{e}*nm*for the width. The extracted source spectrum is plotted in Fig. 11(b). The results show good agreement between the assumed and extracted source spectrum as well as for the emission profile. Compared to the extracted source spectrum in the linear case [see Fig. 6(b)], the result has not much improved. But the extraction of the emission profile in the nonlinear optimization case reproduces the assumed better than in the linear case [see Fig. 6(a)].

### 3.2.4. Multi-emitter organic LEDs

The application of the linear method to multi-emitter OLEDs showed some discrepancies between the extracted and assumed emission profile for the first emitter. We now repeat this multi-emitter test with the nonlinear method. The structure of the device is the open cavity OLED with the same emission profile parameterization as in the test in Sec. 3.1.5. Figure 12(b) presents the assumed and extracted emission profiles for the two emitters by the nonlinear method. It shows perfect agreement between the extracted emission profiles and assumed ones. Also the comparison between the emission spectrum used as data for the nonlinear method and the fitted emission spectrum displays perfect alignment as plotted in Fig. 12(a). This result (see Fig. 12) is superior to the linear fitting result above (see Fig. 9) since the emission profiles match perfectly.

### 3.2.5. Experimental noise on the emission spectrum

As discussed in the linear method section (see Sec. 3.1.6), the investigation with noise on the emission spectrum is also done for the nonlinear fitting method. The definition of the signal to noise ratio can be seen in Eq. (9). Figure 13(a) shows the extracted emission profiles as a function of different signal to noise ratios. Remarkably, the nonlinear fitting method is less sensitive to noise on the emission spectrum than the linear method, as depicted also in Fig. 13(b), which shows the signal to noise ratio dependence of the relative error between the extracted and assumed emission profile. Unlike the linear method, theoretically the nonlinear method is able to achieve a minimal error between the assumed and extracted emission profile which is zero if numerical round-off errors and noise on the emission spectrum can be neglected and the optical dipole model is accurate.

## 4. Conclusion and discussion

Two numerical methods to determine a priori unknown emission profiles and source spectra in organic LEDs were presented in this paper. One method is linear, whereas the other is nonlinear. The terminology linear and nonlinear refers to the parameters of the emission profile and how they are related to the emission spectrum. The methods allow the extraction of the emission profile and source spectrum from a measured emission spectrum of an OLED. The application and reliability of both methods were illustrated on the basis of consistency checks for an open cavity and a cavity device. For that purpose, an emission profile and source spectrum were assumed. The emission was calculated by a dipole model and used as data for the test of the two methods. The algorithms were applied to the data and the thereby extracted emission profile and source spectrum were compared to the assumed.

The big advantage of the linear method is that it makes no assumption about the shape of the emission profile. With the linear method, we demonstrated that the results of the extracted emission profiles improve when angular information is incorporated. The nonlinear method was able to reproduce the shape of the emission profile in all examples and uses a parameterized emission profile which results in an estimation of fewer parameters. We used gaussian shapes of the emission profile for this nonlinear method. The restriction of the nonlinear method is the prior assumption about the type of the emission profile. We have also shown that the fitting methods are robust against noise in the emission spectra to be analyzed. One challenge is the uniqueness of a solution when solving inverse problems as stated in Eq. (6) and Eq. (8). As the consistency checks reveal, the fitted emission spectra generally match the assumed emission spectra very accurately but the fitting algorithms can find emission profiles that may differ from the assumed one. As with any ill-posed mathematical problem, there is not one unique solution. However, with the presented consistency checks we demonstrated the power of the algorithms for realistic OLED device structures.

## Acknowledgments

The authors wish to thank R. Coehoorn and M. Carvelli from Philips Research Eindhoven for fruitful discussions regarding their algorithm for emission profile extraction. Financial support through the European FP7 project AEVIOM.eu under grant no. 213708 is gratefully appreciated.

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