Theoretical comparison of the excitation efficiency of waveguide and surface plasmon modes between quantum-mechanical and electromagnetic optical models of organic light-emitting diodes

Kyungnam Kang; Kyoung-Youm Kim; Jungho Kim

doi:10.1364/OE.26.00A955

1. Introduction

Optical modeling of organic light-emitting diodes (OLEDs) is important to improve the out-coupling efficiency because the micro-cavity effect plays an important role in the light emission characteristics. The micro-cavity effect, caused by the optical interference within multiple thin layers on the order of micrometer, determines the optical mode distribution, the quantum efficiency, the angular emission distribution, and the spectral linewidth [1–5]. In general, there are two different theoretical approaches to consider the micro-cavity effect in the optical modeling of OLEDs. The first approach corresponds to the quantum mechanics, which originates from Purcell who theoretically predicted the modification of the spontaneous emission rate for radio frequencies in a one-dimensional cavity [6]. In the quantum treatment, a radiated optical field from an exciton is considered as the superposition of eigenmodes whose transition probabilities are determined by Fermi’s golden rule [7,8]. In this respect, the quantum-mechanical treatment is similar to the mode expansion method [9].

The second picture is based on the classical electromagnetism, where a radiating exciton is treated as a classical oscillating dipole [10–16]. The classical electromagnetic approach was first presented by Chance, Prock, and Sibley (CPS), where the radiation field of the oscillating dipole was described by the dyadic Green’s function [10,11]. This CPS theory had the difficulty in not being able to separate the polarization-dependent emission characteristics. More-convenient theoretical formulations applied the Sommerfeld identity to expand the Green’s function as an integral summation of Bessel functions in cylindrical coordinates in the transverse direction times a plane-wave in the longitudinal direction over all transverse wave numbers [12,13]. Another formulation, a so-called point dipole model, expanded the radiating field by using a Fourier integral over the transverse wave vectors, where the Fourier components represented the propagating and evanescent plane waves [14–16]. Quantification of the excitation efficiency into four optical pathways of the radiation, substrate, waveguide, surface plasmon modes was performed based on the power dissipation spectrum obtained by the classical electromagnetic model [17,18].

To obtain a comprehensive understanding of the micro-cavity effect in OLEDs, a combination of the two approaches is essential. Several research reports have attempted to use a combined model to describe the emission characteristics of the radiation mode [7,8]. However, the hybrid approach has not yet been applied to optical modeling of waveguide and surface plasmon modes in OLEDs. In addition, a more sophisticated optical model for trapped waveguide and surface plasmon modes is required to provide the better light extraction structure of OLEDs. There are a few reports to separately quantify the excitation efficiencies of each wave and surface plasmon modes in OLED structures [19,20]. Nonetheless, there has been no direct comparison of the calculated excitation efficiency into waveguide and surface plasmon modes between the quantum-mechanical and classical electromagnetic models of OLEDs. In addition, the relative contribution of the near-field absorption to the power dissipation spectrum has not yet been investigated in a multilayer OLED structure [21].

In this paper, we present a theoretical comparison of the excitation efficiency into waveguide and surface plasmon modes between the quantum-mechanical and classical electromagnetic approaches in a bottom-emitting OLED. After theoretical formulations of the two approaches are introduced, the excitation efficiencies based on the two methods are calculated and compared with respective to the position of a dipole emitter. In the quantum approach, the mode expansion method is used to quantify the excitation efficiency of each waveguide and surface plasmon modes. In the classical approach, the power dissipation spectrum calculated by the point dipole model is approximated by the summation of Lorentzian line shape functions, which provides the excitation efficiency of each waveguide and surface plasmon mode. By comparing the calculated excitation efficiencies, we verify the equivalence of two approaches and obtain more comprehensive understanding on the variation of the spectral power density in the waveguide and surface plasmon modes with respect to the emitter position. The ratios of the mode excitation efficiency obtained by the two approaches are in good agreement except the contribution of the near-field absorption component, which has been neglected in the optical modeling of the OLED devices.

2. Theory

2.1 Quantum mechanical approach

In the quantum mechanics, the spontaneous emission in an organic electroluminescent media is described by the optical transition rate determined by the Fermi’s golden rule [16]. The spontaneous relaxation of a recombining exciton to its lower-energy state leads to certain photonic modes. According to the Fermi’s golden rule, the optical transition rate ( $Γ$ ) between the exited-energy state $| j 〉$ and the lower-energy state $| i 〉$ is expressed as [22]

Γ = \frac{2 π}{ℏ} ρ (ν) {| 〈 j | \vec{μ} \cdot \vec{E} ({\vec{r}}_{e}) | i 〉 |}^{2},

where

ℏ

is the Planck constant and

\vec{μ}

is the dipole moment. The total spontaneous emission rate can be obtained by integrating Eq. (1) over all the final state. In terms of quantum-mechanical picture, the optical transition rate of each photonic mode is proportional to the following three factors. The photonic density of states

ρ (ν)

represents the number of photonic modes per unit frequency interval at the emitter frequency

ν

and depends on the optical environment where the photon emission occurs. The term

\vec{E} ({\vec{r}}_{e})

indicates the electric field amplitude at the emitter position (

{\vec{r}}_{e}

). Finally,

\vec{μ} \cdot \vec{E}

is related with the orientation of the electric field with respect to the electric dipole direction.

Fermi’s golden rule is optically described by the mode expansion method. When the dipole orientation is assumed to be isotropic, the optical transition rate of waveguide or surface plasmon modes is given by [9]

Γ = Γ_{0} \cdot \frac{λ}{2} \cdot {\sum_{l} n_{p h a s e}^{T E} (l) \cdot n_{g r o u p}^{T E} (l) \cdot {| \vec{E} ({\vec{r}}_{e}) |}^{2} + \sum_{m} n_{p h a s e}^{T M} (m) \cdot n_{g r o u p}^{T M} (m) \cdot {| \vec{E} ({\vec{r}}_{e}) |}^{2}},

where

Γ_{0}

is the optical transition rate in free space and λ is the wavelength of light. The terms

n_{p h a s e}^{T E} (l)

and

n_{g r o u p}^{T E} (l)

represent the phase and group indices the l-th order transverse electric (TE) mode. Similarly,

n_{p h a s e}^{T M} (m)

and

n_{g r o u p}^{T M} (m)

are the phase and group indices of the m-th order transverse magnetic (TM) mode, respectively. When only waveguide and surface plasmon modes are considered, the photonic density of states is proportional to the multiplication of the phase and group indices of the corresponding modes.

2.2 Classical electromagnetic approach

The excitation efficiency into eigenmodes such as radiation, substrate, waveguide, and surface plasmon modes in OLEDs has been calculated on the basis of the so-called point dipole model in the classical electromagnetic theory [14–16]. The total optical power radiated by an oscillating electric dipole within a multilayer thin-film structure is obtained with use of the superposition of plane and evanescent waves. The spectral power density radiated by the dipole emitter per unit wavelength and per unit in-plane component of the normalized wave vector u is expressed as [16]

K_{v}^{T M} (λ, u) = \frac{3}{4} Re [\frac{u^{2}}{\sqrt{1 - u^{2}}} \frac{(1 + a_{+}^{T M}) (1 + a_{-}^{T M})}{1 - a^{T M}}],

K_{h}^{T M} (λ, u) = \frac{3}{8} Re [\sqrt{1 - u^{2}} \frac{(1 - a_{+}^{T M}) (1 - a_{-}^{T M})}{1 - a^{T M}}],

K_{h}^{T E} (λ, u) = \frac{3}{8} Re [\frac{1}{\sqrt{1 - u^{2}}} \frac{(1 + a_{+}^{T E}) (1 + a_{-}^{T E})}{1 - a^{T E}}],

where

v

(

h

) represents the dipole orientation vertically-aligned (horizontally-aligned) to the interfaces of the multilayer system. As shown in the inset of Fig. 1, the normalized in-plane wave vector is calculated by the

u = | {\vec{k}}_{t, E M L} | / | {\vec{k}}_{E M L} | = \sin θ

, where

{\vec{k}}_{E M L}

is the total wave vector in the emission layer (EML),

{\vec{k}}_{t, E M L}

is the transverse component of

{\vec{k}}_{E M L}

, and

θ

is the emission angle of the dipole emitter [23]. In addition,

a_{+ (-)}^{T E (T M)} = r_{+ (-)}^{T E (T M)} \exp (2 j k_{z, E M L} z^{+ (-)}),

where

r_{+ (-)}^{T E (T M)}

represents the reflection coefficients of the TE(TM)-polarized wave travelling from the dipole emitter to the upward in the + z direction (downward in the –z direction),

k_{z, E M L}

is the longitudinal component of

{\vec{k}}_{E M L}

, and

z^{+ (-)}

denotes the distance from the dipole emitter to the upper (lower) interface of the EML. Then,

a^{T E (T M)} = a_{+}^{T E (T M)} a_{-}^{T E (T M)}

is written as

a^{T E (T M)} = r_{+}^{T E (T M)} r_{-}^{T E (T M)} \exp (2 j k_{z, E M L} d_{E M L}),

where

d_{E M L}

is the thickness of the EML.

Fig. 1 Device structure of a bottom-emitting OLED along with the corresponding layer thickness and refractive index. The Alq₃ layer works as both the electron-transport layer and EML. The dipole emitter with the wavelength of 520 nm is assumed to be isotropic and have the δ-distributed emission zone. In the inset, the configurations of the wave vectors in the EML and the definition of the in-plane component of the normalized wave vector u are shown.

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When the orientation of dipole emitter is isotropic, the total spectral power density at the wavelength of λ is given by

K (λ, u) = \frac{2}{3} K_{h}^{T E} (λ, u) + \frac{1}{3} [K_{v}^{T M} (λ, u) + 2 K_{h}^{T M} (λ, u)] .

Therefore, the total spectral power radiated by the dipole emitter is expressed as

F (λ) = \int_{0}^{\infty} K (λ, u) d u^{2} = 2 \int_{0}^{\infty} u K (λ, u) d u .

3. Calculation results

We numerically compare the excitation efficiency of waveguide and surface plasmon modes between quantum-mechanical and classical electromagnetic models in a well-known bottom-emitting OLED structure [24]. Figure 1 shows a device structure of the OLED, which consists of a thick glass substrate, 140-nm indium tin oxide (ITO) as a bottom anode, 30-nm poly(3,4)-ethylendioxythiophene doped with poly(styrene sulfonate) (PEDOT:PSS) as a hole injection layer, 80-nm N,N’-diphenyl-N,N’-bis(3-methylphenyl)-1,1’-biphenyl-4,4-diamine (TPD) as a hole transporting layer, 342-nm tris-(8-hydroxyquinoline) aluminum (Alq₃) as both the electron-transport layer and EML, 15-nm calcium as an electron injecting layer, and 100-nm aluminum as a top cathode. The thickness of the electron transport plus emission layers can be on the order of 300 nm for the efficient third-order micro-cavity OLED, which may be very crucial for the large size TV applications that require the insensitivity to surface particle and roughness to obtain high production yield [25]. In addition, the thickness of the Alq₃ layer can be large with a thickness of 342 nm because Alq₃ is used as the electron transport layer as well as the EML. Thus, the OLED device with the Alq₃ thickness of about 350 nm can be a feasible and acceptable structure in the optical simulation study of OLEDs. In Fig. 1, the hole injection layer of PEDOT:PSS is spin coated whereas the other stacks of TPD/Alq3/CS/Al are thermally evaporated. This mixture of the spin-coated PEDOT:PSS and the thermally-evaporated organic multilayer is not popular in the OLED fabrication, but has been applied to several OLED devices with highly-efficient output performance [26–29].

All the simulations are performed at the wavelength of λ = 520 nm, which corresponds to the photoluminescence peak of Alq₃ in the literature [5]. The refractive indices of the materials shown in Fig. 1 are taken from the literature [30] or experimental date measured by the ellipsometry. Thickness of the EML is determined to maximize the multi-beam interference of the micro-cavity effect [5]. The dipole emitter located at the EML is assumed to be isotropic and have the δ-distributed emission zone. On the basis of the two-beam interference of the micro-cavity effect for the radiation mode, the excitation efficiencies of the dipole emitter are calculated at three positions in reference to the interface between the Ca layer and EML. When the refractive index of the EML is denoted as n_EML, the position A (λ/4n_EML = 50 nm) and C (3λ/4n_EML = 198 nm) correspond to the first and second resonance conditions of the two-beam interference term. On the other hand, the position B (λ/2n_EML = 124 nm) satisfies the destructive interference condition.

3.1 Excitation efficiency calculation of waveguide and surface plasmon modes based on the quantum approach

According to the mode expansion method in Eq. (2), three mode characteristics, which are the phase and group indices together with the electric field intensity at the emitter position, are required to calculate the excitation efficiency of the waveguide or surface plasmon mode. These mode characteristics of the bottom-emitting OLED structure are calculated through the boundary mode analysis performed by the finite element method (FEM), where a commercial software of COMSOL MULTIPHYSICS is used as an FEM solver [31]. The eigenvalue and eigenfunction obtained by the boundary mode analysis corresponds to the effective mode index and electric field amplitude, respectively. The validity of calculated eigenvalues and eigenfunctions obtained by the boundary mode analysis of the COMSOL MULTIPHYSICS are confirmed through the agreement of the calculation results obtained by our home-made simulation code based on the transfer-matrix-method-based waveguide analysis of the OLED [32].

According to the boundary mode analysis, there are two TE and three TM confined modes, as shown in Table 1. The polarization-dependent mode number is designated on the basis of the ascending order of the phase index, which is the real part of the complex effective mode index. The imaginary part of the effective mode index is related with the extinction coefficient of the respective waveguide or surface plasmon mode. Two TE confined modes correspond to waveguide modes because TE-polarized light cannot excite the surface plasmon mode [22]. According to the point dipole model [15], the surface plasmon mode is a purely evanescent wave in the z direction, which contains a pure imaginary value of the longitudinal wave vector in the EML ( $k_{z, E M L}$ ). Thus, the electric field distribution of the surface plasmon mode exponentially decays on both + z and -z longitudinal directions in reference to the metal-dielectric interface, as will be shown in Fig. 3. In the inset of Fig. 1, the value of the longitudinal wave vector can be obtained by

k_{z, E M L} = \sqrt{{| {\vec{k}}_{E M L} |}^{2} - {| {\vec{k}}_{t, E M L} |}^{2}} = k_{0} \sqrt{{(n_{E M L})}^{2} - {(n_{s p})}^{2}},

where

k_{0} = 2 π / λ

is the wave vector in free space and n_sp is the phase index of the surface plasmom mode. In Eq. (10), the longitudinal wave vector of

k_{z, E M L}

can be an imaginary value if

n_{s p} > n_{E M L}

is satisfied. Among three TM confined modes, the TM2 mode is the surface plasmon mode because its phase index of 1.9209 is larger than the EML refractive index of 1.7690. Because of a pure imaginary value of the longitudinal wave vector in the EML, the TM2 surface plasmon mode is strongly confined at the interface between the metal Ca layer and dielectric EML. Thus, the spatial distribution of the normalized electric field intensity of the TM2 surface plasmon mode is much stronger when close to the Ca/EML boundary. In addition, the TM2 mode has a significantly large imaginary part of the effective mode index, which results from the evanescent wave nature of the surface plasmon mode propagating in the z direction. Figure 2 shows the calculated dispersion curves around the emission frequency for TE- and TM-polarized confined modes. The group index is proportional to the reciprocal of the group velocity, which is calculated from the slope of the dispersion curve [33]. Finally, the value of

n_{p h a s e}^{} \cdot n_{g r o u p}^{}

, which represents the photonic density of states, is also designated in Table 1.

Table 1. Calculated effective mode index, phase index, group index, and photonic density of states of five confined modes in the bottom-emitting OLED shown in Fig. 1.

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Fig. 2 Calculated dispersion curve of (a) TE-polarized and (b) TM-polarized waveguide or surface plasmon modes. The short-dotted line in orange color indicates the dispersion curve of light in vacuum.

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Figure 3 shows the calculated spatial distribution of the normalized electric field intensity of five confined modes in the bottom-emitting OLED. According to the inset of Fig. 1, TM-polarized light has both x- and z-directional components of the electric field intensities. Thus, the electric field intensities of the three TM modes (TM0, TM1, TM2) are obtained by ${| E |}^{2} = {| E_{x} |}^{2} + {| E_{z} |}^{2}$ . On the other hand, the electric field intensities of the two TE modes (TE0, TE1) are calculated by ${| E |}^{2} = {| E_{y} |}^{2}$ . Both two TE modes have the waveguide nature, where the peaks of the electric field intensity are located at the EML for the TE0 mode and the ITO layer for the TE1 mode, respectively. Similarly, TM0 and TM1 modes correspond to the waveguide mode whose electric field intensity is mostly positioned at the EML and ITO layer, respectively. On the other hand, the TM2 mode, identified as the surface plasmon mode, is strongly confined at the interface between the metal Ca layer and dielectric EML.

Fig. 3 Spatial distribution of the normalized electric field intensity of waveguide and surface plasmon modes in the bottom-emitting OLED. The purple dashed lines indicate three dipole emitter positions of A (50 nm), B (124 nm), and C (198 nm).

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Because of the boundary condition of the Maxwell equation, the x- and y-directional tangential components of the electric field amplitude should be continuous at the interface boundary, but the z-directional normal component is discontinuous due to the relation of ${(n_{j} + i κ_{j})}^{2} E_{z, j} = {(n_{j + 1} + i κ_{j + 1})}^{2} E_{z, j + 1}$ , where n_{j(j + 1)} and κ_{j(j + 1)} are the refractive index and extinction coefficient of the adjacent j(j + 1) layers, respectively. The electric field intensity for TM polarization is discontinuous at the interfaces because the electric field has a component normal to the interface [34]. Especially, the sudden jump up of the TM0-mode electric field intensity at the Al/Ca boundary results from the large difference of the complex refractive index between the Al (0.84 + 6.33i) and Ca (1.5 + 1.986i).

Table 2 shows the calculated electric field intensity of five confined modes at three emitter positions, which are designated in the purple dash lines in Fig. 3. The value of the electric field intensity of the TE0 mode is much larger than that of the TE1 mode at all the dipole positions because the TE0 mode is mainly distributed at the EML and the TE1 mode is mainly positioned at the ITO layer. The value of the electric field intensity of the TM2 surface plasmon mode is dominant over the other two TM waveguide modes at the emitter position A, which is close to the Ca layer. As the emitter position moves away from the Ca layer, the value of the electric field intensity of the TM2 mode dramatically decreases due to the exponentially-decaying electric field profile of the surface plasmon mode. Then, the value of the electric field intensity of the TM0 mode becomes dominant at the emitter position C, which is close to the peak of the electric field intensity of the TM0 waveguide mode.

Table 2. Calculated electric field intensity of five confined modes at three dipole positions

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According to Eq. (2), the excitation efficiency in the quantum approach depends on the photonic density of states obtained by $n_{p h a s e} \cdot n_{g r o u p}$ and the electric field intensity at the emitter position. Table 3 shows the calculated relative excitation efficiencies of five confined modes at three emitter positions based on the calculated values in Table 1 and 2, where the interaction between the bottom-emitting OLED micro-cavity and the dipole emitter is assumed to be in the weak-coupling quantum-electrodynamics regime [35]. Correspondingly, we assume that the photonic density of states in Table 1 and the electric field profiles of the five confined modes in Fig. 3 are not affected by the existence of the dipole emitter and independent of the emitter position. According to Table 1, the magnitude of the photonic density of states has no big difference among five modes. However, the value of electric field intensity in Table 2 is greatly changed with respect to the mode type and emitter position. Thus, the value of ${| \vec{E} ({\vec{r}}_{e}) |}^{2}$ is a dominant factor to determine the excitation efficiency of waveguide and surface plasmon modes in the bottom-emitting OLEDs.

Table 3. Calculated relative excitation efficiencies based on the quantum-mechanical mode expansion method

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3.2 Excitation efficiency calculation of waveguide and surface plasmon modes based on the electromagnetic approach

In the classical electromagnetic approach, the total optical power radiated by a dipole emitter is expressed as the spectral power density with respect to the normalized in-plane wave vector, which is described by Eqs. (3)-(9). The spectral power densities of the OLED are calculated using the commercial software of the SETFOS [36]. The calculation results of the spectral power densities obtained by the SETFOS have been already verified through the good agreement of the calculation results obtained by our home-made simulation code based on the point dipole model, which was described in our previous publication [23]. Figure 4 shows the calculation results of the polarization-dependent spectral power densities at three emitter positions. In the case of the bottom-emitting OLED, there are four light coupling mechanisms depending on the range of the normalized in-plane wave vector u. In general, they are classified into the radiation mode $(0 \leq u \leq n_{a i r} / n_{E M L}),$ substrate mode $(n_{a i r} / n_{E M L} \leq u \leq n_{s u b} / n_{E M L}),$ waveguide mode $(n_{s u b} / n_{E M L} \leq u \leq 1),$ and surface plasmon mode $(1 \leq u \leq \infty)$ [16]. Here, n_air = 1 and n_sub = 1.547 is the refractive index of the air and substrate, respectively. In Fig. 4, these four light coupling regions of the radiation, substrate, waveguide, surface plasmon modes are designated in the different shaded colors of sky blue, gray, yellow, and pink, respectively. There is no surface plasmon mode region for TE polarization because it is only generated for TM polarization. In the x axis, the normalized in-plane wavevector u is converted into $n_{E M L} \cdot u$ , which corresponds to the phase index of waveguide and surface plasmon modes.

Fig. 4 The calculated spectral power densities at three emitter positions for (a) TE and (b) TM polarizations. Five spectral peaks in the waveguide or surface plasmon modes are fitted to the summation of the respective Lorentzian line shape function.

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In Fig. 4, five spectral peaks found in the region of waveguide or surface plasmon modes can be fitted to the summation of the respective Lorentzian line shape function [19]. The total spectral power density can be approximated by

K (u) \approx \sum_{l = 1}^{2} \frac{σ_{l}^{T E}}{{(n_{E M L} u - n_{p h a s e, l}^{T E})}^{2} + {(κ_{l}^{T E})}^{2}} + \sum_{m = 1}^{3} \frac{σ_{m}^{T M}}{{(n_{E M L} u - n_{p h a s e, m}^{T M})}^{2} + {(κ_{m}^{T M})}^{2}},

where

n_{p h a s e, l (m)}^{T E (T M)}

and

κ_{l (m)}^{T E (T M)}

indicate the phase index and extinction coefficient of the l(m)-th confined mode for TE(TM) polarizations. The excitation magnitude

σ_{l (m)}^{T E (T M)}

of each Lorentzian line shape function is determined to give the best fitting to the total spectral power density. Table 4 shows the values of phase indices and extinction coefficients of five Lorentzian line shape functions that are obtained by the best fitting to the total spectral power density in Fig. 4. Two peaks in Fig. 4(a) indicate the TE0 and TE1 waveguide modes. Three peaks in Fig. 4(b) correspond to the TM0 and TM1 waveguide modes and the TM2 surface plasmon mode, respectively. A certain spectral peak, such as the TM1 mode at the emitter position of 50 nm or the TM2 mode at the emitter position of 198 nm, is not distinguishable due to its low emission intensity. It is noticeable that the phase indices and extinction coefficients obtained by the fitting in Table 4 are nearly same as the effective mode indices calculated by the boundary mode analysis in Table 1. This verifies the equivalence between the electromagnetic and quantum approaches of the optical modeling of OLEDs.

Table 4. The values of the phase index and extinction coefficient of five confined modes obtained by the fitting to the spectral power density

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In this calculation, the emitter position A and C correspond to the first and second resonance conditions, but the emitter position B satisfies the destructive interference condition of the two-beam interference for the radiation mode. Thus, the spectral power intensity for the emitter position A and C is much larger than that for the emitter position B in the radiation mode region. However, the variation of the spectral power density in the waveguide and surface plasmon mode regions is not similar to that in the radiation mode region. To quantify the emission efficiency of each waveguide or surface plasmon mode, the area of each Lorentzian line shape function is calculated and shown in Table 5. As the dipole emitter moves from position A to C, the excitation efficiency of the TE0, TE1, TM0, and TM1 waveguide modes increases, but that of the TM2 surface plasmon mode dramatically decreases. This variation of the excitation efficiencies extracted from the spectral power density agrees with the variation of the electric field intensity at the emitter position shown in Table 2. This similarity also confirms the equivalence between the quantum-mechanical and electromagnetic methods of the optical modeling of OLEDs.

Table 5. Calculated relative excitation efficiency based on the electromagnetic point dipole model

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In Fig. 4, the peak value of the spectral power density of the TE0 mode is much larger than those of the three TM modes. However, according to the Table 5, the TE0-mode excitation efficiency of 7.3082 at the emitter position A is comparable to the TM0-mode excitation efficiency of 4.4193 or smaller than the TM2-mode excitation efficiency of 79.2874. It is noticeable that, as shown in Table 4, the TE0-mode extinction coefficient of 0.0021 is much smaller than the TM0-mode extinction coefficient of 0.0168 or the TM2-mode extinction coefficient of 0.2391. Because the area of the Lorentzian lineshape function is approximated as the peak value multiplied by the extinction coefficient, the peak value of the TE0 mode in Fig. 4 becomes larger at least by a factor of ten than those of TM0 or TM2 modes.

3.3 Comparison of the mode coupling ratio between quantum and electromagnetic approaches

To quantitatively compare the calculated relative excitation efficiency of OLEDs in waveguide and surface plasmon modes between the quantum and electromagnetic approaches, the ratios of the mode coupling efficiency are calculated for the two methods. The mode coupling ratios are obtained for separate TE and TM polarizations to analyze the difference of polarization-dependent excitation efficiency. Table 6 shows the ratio of the mode coupling efficiency calculated by the mode expansion method, which is based on the relative excitation efficiency in Table 3. Similarly, the ratio of the mode coupling efficiency obtained by the point dipole model is shown in Table 7, which refers to the relative excitation efficiency in Table 5. The calculated mode coupling ratios between the quantum and electromagnetic approaches agree to each other for TE polarization. On the other hand, the difference of the calculated mode coupling ratios is relatively large for TM polarization. This polarization-sensitive difference of the calculated mode coupling ratio is ascribed to the polarization-dependent contribution of the near-field absorption that is only included in the relative excitation efficiency calculated by the point dipole method [21].

Table 6. The ratio of mode coupling efficiency calculated by the mode expansion method based on the quantum approach.

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Table 7. The ratio of mode coupling efficiency calculated by the point dipole method based on the classical electromagnetic approach.

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In Fig. 4, the Lorentzian line shape functions are well fitted to the calculated spectral power densities around the peak positions at all three emitter positions. In contrast, the fitted Lorentzian line shape functions become different from the spectral power density at the large spectral value of $n_{E M L} u$ > 2. This difference is speculated to result from the contribution of the near-field absorption, which depends on the light polarization and emitter position. The analytical formula for the dipole excitation coupling to the near-field absorption was investigated in a planar dielectric-metal interface [21]. The effect of the near-field absorption was dramatically reduced as the distance between the emitter dipole and the absorbing metal increases. The excitation coupling to the near-field absorption was noticeable at a distance up to 50 nm. In addition, the coupling efficiency to the near-field absorption for TM polarization was much larger than that for TE polarization.

In Fig. 4, the magnitudes of the spectral power densities at $n_{E M L} u$ > 2 are very large for both polarizations when the emitter position from the Ca metal interface is 50 nm. Then, the spectral power density at the large value of $n_{E M L} u$ significantly decreases as the emitter position further increases. This dramatic variation of the spectral power density at the large spectral range is caused by the dependence of the near-field absorption on the emitter position. Thus, the spectral power density at $n_{E M L} u$ > 2 is larger than, comparable to, or smaller than the total fitted Lorentzian line shape functions when the emitter position is 50, 124, and 198 nm, respectively. In addition, the TM-polarized spectral power density at $n_{E M L} u$ > 2 is larger by a factor of about 10 than the TE-polarized spectral power density when the emitter positions are 50 and 124 nm, where the contribution of the near-field absorption is noticeable in this large spectral range. Furthermore, the broad light emission from the TM2 surface plasmon mode ( $n_{p h a s e}^{}$ = 1.92) is overlapped with light emission from the near-field absorption at the large spectral range of $n_{E M L} u$ > 2. Thus, the calculated mode coupling ratio for TE polarization is in a good agreement between the quantum and electromagnetic approaches, as shown in Tables 6 and 7. On the other hand, TM-polarized light has the relatively large difference of the calculated mode coupling ratio between the two methods.

Figure 5 shows the comparison of the polarization-dependent mode coupling ratios of five waveguide or surface plasmon modes between the quantum-mechanical mode expansion and electromagnetic point dipole methods at various dipole emitter positions. The overall variation of the mode coupling ratio depending on the emitter position is matched between quantum-mechanical and electromagnetic approaches. The mode coupling ratio for TE polarization shows the better agreement than that for TM polarization. The calculation results in Fig. 5 are in accordance with the previously extracted values in Tables 6 and 7. In addition, the identification of the mode-dependent coupling ratio variation with respect to the emitter position will be helpful to design the better light extraction structure of OLEDs.

Fig. 5 Calculated mode coupling ratio of five waveguide or surface plasmon modes between the quantum-mechanical mode expansion and electromagnetic point dipole methods at various dipole emitter positions for (a) TE and (b) TM polarizations.

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In general, the optical modeling of OLEDs can be classified into analytical and numerical methods. Both methods solve the same governing equation in a different way, but have the same output results when an identical OLED structure is optically modeled. The analytical method uses the exact analytical solutions of the governing differential equations, and has the advantage of a small computation power and the easiness to gain a physical insight. However, the analytical method has the limitation that it can be only applicable to a one-dimensional planar multilayer structure. The numerical method approximately solves the governing equations in discrete tiny meshes. This method requires a relatively large computation power, but can perform the optical modeling of even three-dimensional complex and non-planar structures beside the one-dimensional planar multilayer structure.

The quantum-mechanical mode expansion and electromagnetic point dipole methods correspond to the analytical method. Two analytical methods have different mathematical forms of the exact analytical solutions due to the difference in the theoretical approach, explained in Eq. (1)-(9). The CPS theory is another type of the analytical methods based on the electromagnetic approach [10,11]. As explained in the introduction, the difference between the CPS theory and point dipole model results from the use of the different basis functions to expand the radiating field, which are mathematically equivalent. Among several numerical techniques, finite-difference time-domain (FDTD) has been widely used in the optical modeling of OLEDs based on the electromagnetic approach. The FDTD solves the Maxwell’s equations with central difference approximations by discretizing both in time and space [37]. The FDTD method has been applied to the optical modeling of both planar and surface-textured OLEDs [38,39].

4. Discussion

In Fig. 5, the mode coupling ratios of waveguide and surface plasmon modes based on the quantum-mechanical and electromagnetic optical models are calculated in a thick OLED with the Alq₃ EML thickness of 342 nm. In such a thick OLED, five TE and TM confined modes exist, whereas in thinner and more typical OLED, where the Alq₃ EML is 40 nm thick for example, much fewer waveguide or surface plasmon modes exist. It is worthy of investigating the consequence on the comparison of the mode coupling ratio between the quantum-mechanical and classical electromagnetic approaches in the thinner bottom-emitting OLED.

Figure 6(a) shows the device structure of the thin OLED, whose layer structure and refractive index is the same as that of the thick OLED in Fig. 1 except that the thickness of the Alq₃ layer is reduced from 342 to 40 nm. When the boundary mode analysis is applied to the thin OLED, we obtain one TE and two TM confined modes, whose effective mode index, phase index, group index, and photonic density of states are summarized in Table 8. Figure 6(b) shows the calculation results of the normalized electric field intensity of three confined modes in the thin bottom-emitting OLED. According to Fig. 6(b), the TE0 and TM0 modes of the thick OLED in Fig. 3, which have the peak electric field intensity located at the thick EML, disappear due to the reduction of the Alq₃ EML thickness from 342 to 40 nm. Thus, the TE0, TM0, and TM1 modes of the thin OLED in Fig. 6(b) are matched with the TE1, TM1, and TM2 modes of the thick OLED in Fig. 3, respectively. Regarding the mode characteristics such as the effective mode index, phase index, group index, and photonic density of states, the values of the TE0, TM0, and TM1 modes of the thin OLED in Table 7 are close to those of TE1, TM1, and TM2 modes of the thick OLED in Table 1, respectively.

Fig. 6 (a) Device structure of the thin bottom-emitting OLED, whose layer structure and refractive index is the same as that of the thick OLED in Fig. 1 except that the thickness of the Alq₃ layer is reduced from 342 to 40 nm. (b) Calculation results of the normalized electric field intensity of waveguide and surface plasmon modes in the thin bottom-emitting OLED. The TE0, TM0, and TM1 modes of the thin OLED are matched with the TE1, TM1, and TM2 modes of the thick OLED in Fig. 3, respectively. The purple dashed lines indicate three dipole emitter positions of D (10 nm), E (20 nm), and F (30 nm).

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Table 8. Calculated effective mode index, phase index, group index, and photonic density of states of three confined modes in the thin bottom-emitting OLED shown in Fig. 6(a).

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Table 9 shows the calculated electric field intensities of three confined modes at three emitter positions, which are denoted in the purple dash lines in Fig. 6(b). The value of the electric field intensity of the TE0 waveguide mode, which has the sinusoidally-increasing electric field profile at the EML, decreases as the emitter position moves away from the Ca/EML interface. On the other hand, the values of the electric field intensity of the two TM modes keep decreasing when the emitter position moves away from the Ca/EML interface. This can be explained by the decreasing behaviors of the electric field profiles of the TM0 waveguide and TM1 surface plasmon modes inside the EML. It is noticeable that the value of the electric field intensity of the TM1 surface plasmon mode is dominant over the TE0 and TE1 waveguide modes at all the emitter positions of D, E, and F, all of which are close to the peak of the electric field intensity of the TM1 surface plasmon mode located at the Ca/EML interface. Table 10 shows the calculated relative excitation efficiencies of three confined modes at three emitter positions in the thin bottom-emitting OLED, which are obtained on the basis of the calculated values in Tables 8 and 9. Similarly to the calculated relative excitation efficiencies of the thick bottom-emitting OLED shown in Table 3, the value of ${| \vec{E} ({\vec{r}}_{e}) |}^{2}$ is dominant to determine the excitation efficiency of three confined modes in the thin bottom-emitting OLEDs. Thus, the excitation efficiency of the TM1 surface plasmon mode is much larger than the other two TE0 and TM0 modes.

Table 9. Calculated electric field intensity of three confined modes at three dipole positions in the thin bottom-emitting OLED

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Table 10. Calculated relative excitation efficiencies based on the quantum-mechanical mode expansion method in the thin bottom-emitting OLED

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Figure 7 shows the calculated polarization-dependent spectral power densities of the thin bottom-emitting OLED at three emitter positions, which are marked in the purple dash lines in Fig. 6(b). On the basis of Eq. (11), three spectral peaks located at the region of waveguide or surface plasmon modes are fitted to the summation of the respective Lorentzian line shape function. Table 11 shows the values of phase indices and extinction coefficients of three Lorentzian line shape functions that are extracted from the best fitting to the total spectral power density in Fig. 7. One peak in Fig. 7(a) corresponds to the TE0 waveguide mode. Two peaks in Fig. 7(b) represent the TM0 waveguide mode and the TM1 surface plasmon mode, respectively. The phase indices and extinction coefficients obtained by the fitting in Table 11 are equal to the effective mode indices calculated by the boundary mode analysis in Table 8. This also demonstrates the equivalence of the optical modeling methods between the electromagnetic and quantum approaches in the thin bottom-emitting OLED.

Fig. 7 The calculated spectral power densities of the thin bottom-emitting OLED at three emitter positions for (a) TE and (b) TM polarizations. Three spectral peaks in the waveguide or surface plasmon modes are fitted to the respective Lorentzian line shape function.

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Table 11. The values of the phase index and extinction coefficient of three confined modes obtained by the fitting to the spectral power density in the thin bottom-emitting OLED shown in Fig. 6(a).

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Table 12 shows the calculation results of the relative excitation efficiency based on the electromagnetic point dipole model in the thin bottom-emitting OLED, where the area of each Lorentzian line shape function in Fig. 7 is used to calculate the relative excitation efficiency. As the dipole emitter shifts from position D to F, the excitation efficiency of the TE0 increases, whereas those of the TM0 and TM1 modes decrease. This variation of the relative excitation efficiencies based on the electromagnetic point dipole model is matched with the variation of the relative excitation efficiencies based on the quantum-mechanical mode expansion method in Table 10.

Table 12. Calculated relative excitation efficiency based on the electromagnetic point dipole model in the thin bottom-emitting OLED.

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The ratios of the dipole-position-dependent mode coupling efficiency based on the quantum and electromagnetic approaches in the thin bottom-emitting OLED are shown in Tables 13 and 14, which are obtained in reference to the calculated relative excitation efficiencies in Tables 10 and 12, respectively. The mode coupling ratio for TE polarization is assumed to be 100% because there is only one TE waveguide mode in the thin bottom-emitting OLED. For TM polarization, a relatively large difference of the calculated mode coupling ratio is observed between quantum and electromagnetic approaches. This fact results from the considerable contribution of the near-field absorption to the spectral power density at the large spectral value of $n_{E M L} u$ > 2, which is only considered in the relative excitation efficiency calculated by the point dipole method, as explained for the thick bottom-emitting OLED. Furthermore, the mode coupling ratios for TM polarization nearly remain the same with respective to all the emitter positions. Because the thin bottom-emitting OLED has the 40-nm-thick EML, the position of the dipole emitter from the Ca/EML interface lies within 40 nm, which is smaller than the critical distance of 50 nm that the excitation coupling to the near-field absorption is predominant [21].

Table 13. The ratio of mode coupling efficiency calculated by the mode expansion method based on the quantum approach in the thin bottom-emitting OLED.

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Table 14. The ratio of mode coupling efficiency calculated by the point dipole method based on the classical electromagnetic approach in the thin bottom-emitting OLED.

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Figure 8 shows the comparison of the calculated mode coupling ratio of two TM-polarized confined modes between the quantum-mechanical mode expansion and electromagnetic point dipole methods at various dipole emitter positions in the thin bottom-emitting OLED. The mode coupling ratio for TE polarization is assumed to be 100% and not shown here because there is only one TE waveguide mode in the thin bottom-emitting OLED. The overall variation of the mode coupling ratio with respect to the emitter position is not well matched between quantum-mechanical and electromagnetic approaches. This discrepancy indicates that the effect of the near-field absorption plays a more important role in the excitation efficiency of the dipole emitter in the thin bottom-emitting OLED than in the thick bottom-emitting OLED.

Fig. 8 Comparison of the calculated mode coupling ratio of two TM-polarized confined modes between the quantum-mechanical mode expansion and electromagnetic point dipole methods at various dipole emitter positions in the thin bottom-emitting OLED. The mode coupling ratio for TE polarization is assumed to be 100% and not shown here because there is only one TE waveguide mode in the thin bottom-emitting OLED.

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This hybrid approach can be also applied to analyze the excitation efficiency of waveguide and surface plasmon modes in the top-emitting OLED, which has an additional semi-transparent thin metal cathode besides one thick metal anode. The characteristics and analysis of the TE-polarized waveguide modes in the top-emitting OLED will be similar to that of the bottom-emitting OLED. However, the effect of the near-field absorption of the top-emitting OLED will be enhanced due to the additional semi-transparent thin metal cathode compared with that of the bottom-emitting OLED. In addition, the influence of the TM-polarized surface plasmon modes will increase in the top-emitting OLED because the surface plasmon mode is generated at the metal-dielectric boundary.

5. Conclusion

We theoretically compared the excitation efficiency of waveguide and surface plasmon modes between quantum-mechanical and electromagnetic approaches to the optical modeling of OLEDs. In the quantum-mechanical approach, the excitation efficiency was calculated on the basis of the mode expansion method, where the photonic density of states, obtained by the product of phase and group indices, and the electric field intensity at the emitter position determine the excitation efficiency. In the classical electromagnetic approach, the spectral power density obtained by the point dipole model was fitted by the summation of Lorentzian line shape functions, from which the excitation probability of each waveguide and surface plasmon mode was extracted.

The relative mode excitation efficiencies based on the two approaches were calculated in the bottom-emitting OLED as the emitter position moved away from the interface of the metal Ca layer. The real and imaginary part of the effective mode index obtained by the quantum-mechanical approach was well matched with the spectral position and width of the respective Lorentzian line shape function in the spectral power density. The variation of the calculated spectral power density with respect to the emitter position was qualitatively well explained by the electric field intensity at the emitter position in the quantum-mechanical approach, which has not yet been explained on the basis of the point dipole model in the electromagnetic approach. Quantitatively, the ratio of mode excitation efficiencies calculated by two approaches was in a good agreement except the contribution of the near-field absorption at the large spectral range, which has not considered in the optical modeling of OLEDs. By comparing the calculation results, we confirm the equivalence of two approaches and obtain the better physical interpretation to the calculated excitation efficiency of waveguide and surface plasmon modes in OLEDs.

Funding

National Research Foundation of Korea (NRF) (No. 2018R1D1A1B07047249).

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Mode list	Effective mode index ( $n_{e f f}^{}$ )	Phase index ( $n_{p h a s e}^{}$ )	Group index ( $n_{g r o u p}^{}$ )	Photonic density of states ( $n_{p h a s e}^{} \cdot n_{g r o u p}^{}$ )
TE0	1.6734 + 0.0021i	1.6734	2.4365	4.0773
TE1	1.8381 + 0.0048i	1.8381	2.3418	4.3045
TM0	1.6120 + 0.0168i	1.6120	2.3754	3.8292
TM1	1.7668 + 0.0055i	1.7668	2.2709	4.0122
TM2	1.9209 + 0.2391i	1.9209	2.4241	4.6565

Dipole position (nm)	TE0 (arb. unit)	TE1 (arb. unit)	Total (TE) (arb. unit)	TM0 (arb. unit)	TM1 (arb. unit)	TM2 (arb. unit)	Total (TM) (arb. unit)
50 (A)	9.51 × 10⁵	5.00 × 10³	9.56 × 10⁵	8.50 × 10⁵	4.71 × 10⁴	6.73 × 10⁶	7.63 × 10⁶
124 (B)	2.94 × 10⁶	2.46 × 10⁴	2.96 × 10⁶	1.30 × 10⁶	7.71 × 10⁴	1.39 × 10⁶	2.77 × 10⁶
198 (C)	4.07 × 10⁶	7.57 × 10⁴	4.14 × 10⁶	2.72 × 10⁶	2.39 × 10⁵	2.89 × 10⁵	3.25 × 10⁶

Dipole position (nm)	TE0(arb. unit)	TE1 (arb. unit)	Total (TE) (arb. unit)	TM0 (arb. unit)	TM1 (arb. unit)	TM2 (arb. unit)	Total (TM) (arb. unit)
50 (A)	3.72 × 10⁶	2.24 × 10⁴	3.74 × 10⁶	3.33 × 10⁶	2.02 × 10⁵	3.07 × 10⁷	3.42 × 10⁷
124 (B)	1.19 × 10⁷	1.18 × 10⁵	1.20 × 10⁷	5.39 × 10⁶	3.55 × 10⁵	5.84 × 10⁶	1.15 × 10⁷
198 (C)	1.59 × 10⁷	3.39 × 10⁵	1.62 × 10⁷	1.07 × 10⁷	1.03 × 10⁶	1.32 × 10⁶	1.30 × 10⁷

Dipole position (nm)		TE0	TE1	TM0	TM1	TM2
50 (A)	n_phase	1.6735	1.8380	1.6095	1.7640	1.9770
	κ	0.0021	0.0048	0.0168	0.0055	0.2391
124 (B)	n_phase	1.6735	1.8380	1.6150	1.7680	1.9209
	κ	0.0021	0.0048	0.0168	0.0055	0.2391
198 (C)	n_phase	1.6735	1.8380	1.6135	1.7675	1.9209
	κ	0.0021	0.0048	0.0168	0.0055	0.2391

Dipole position (nm)	TE0 (arb. unit)	TE1 (arb. unit)	Total (TE) (arb. unit)	TM0 (arb. unit)	TM1 (arb. unit)	TM2 (arb. unit)	Total (TM) (arb. unit)
50 (A)	7.3082	0.0748	7.3830	4.4193	0.2489	79.2874	83.9556
124 (B)	22.6383	0.2013	22.8396	10.1798	0.6830	19.5083	30.3711
198 (C)	31.2807	0.5789	31.8596	23.4505	2.0458	5.0612	30.5575

Theoretical comparison of the excitation efficiency of waveguide and surface plasmon modes between quantum-mechanical and electromagnetic optical models of organic light-emitting diodes

Abstract

1. Introduction

2. Theory

2.1 Quantum mechanical approach

2.2 Classical electromagnetic approach

3. Calculation results

3.1 Excitation efficiency calculation of waveguide and surface plasmon modes based on the quantum approach

3.2 Excitation efficiency calculation of waveguide and surface plasmon modes based on the electromagnetic approach

3.3 Comparison of the mode coupling ratio between quantum and electromagnetic approaches

4. Discussion

5. Conclusion

Funding

References

Cited By

Figures (8)

Tables (14)

Equations (11)

Optics Express

Dipole position (nm)	TE0 (%)	TE1 (%)	Total (TE) (%)	TM0 (%)	TM1 (%)	TM2 (%)	Total (TM) (%)
50 (A)	99.45	0.55	100.00	9.35	0.54	90.10	100.00
124 (B)	99.08	0.92	100.00	42.66	2.77	54.57	100.00
198 (C)	97.95	2.05	100.00	82.42	7.40	10.18	100.00

Dipole position (nm)	TE0 (%)	TE1 (%)	Total (TE) (%)	TM0 (%)	TM1 (%)	TM2 (%)	Total (TM) (%)
50	98.99	1.01	100.00	5.26	0.30	94.44	100.00
124	99.12	0.88	100.00	33.52	2.25	64.23	100.00
198	98.18	1.82	100.00	76.74	6.70	16.56	100.00

Mode list	Effective mode index ( $n_{e f f}^{}$ )	phase index ( $n_{p h a s e}^{}$ )	group index ( $n_{g r o u p}^{}$ )	photonic density of states ( $n_{p h a s e}^{} \cdot n_{g r o u p}^{}$ )
TE0	1.8318 + 0.0055i	1.8318	2.3817	4.3628
TM0	1.7226 + 0.0326i	1.7226	2.5546	4.4006
TM1	1.8455 + 0.2277i	1.8455	2.0411	3.7669

Dipole position (nm)	TE0 (arb. unit)	Total (TE) (arb. unit)	TM0 (arb. unit)	TM1 (arb. unit)	Total (TM) (arb. unit)
10	5.44 × 10⁴	5.44 × 10⁴	2.39 × 10⁶	1.41 × 10⁷	1.65 × 10⁷
20	9.29 × 10⁴	9.29 × 10⁴	2.02 × 10⁶	1.15 × 10⁷	1.35 × 10⁷
30	1.43 × 10⁵	1.43 × 10⁵	1.71 × 10⁶	9.31 × 10⁶	1.10 × 10⁷

Dipole position (nm)	TE0 (arb. unit)	Total (TE) (arb. unit)	TM0 (arb. unit)	TM1 (arb. unit)	Total (TM) (arb. unit)
10	2.37 × 10⁵	2.37 × 10⁵	1.05 × 10⁷	5.31 × 10⁷	6.36 × 10⁷
20	4.05 × 10⁵	4.05 × 10⁵	8.89 × 10⁶	4.33 × 10⁷	5.22 × 10⁷
30	6.24 × 10⁵	6.24 × 10⁵	7.53 × 10⁶	3.51 × 10⁷	4.26 × 10⁷

Dipole position (nm)	TE0 (arb. unit)	Total (TE) (arb. unit)	TM0 (arb. unit)	TM1 (arb. unit)	Total (TM) (arb. unit)
10	0.47	0.47	10.46	182.80	193.26
20	0.76	0.76	8.68	143.59	152.27
30	1.14	1.14	7.19	112.92	120.11

Dipole position (nm)	TE0 (%)	Total (TE) (%)	TM0 (%)	TM1 (%)	Total (TM) (%)
10	100.00	100.00	16.48	83.52	100.00
20	100.00	100.00	17.03	82.97	100.00
30	100.00	100.00	17.70	82.30	100.00

Dipole position (nm)	TE0 (%)	Total (TE) (%)	TM0 (%)	TM1 (%)	Total (TM) (%)
10	100.00	100.00	5.41	94.59	100.00
20	100.00	100.00	5.70	94.30	100.00
30	100.00	100.00	5.99	94.01	100.00