Theoretical/Numerical Studies of the Nanoscale-cavity Effects on Dipole Emission, Förster Resonance Energy Transfer and Surface Plasmon Coupling

doi:10.21203/rs.3.rs-3144087/v1

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Theoretical/Numerical Studies of the Nanoscale-cavity Effects on Dipole Emission, Förster Resonance Energy Transfer and Surface Plasmon Coupling

https://doi.org/10.21203/rs.3.rs-3144087/v1

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published 11 Aug, 2023

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The electric field and radiated power of a radiating dipole located inside a spherical nano-cavity are formulated to show that the nano-cavity structure or nanoscale-cavity effect can enhance the near-field intensity inside the cavity and the far-field radiated power of the dipole. Such enhancements are caused by two contributing factors, including the classical electromagnetic scattering as formulated and the Purcell effect, which is implemented through a numerical feedback process by assuming a two-level system for the radiating dipole. The enhancement of near-field intensity results in the efficiency increase of Förster resonance energy transfer when both energy donor and acceptor are located inside the nano-cavity. By combining the enhancements of the field intensity of the donor and the radiated power of the acceptor, the color conversion efficiency can be increased through the nanoscale-cavity effect. We also numerically demonstrate that the nanoscale-cavity effect can enhance surface plasmon coupling for increasing the radiated power of a dipole located nearby an Ag nanoparticle inside a nano-cavity.

The electromagnetic field generated by an infinitesimal radiating dipole source can be classified into the categories of far and near fields depending on the distance between the observation point and source position. Generally, near field distributes in the region within the distance of one-half the wavelength. Outside this region, far field dominates. From a radiating dipole source, the far field strength is proportional to the inverse of the distance. In the near field regime, more contributing terms inversely proportional to the higher powers of distance exist. Resonance modes in a cavity structure can be built with far field when its propagation distance in a round-trip is equal to an integral number of wavelength. The cavity resonance behavior is sensitive to cavity geometry. However, when the cavity dimension is smaller than one-half the wavelength, the near field distribution inside the cavity can produce certain effects on the radiation behavior of a dipole other than the aforementioned cavity resonance. On the other hand, the Purcell effect has been used for generally describing the influence of a cavity structure on the radiating dipole source inside the cavity [1]. The electromagnetic field distribution built at the dipole position produced by the scattering of the surrounding structure can change the radiation behavior of the dipole source and hence the overall radiation result. Either near or far field portion produced by a dipole source can lead to the Purcell effect for changing the near field distribution and far field radiated power.

With the development of nano-fabrication technique, nanoscale cavities of various geometries can be produced in light-emitting devices. The dimension of such a nanoscale cavity ranges from a couple tens through a few hundred nm. Light-emitting nanoparticles (NPs) such as colloidal quantum dots (QDs) can be inserted into such a nanoscale cavity. Such an inserted QD can produce a strong near field distribution inside such a cavity. The examples of nanoscale cavity for inserting QDs in a light-emitting device include a GaN porous structure and a surface nano-hole. By growing a highly conductive GaN layer, such as a highly Si-doped GaN layer, in a GaN-based device, a subsurface GaN porous structure of essentially laterally-extended pores of several tens nm in cross-sectional size can be fabricated in the highly conductive layer through an electrochemical etching process [2–6]. QDs can be inserted into the pores of such a porous structure for observing the enhanced QD emission efficiency, the stronger Förster resonance energy transfer (FRET) between the QDs of different emission wavelengths, and the stronger FRET from a nearby InGaN/GaN quantum well (QW) into the inserted QDs [7, 8]. Nanoscale cavities can also be implemented by fabricating surface nano-holes on a GaN template. Through nano-imprint lithography and reactive ion etching, a surface nano-hole array of a couple hundred nm in hole width and several hundred nm in hole depth can be obtained for filling with QDs and chemically synthesized metal NPs. Such a surface nano-hole structure has been used in a light-emitting device for enhancing the FRET from the embedded QWs into the filled-in QDs based on the advantageous short distance between the QWs and QDs [9–14]. As a matter of fact, a nano-hole structure can produce more advantageous results through the nanoscale-cavity effect, including a higher QD emission efficiency, a more effective FRET process, and a stronger surface plasmon (SP) coupling effect [15].

FRET and SP coupling are two important near-field interactions for enhancing light emission and color conversion. From the viewpoint of the electromagnetic theory, FRET is an absorption process of the near-field energy from an energy donor by an acceptor. Such an energy transfer is effective when the absorption spectrum of the acceptor overlaps the emission wavelength of the donor and the distance between the donor and acceptor is shorter than several tens nm [16–21]. FRET is a useful mechanism for color conversion, particularly when the energy donor and acceptor are placed inside or near a nanoscale cavity and hence their distance is small [7, 8, 16, 22]. The SP coupling between a light emitter and an SP resonance mode of a metal nanostructure can be understood as the process of energy transfer from the light emitter into the SP resonance mode for radiation before the energy is consumed by the non-radiative process of the light emitter. Through SP radiation, the overall emission efficiency of the coupled system can be enhanced [23, 24]. For producing an effective SP coupling process, the distance between the light emitter and the metal nanostructure needs to be smaller than ~ 100 nm. Either deposited surface metal NPs or chemically synthesized metal NPs can be placed close to a QW or QD for inducing an effective SP coupling and hence a higher emission or FRET efficiency [20, 21, 25–29].

The result of the Purcell effect in the near field regime is quite different from that caused by far field. Inside a nanoscale cavity, the electromagnetic behavior produced by a dipole source is expected to differ from that of cavity resonance. Such a nanoscale-cavity effect is useful for enhancing the efficiencies of emission and color conversion in a light-emitting device for illumination and display applications. However, the mechanism behind the nanoscale-cavity effect has not been well investigated yet. In particular, the cavity effect in the transition from the far-field into near-field regime is of great interest for understanding the difference between the two regimes. Meanwhile, a nanoscale cavity structure is expected to influence other near-field interactions, including FRET and SP coupling. In this paper, we theoretically/numerically study those issues based on a simple structure of a closed spherical nano-cavity. Although it is difficult to practically implement such a closed spherical nano-cavity, it provides us with a theoretical platform for demonstrating the important nanoscale-cavity effects. In section 2 of this paper, the sample structure under study is described. Also, the formulations in the theoretical study are demonstrated. The numerical results and discussions based on the theoretical study are shown in section 3. Then, the numerical study results of the nanoscale-cavity effect on SP coupling are reported in section 4. Further discussions about the results are given in section 5. Finally, conclusions are drawn in section 6.

Figure 1(a) shows the fundamental structure (structure A) for theoretical study with a spherical nano-cavity of R in radius embedded in an infinitely large material of n₂ in refractive index. The refractive index of the medium filling the nano-cavity is n₁. Here, the origin of the coordinate system is set at the center of the spherical cavity, at which an x- or z-oriented source dipole is located. In Fig. 1(b) we illustrate the structure (structure B) for studying the FRET behavior when both energy donor and acceptor are located inside a spherical nano-cavity at the coordinate origin and (0, 0. d) respectively. Structures C-E illustrated in Figs. 1(c)-1(e) are used for investigating the nanoscale-cavity effect on SP coupling. Figure 1(f) illustrates the reference structure (structure R) in which a radiating dipole is situated in a homogeneous medium of n₁ in refractive index.

To study the nanoscale-cavity effect, we first derive the electromagnetic field distribution produced by an x-oriented dipole with the intrinsic dipole moments ${\vec {p}_0}={p_0}\hat {x}$ in a spherical nano-cavity structure (structure A) based on the classical electromagnetic theory. It is noted that the electromagnetic derivation does not include the Purcell effect, which describes the emission behavior change of a source dipole caused by the scattered field from its surrounding structure. In other words, the scattered field of the source dipole can change its own dipole moment. In this situation, the derived results are still valid if the source dipole moment ${\vec {p}_0}={p_0}\hat {x}$ is replaced by a modified dipole moment ${\vec {p}_m}={p_m}\hat {x}$. The modified dipole moment can be obtained by assuming that the source dipole acts as a two-level oscillation system and solving its optical Bloch equations under the influence of the scattered field [23, 30]. Generally, the modified dipole moment ${\vec {p}_m}={\vec {p}_0}+\Delta \vec {p}$ can be derived from the steady-state solution of the optical Bloch equations to give [30]

$$\Delta \vec {p}= - \frac{{i{T_2}\left( {\rho _{{11}}^{{(0)}} - \rho _{{00}}^{{(0)}}} \right)}}{\hbar }\frac{{{{\left| {\frac{{{{\vec {p}}_0} \cdot \vec {E}}}{2}} \right|}^2}}}{{{{\left| {\vec {E}} \right|}^2}}}\frac{{\vec {E}}}{{1+\frac{{{T_1}{T_2}}}{\hbar }{{\left| {\frac{{{{\vec {p}}_0} \cdot \vec {E}}}{2}} \right|}^2}}}$$

Here, $\vec {E}$ is the scattered electric field at the dipole position, $\rho _{{11}}^{{(0)}}$ and $\rho _{{00}}^{{(0)}}$ are the density matrix elements at zero field of the upper and lower states in the two-level system, respectively. Also, T₁ and T₂ are the population relaxation time and the dipole moment dephasing time, respectively. We can set T₂ = 2T₁ in the absence of collision broadening [31]. According to the Wigner-Weisskopf theory of spontaneous emission, T₁ is equal to the reciprocal of Einstein’s A coefficient as [32]

$${T_1}=\frac{{6{\varepsilon _0}h{c^3}}}{{{\omega ^3}{n_b}{{\left| {{{\vec {p}}_0}} \right|}^2}}}$$

Here, h is the Planck constant, ω is the angular frequency of the electromagnetic field, n_b is the background refractive index, ε₀ and c are the dielectric permittivity and light speed, respectively, in vacuum. From the last factor of Eq. (1) we can see the saturation effect, indicating that $\Delta \vec {p}$ is small when the scattered electric field is very strong. The radiation of the modified source dipole produces a new scattered field, which leads to a new solution of the optical Bloch equations and hence a new modified dipole moment. Such a procedure is iterated until a steady state is reached for giving the final modified dipole moment. The numerical implementation of the Purcell effect is called the feedback process [23]. In numerical computations, we choose $\rho _{{11}}^{{(0)}} - \rho _{{00}}^{{(0)}}=1$ and $\left| {{{\vec {p}}_0}} \right|$= 2 x 10^− 10/ω (Ω-m). The choice for the $\left| {{{\vec {p}}_0}} \right|$ value is actually immaterial because it will be normalized later in the numerical results shown below. Depending on the surrounding structure, the orientation of the source dipole can also be changed after the feedback process is applied. However, in a cavity structure of circular symmetry with respect to the dipole orientation, such as structure A illustrated in Fig. 1(a) the feedback process does not change the dipole orientation. Because of the strong scattering by the nano-cavity structure, the Purcell effect or feedback process is important in the current study. In particular, with the nanoscale dimension, the scattering of near field for producing the Purcell effect can provide us with novel emission behaviors for useful applications.

In a spherical nano-cavity structure, the radiated fields, including near and far fields, from a dipole at an arbitrary position inside the cavity can be analytically derived. However, to simplify the demonstration of formulations, we only show the derivations in a simple case, in which the source dipole is placed at the center of the spherical cavity, as illustrated in Fig. 1(a). In terms of the spherical coordinates (r, θ, φ) the total fields at $\overrightarrow{r}$ inside the spherical cavity generated by an x-dipole with the modified dipole moment ${\vec {p}_m}={p_m}\hat {x}$ are derived to give

$$\begin{gathered} E_{r}^{{in}}=\frac{{i{k^2}{p_m}}}{{2\pi {\varepsilon _1}{\varepsilon _0}r}}\left\{ {h_{1}^{{(1)}}(kr) - \alpha {j_1}\left( {kr} \right)} \right\}\sin \theta \cos \phi \\ =\frac{{i{k^3}{p_m}}}{{2\pi {\varepsilon _1}{\varepsilon _0}}}\left\{ {\left( {2\alpha - 1} \right)\frac{{\cos (kr)}}{{{{(kr)}^2}}} - (2\alpha +1)\frac{{\sin (kr)}}{{{{(kr{\text{)}}}^3}}} - i\left( {\frac{{\sin (kr)}}{{{{(kr)}^2}}} - \frac{{\cos (kr)}}{{{{(kr{\text{)}}}^3}}}} \right)} \right\}\sin \theta \cos \phi \\ \end{gathered}$$

$$\begin{gathered} E_{\theta }^{{in}}=\frac{{i{k^2}{p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}r}}\left\{ {\frac{{d\left[ {rh_{1}^{{(1)}}(kr)} \right]}}{{dr}} - \alpha \frac{{d\left[ {r{j_1}(kr)} \right]}}{{dr}}} \right\}\cos \theta \cos \phi \\ = - \frac{{i{k^3}{p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}}}\left\{ {(\alpha - 1)\left( {\frac{{\sin (kr)}}{{kr}}+\frac{{\cos (kr)}}{{{{(kr)}^2}}} - \frac{{\sin (kr)}}{{{{(kr)}^3}}}} \right)+i\left( {\frac{{\cos (kr)}}{{kr}} - \frac{{\sin (kr)}}{{{{(kr)}^2}}} - \frac{{\cos (kr)}}{{{{(kr)}^3}}}} \right)} \right\}\cos \theta \cos \phi \\ \end{gathered}$$

$$\begin{gathered} E_{\phi }^{{in}}= - \frac{{i{k^2}{p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}r}}\left\{ {\frac{{d\left[ {rh_{1}^{{(1)}}(kr)} \right]}}{{dr}} - \alpha \frac{{d\left[ {r{j_1}(kr)} \right]}}{{dr}}} \right\}\sin \phi \\ =\frac{{i{k^3}{p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}}}\left\{ {(\alpha - 1)\left( {\frac{{\sin (kr)}}{{kr}}+\frac{{\cos (kr)}}{{{{(kr)}^2}}} - \frac{{\sin (kr)}}{{{{(kr)}^3}}}} \right)+i\left( {\frac{{\cos (kr)}}{{kr}} - \frac{{\sin (kr)}}{{{{(kr)}^2}}} - \frac{{\cos (kr)}}{{{{(kr)}^3}}}} \right)} \right\}\sin \phi \\ \end{gathered}$$

with

$$\alpha =\frac{{{\zeta ^2}h_{1}^{{(1)}}(\kappa R){{\left\{ {\frac{{d\left[ {rh_{1}^{{(1)}}(kr)} \right]}}{{dr}}} \right\}}_{r=R}} - h_{1}^{{(1)}}(kR){{\left\{ {\frac{{d\left[ {rh_{1}^{{(1)}}(\kappa r)} \right]}}{{dr}}} \right\}}_{r=R}}}}{{{\zeta ^2}h_{1}^{{(1)}}(\kappa R){{\left\{ {\frac{{d\left[ {r{j_1}(kr)} \right]}}{{dr}}} \right\}}_{r=R}} - {j_1}(kR){{\left\{ {\frac{{d\left[ {rh_{1}^{{(1)}}(\kappa r)} \right]}}{{dr}}} \right\}}_{r=R}}}}$$

Here, $h_{1}^{{(1)}}(x)$ is a spherical Hankel function of the first kind with order 1 and ${j_1}(x)$ is a spherical Bessel function of the first kind with order 1. Also, ${\varepsilon _1}=n_{1}^{2}$ and ${\varepsilon _2}=n_{2}^{2}$ (k = 2πn₁/λ and κ = 2πn₂/λ) are the dielectric constants (propagation constants) inside and outside the cavity, respectively. Meanwhile, ζ = n₂/n₁. In addition, λ is the wavelength in vacuum. The electric fields outside the cavity are derived to give

$$E_{r}^{{out}}=\frac{{i{k^2}\beta {p_m}}}{{2\pi {\varepsilon _1}{\varepsilon _0}r}}h_{1}^{{(1)}}\left( {\kappa r} \right)\sin \theta \cos \phi = - 2\frac{{{k^3}\zeta {p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}}}\beta \left( {\frac{i}{{{{\left( {\kappa r} \right)}^2}}} - \frac{1}{{{{\left( {\kappa r} \right)}^3}}}} \right){e^{i\kappa r}}\sin \theta \cos \phi$$

$$E_{\theta }^{{out}}=\frac{{i{k^2}\beta {p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}r}}\frac{{dh_{1}^{{(1)}}(\kappa r)}}{{dr}}\cos \theta \cos \phi =\frac{{{k^3}\zeta {p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}}}\beta \left( {\frac{1}{{\kappa r}}+\frac{i}{{{{\left( {\kappa r} \right)}^2}}} - \frac{1}{{{{\left( {\kappa r} \right)}^3}}}} \right){e^{i\kappa r}}\cos \theta \cos \phi$$

$$E_{\phi }^{{out}}= - \frac{{i{k^2}\beta {p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}r}}\frac{{d\left[ {rh_{1}^{{(1)}}(\kappa r)} \right]}}{{dr}}\sin \phi = - \frac{{{k^3}\zeta {p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}}}\beta \left( {\frac{1}{{\kappa r}}+\frac{i}{{{{\left( {\kappa r} \right)}^2}}} - \frac{1}{{{{\left( {\kappa r} \right)}^3}}}} \right){e^{i\kappa r}}\sin \phi$$

with

$$\begin{gathered} \beta =\frac{{h_{1}^{{(1)}}(kR){{\left\{ {\frac{{d[r{j_1}(kr)]}}{{dr}}} \right\}}_{r=R}} - {j_1}(kR){{\left\{ {\frac{{d[rh_{1}^{{(1)}}(kr)]}}{{dr}}} \right\}}_{r=R}}}}{{{\zeta ^2}h_{1}^{{(1)}}(\kappa R){{\left\{ {\frac{{d[r{j_1}(kr)]}}{{dr}}} \right\}}_{r=R}} - {j_1}(kR){{\left\{ {\frac{{d[rh_{1}^{{(1)}}(\kappa r)]}}{{dr}}} \right\}}_{r=R}}}} \\ = - i{e^{ - i\kappa R}}{\left\{ \begin{gathered} \frac{1}{\zeta }\left[ { - {\zeta ^2}\left( {\sin (kR)+\frac{{\cos (kR)}}{{kR}} - \frac{{\sin (kR)}}{{{{(kR)}^2}}}} \right) - \left( { - \frac{{\cos (kR)}}{{kR}}+\frac{{\sin (kR)}}{{{{(kR)}^2}}}} \right)} \right] \hfill \\ - i\left[ {{\zeta ^2}\left( {\sin (kR)+\frac{{\cos (kR)}}{{kR}} - \frac{{\sin (kR)}}{{{{(kR)}^2}}}} \right)\frac{{kR}}{{{{(\kappa R)}^2}}}+\left( {\cos (kR) - \frac{{\sin (kR)}}{{kR}}} \right)\left( {1 - \frac{1}{{{{(\kappa R)}^2}}}} \right)} \right] \hfill \\ \end{gathered} \right\}^{ - 1}} \\ \end{gathered}$$

For obtaining the expressions after the second equality signs in Eqs. (3)-(5) and (7)-(10) we have used the expansions of the two special functions as

$$h_{1}^{{(1)}}(kr)= - \frac{{{e^{ikr}}}}{{kr}}\left( {1+\frac{i}{{kr}}} \right)$$

$${j_1}(kr)= - \frac{{\cos (kr)}}{{kr}}+\frac{{\sin (kr)}}{{{{(kr)}^2}}}$$

In the equations above, the nanoscale-cavity effect plays its roles through α and β besides the modified dipole moment, p_m. If the cavity does not exist, i.e., ${n_1}={n_2}$, ${\varepsilon _1}={\varepsilon _2}$, and k = κ, α = 0 and β = 1. In this situation, Eqs. (3)-(5) and (7)-(9) are reduced to the well-known near-field expressions produced by a radiating dipole in a homogeneous medium [33].

The FRET efficiency is determined by the near-field energy of the donor absorbed by the acceptor. This absorbed donor energy is proportional to the total field intensity at the position of the acceptor produced by the donor if the acceptor absorption coefficient is not affected by the donor intensity. In structure B illustrated in Fig. 1(b) in which the donor and acceptor are located at the coordinates of (0, 0, 0) and (0, 0, d) respectively, the enhancement factor of the FRET efficiency caused by the nanoscale-cavity effect is proportional to the normalized field intensity at the position of the acceptor. With an x-dipole, the normalized field intensity, I_n(x) is given by

$${I_{n(x)}}=\frac{{{{\left| {E_{\theta }^{{in}}} \right|}^2}}}{{{{\left| {E_{\theta }^{0}} \right|}^2}}}=\frac{{{{\left| {\frac{{i{k^2}{p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}}}\frac{1}{r}\left\{ { - \alpha \frac{{dr{j_1}(kr)}}{{dr}}+\frac{{drh_{1}^{{(1)}}(kr)}}{{dr}}} \right\}} \right|}^2}}}{{{{\left| {i\frac{{{k^2}{p_0}}}{{4\pi {\varepsilon _1}{\varepsilon _0}}}\frac{1}{r}\left( {\frac{{drh_{1}^{{(1)}}(kr)}}{{dr}}} \right)} \right|}^2}}}={\left| {\frac{{{p_m}}}{{{p_0}}}} \right|^2}\left\{ {1+{{\left| {\frac{{\frac{{dr{j_1}(kr)}}{{dr}}}}{{\frac{{drh_{1}^{{(1)}}(kr)}}{{dr}}}}} \right|}^2}{{\left| \alpha \right|}^2} - \operatorname{Re} \left( {\frac{{\frac{{dr{j_1}(kr)}}{{dr}}}}{{\frac{{drh_{1}^{{(1)}}(kr)}}{{dr}}}}\alpha } \right)} \right\}$$

Here, $E_{\theta }^{0}$ is the θ-component of the electric field with the dipole source in the homogeneous medium of n₁ in refractive index [structure R in Fig. 1(f)]. It can be obtained from $E_{\theta }^{{in}}$ by setting α = 0 and p_m = p₀. From Eq. (6) when kR < < 1 and κR < < 1, we can obtain

$$\alpha \sim \frac{{3i\left( {{\zeta ^2} - 1} \right)}}{{2{\zeta ^2}+1}}\frac{1}{{{{\left( {kR} \right)}^3}}}\left\{ {1+\frac{3}{5}\left( {\frac{{4{\zeta ^2}+1}}{{2{\zeta ^2}+1}}} \right){{\left( {kR} \right)}^2}+O\left[ {{{\left( {kR} \right)}^3}} \right]} \right\}$$

We can see that α diverges when kR→0. Under the condition of kR < < 1 and κR < < 1, the normalized field intensity in Eq. (13) is reduced to

$${I_{n(x)}}\sim {\left| {\frac{{{p_m}}}{{{p_0}}}} \right|^2}\left\{ {{{\left[ {1 - \frac{{2\left( {{\zeta ^2} - 1} \right)}}{{2{\zeta ^2}+1}}{\eta ^3}} \right]}^2} - \frac{6}{5}\left( {\frac{{{\zeta ^2} - 1}}{{2{\zeta ^2}+1}}} \right)\left[ {1 - \frac{{2\left( {{\zeta ^2} - 1} \right)}}{{2{\zeta ^2}+1}}{\eta ^3}} \right]\left[ {{\eta ^2}+2\left( {\frac{{4{\zeta ^2}+1}}{{2{\zeta ^2}+1}}} \right)} \right]{\eta ^3}{{\left( {kR} \right)}^2}} \right\}$$

Here, η = r/R with r < R. When kR→0, Eq. (15) is reduced to

$${I_{n(x)}}\sim {\left| {\frac{{{p_m}}}{{{p_0}}}} \right|^2}{\left[ {1 - \frac{{2\left( {{\zeta ^2} - 1} \right)}}{{2{\zeta ^2}+1}}{\eta ^3}} \right]^2}$$

In Eq. (15) we can see that the normalized field intensity depends on η. When η << 1, i.e., near the source dipole, the normalized field intensity is equal to ${\left| {{p_m}/{p_0}} \right|^2}$. In other words, only the Purcell effect can contribute to the intensity enhancement. As a matter of fact, when η << 1, p_m→p₀, i.e., the Purcell effect is negligibly weak. This is so because near the source dipole the intensity is dominated by the direct radiation field. The scattered field makes an insignificant contribution to the total field near the source dipole.

When a z-oriented source dipole at the cavity center is considered, we can also derive the electric field distribution and hence the normalized intensity. Under the condition of kR < < 1, we can obtain the normalized intensity, I_n(z) as

$${I_{n(z)}}\sim {\left| {\frac{{{p_m}}}{{{p_0}}}} \right|^2}\left\{ {{{\left[ {1+\left( {\frac{{{\zeta ^2} - 1}}{{2{\zeta ^2}+1}}} \right){\eta ^3}} \right]}^2} - \frac{6}{5}\left( {\frac{{{\zeta ^2} - 1}}{{2{\zeta ^2}+1}}} \right)\left[ {1+\left( {\frac{{{\zeta ^2} - 1}}{{2{\zeta ^2}+1}}} \right){\eta ^3}} \right]\left[ {{\eta ^2} - \left( {\frac{{4{\zeta ^2}+1}}{{2{\zeta ^2}+1}}} \right)} \right]{\eta ^3}{{\left( {kR} \right)}^2}} \right\}$$

When kR→0, Eq. (17) is reduced to

$${I_{n(z)}}\sim {\left| {\frac{{{p_m}}}{{{p_0}}}} \right|^2}{\left[ {1+\left( {\frac{{{\zeta ^2} - 1}}{{2{\zeta ^2}+1}}} \right){\eta ^3}} \right]^2}$$

The limiting result in Eq. (18) for z-dipole is different from that in Eq. (16) for x-dipole. In Eq. (16) besides the factor of dipole moment ratio, the normalized intensity is proportional to a factor smaller than unity. However, in Eq. (18) this proportion factor is always larger than unity. This comparison result implies that the nanoscale-cavity effect on FRET efficiency depends on the alignment of the donor and acceptor positions with respect to the donor dipole orientation. When the donor and acceptor positions are aligned with the donor dipole orientation, FRET efficiency can more likely be enhanced through the nanoscale-cavity effect.

In the far field limit, i.e., κr > > 1, the electrical fields outside the cavity shown in Eqs. (7)-(9) can be simplified to $E_{r}^{{out}}\sim 0$,

$$E_{\theta }^{{out}}\sim \frac{{{k^3}\zeta {p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}}}\beta \frac{{{e^{i\kappa r}}\cos \theta \cos \phi }}{{\kappa r}}$$

$$E_{\phi }^{{out}}\sim - \frac{{{k^3}\zeta {p_m}}}{{4\pi {\varepsilon _1}{\varepsilon _0}}}\beta \frac{{{e^{i\kappa r}}\sin \phi }}{{\kappa r}}$$

Equations (19) and (20) are used for evaluating the total radiated power, P, of a dipole inside the cavity to give

$$\begin{gathered} P=\frac{1}{2}\operatorname{Re} \left\{ {\int_{0}^{{2\pi }} {\int_{0}^{\pi } {\left( {{{{\mathbf{\vec {E}}}}^{out}} \times {{{\mathbf{\vec {H}}}}^{out*}}} \right) \cdot {\mathbf{\hat {r}}}{r^2}\sin \theta d\theta d\phi } } } \right\}=\frac{{{n_2}}}{{2{Z_0}}}\int_{0}^{{2\pi }} {\int_{0}^{\pi } {\left( {{{\left| {E_{\theta }^{{out}}} \right|}^2}+{{\left| {E_{\phi }^{{out}}} \right|}^2}} \right){r^2}\sin \theta d\theta d\phi } } \\ =\frac{{k_{0}^{4}{{\left| {{p_m}} \right|}^2}{n_2}}}{{12\pi \varepsilon _{0}^{2}{Z_0}}}{\left| \beta \right|^2} \\ \end{gathered}$$

Here, Z₀ is the intrinsic impedance in vacuum. Without the Purcell effect or feedback process, the total radiated power, P₀, of a dipole ${\vec {p}_0}={p_0}\hat {x}$ in a homogeneous medium of n₁ in refractive index is given by

$${P_0}=\frac{{k_{0}^{4}p_{0}^{2}{n_1}}}{{12\pi \varepsilon _{0}^{2}{Z_0}}}$$

Therefore, the normalized radiated power P_n is given by

$${P_n}=\frac{P}{{{P_0}}}=\zeta {\left| \beta \right|^2}{\left| {\frac{{{p_m}}}{{{p_0}}}} \right|^2}$$

Here, we can see that the radiated power enhancement caused by the nanoscale-cavity effect is determined by two factors, including ${\left| {{p_m}/{p_0}} \right|^2}$ and $\zeta {\left| \beta \right|^2}$. The former (latter) is caused by the Purcell effect (classical scattering). In the limiting case of kR > > 1,

$${\left| \beta \right|^2}\sim \frac{1}{{\left( {{\zeta ^2} - 1} \right){{\sin }^2}(kR)+1}}$$

Therefore, in the far-field limit, when R is large, the normalized radiated power shows a periodical function of photon energy with hc/(2n₁R) in period. By using n₁ = 1.577 and R = 250 (200) nm, the variation period is 1.576 (1.970) eV. In another limiting case of kR < < 1,

$${\left| \beta \right|^2}\sim \frac{{9{\zeta ^4}}}{{{{\left( {2{\zeta ^2}+1} \right)}^2}}}\left\{ {1 - \frac{1}{5}\left( {\frac{{10{\zeta ^4} - 9{\zeta ^2}+1}}{{2{\zeta ^2}+1}}} \right){{(kR)}^2}} \right\}+O({(kR)^4})$$

When kR→0, we can obtain

${\left| \beta \right|^2} \to {\left( {\frac{{3{\zeta ^2}}}{{2{\zeta ^2}+1}}} \right)^2}$ , ${P_n} \to \frac{{9{\zeta ^5}}}{{{{\left( {2{\zeta ^2}+1} \right)}^2}}}{\left| {\frac{{{p_m}}}{{{p_0}}}} \right|^2}$. (26)

It is noted that when kR→0, ${\left| {{p_m}/{p_0}} \right|^2}$→1 because insignificant Purcell effect can be produced under this condition. Therefore, by assuming that n₁ = 1.577 and n₂ = 2.399, ${\left| \beta \right|^2}$→1.523, and P_n →2.317 when kR→0. It is also noted that when the source dipole is placed at the center of a spherical cavity, the normalized radiated power is independent of the dipole orientation.

For numerical computations, without loss of generality, we set n₁ = 1.577 and n₂ = 2.399. The latter (former) implies the use of GaN (the photoresist of SU-8) for embedding (filling) the cavity [15]. Figure 2(a) shows the normalized intensity without the feedback process produced by an x-dipole at the cavity center, I_n(x) under various conditions of cavity radius (R) and observation or acceptor position, which is located at the coordinate of (0, 0, d) [see Fig. 1(b)]. In Fig. 2(a) the R and d values are shown outside and inside the parentheses, respectively, in the labels. Without the feedback process, the results in Fig. 2(a) are obtained from Eq. (13) by removing the factor of ${\left|{p}_{m}/{p}_{0}\right|}^{2}$. Here, we can see that for each R value, when d is small, I_n(x) is almost a constant slightly smaller than unity over the concerned spectral range. Such a result is consistent with Eq. (15) when η = r/R = d/R is small. It has been attributed to the strong radiated field directly from the dipole and hence the effect of scattered field from the cavity structure is weak. When d becomes close to R, the larger η values lead to the significant deviations of I_n(x) from unity. Generally, a lager cavity results in a larger I_n(x). The three vertical dashed lines in Fig. 2(a) indicate the typically used wavelengths of blue, green, and red lights at 450, 530, and 625 nm, corresponding to the photon energies at ~ 2.76, ~ 2.34, and ~ 1.98 eV, respectively. Figure 3(a) shows the results of normalized intensity, I_n(z) similar to Fig. 2(a) (without the feedback process) for the case of a z-dipole at the cavity center. Again, I_n(z) is close to unity when d is small. However, it is slightly larger than unity that is consistent with Eq. (17) or (18) when η is small. When η is larger, I_n(z) becomes significantly larger and approaches a value close to 1.4 at the low-energy limit.

Figures 4(a) and 4(b) show the feedback effects, i.e., ${\left|{p}_{m}/{p}_{0}\right|}^{2}$, for the cases of x- and z-dipole, respectively, under the conditions of R and d values the same as those in Figs. 2(a) plus those of d = 0 for various R values. However, now the d values indicate the positions of the source dipole, i.e., the acceptor. The results of ${\left|{p}_{m}/{p}_{0}\right|}^{2}$ under the condition of d = 0 in the cases of x- and z-dipole are the same. The feedback effect diminishes at the low-energy limit. For any R value, when d is small (3 or 5 nm) the feedback effects are close to the corresponding results of d = 0 and generally decrease with decreasing cavity size. When d is close to the R value, i.e., the source dipole is placed near the cavity boundary, the feedback effect becomes insignificant (${\left|{p}_{m}/{p}_{0}\right|}^{2}\sim1$) unless both R and photon energy are large. The weak feedback effect when the source dipole is close to the cavity boundary is attributed to the saturation of population inversion in the two-level system used for describing the source dipole behavior [30]. In other words, when the source dipole is close to the cavity boundary, the scattered field is so strong such that its feedback effect becomes insignificant [see Eq. (1)]. Such a saturated feedback result obtained based on the assumed two-level system may not perfectly describe the Purcell effect. However, a more complicated model for the source dipole behavior would make the theoretical/numerical studies more difficult. By including the feedback effects or ${\left|{p}_{m}/{p}_{0}\right|}^{2}$ with d = 0 (dipole source at the cavity center) in evaluating I_n(x) and I_n(z) we obtain the results shown in Figs. 2(b) and 3(b) for the cases of x- and z-dipole, respectively. With d = 0 in either Fig. 4(a) or 4(b) the feedback effect can lead to significant enhancements in normalized intensity except the case of R = 10 nm for either x- or z-dipole. In the case of x-dipole, for a small cavity, i.e., R = 10 or 25 nm, the intensity near the cavity boundary (source dipole) is significantly reduced (almost unchanged) through the nanoscale-cavity effect, as shown in Fig. 2(b). However, in the case of z-dipole, for a small cavity, the intensity near the cavity boundary is significantly enhanced through the nanoscale-cavity effect, as shown in Fig. 3(b).

Based on Eqs. (10) and (23) Fig. 5(a) shows the normalized radiated powers of a dipole at the center of a spherical cavity of R in radius as functions of photon energy without the feedback effect, i.e, ζ${\left|\beta \right|}^{2}$. Here, when R is larger than 50 nm, oscillations can be observed, similar to the behavior of far-field cavity resonance. For any R value, ζ${\left|\beta \right|}^{2}$ approaches 2.317 at the low energy limit. By using the results of ${\left|{p}_{m}/{p}_{0}\right|}^{2}$ with d = 0 in Fig. 4(a) we obtain the normalized radiated power, P_n, after the feedback effect is included based on Eq. (23) as shown in Fig. 5(b). Because the feedback effect is insignificant at the low energy limit, P_n shows almost the same behavior as that without feedback for any cavity size at the low energy limit. When R = 250 or 200 nm, three oscillation peaks can be observed. From the low energy end, the photon energy separations between the first and second peaks and between the second and third peaks are 1.697 and 1.600 (2.122 and 1.994) eV, respectively, when R = 250 (200) nm. As shown in Eq. (24) the period of the periodical P_n peaks in the limit of kR > > 1 is 1.576 (1.970) eV when R = 250 (200) nm. These limiting oscillation periods of far-field cavity resonance are slightly smaller than the separations between the second and third peaks mentioned above for the cases of R = 250 and 200 nm. The increased peak separations are caused by the near field scattering in the nano-cavity structure. Such a near field scattering effect becomes stronger when the nano-cavity is smaller. In the case of R = 150 nm, the separation between the two oscillation peaks is increased to 2.835 eV. In the limit of kR < < 1, Eq. (25) predicts a quadratic decreasing trend from the limiting value of P_n = 2.317 with increasing photon energy. With R = 10 or 25 nm, the near field scattering by the nano-cavity structure dominates over the far-field radiation behaviors. The spectral variation behavior of the radiated power in the case of a small R is quite different from that of cavity resonance. Another important observation in Fig. 5(b) is that P_n is always larger than unity in the concerned spectral range when R is smaller than 50 nm, indicating that the nanoscale-cavity effect can enhance the radiated power of a light emitter located inside a nano-cavity. Figure 6 shows the variations of normalized radiated power, P_n, with R for five wavelengths. Here, one can see that when R < 50 nm, the spectral dependence of P_n is relatively weaker, compared with that of a far-field cavity resonance.

Figures 7(a) and 7(b) show the results of normalized radiated powers without and with the feedback effect, similar to those in Figs. 5(a) and 5(b) respectively, when the position of an x-dipole deviates from the cavity center. Here, the numbers outside (inside) the parentheses indicate the values of cavity radius, R (dipole position, d). Between the cases of different dipole positions, in the visible range, the radiated power is always higher when the source dipole is located close to the cavity boundary. In other words, the nanoscale-cavity effect is stronger when the source dipole is close to the cavity boundary. Figures 8(a) and 8(b) show the results for a z-dipole with its position deviated from the cavity center, similar to those in Figs. 7(a) and 7(b) for an x-dipole, respectively. The radiated power behaviors of a z-dipole are similar to those of an x-dipole. As mentioned earlier, the normalized intensities at the acceptor positions shown in Figs. 2(b) and 3(b) can be regarded as the enhancements of FRET efficiency. The color conversion efficiency relies on the two enhancement factors of FRET and acceptor radiated power. The product of these two factors corresponds to the enhancement of color conversion. In column 4 of Table 1, we show the normalized intensities at 530 nm in wavelength for different acceptor positions (d) when the x- or z-oriented donor dipole is located at the center of a spherical nano-cavity with three R values based on the results in Figs. 2(b) and 3(b). In column 5 of Table 1, the corresponding normalized radiated powers of the acceptor at 625 nm in wavelength are shown based on the results in Figs. 7(b) and 8(b). The products of two enhancement factors in columns 4 and 5 of Table 1 are shown in column 6 for the color conversion enhancements. Except the cases of an x-dipole located close to cavity boundary, color conversion can always be significantly enhanced through the nanoscale-cavity effect.

Table 1

Normalized donor field intensity at 530 nm in wavelength, normalized acceptor radiated power at 625 nm, and color conversion enhancement factor under various conditions of different cavity radii, acceptor positions, and donor dipole orientations.
Cavity radius, R (nm)	Acceptor position, d (nm)	Donor dipole orientation	Normalized intensity at 530 nm	Normalized radiated power at 625 nm	Color conversion enhancement
10	3	x	0.974	2.253	2.194
	3	z	1.013	2.253	2.485
	9	x	0.414	2.269	0.939
	9	z	1.375	2.266	3.116
25	5	x	0.991	1.995	1.977
	5	z	1.004	1.994	2.002
	22	x	0.385	2.086	0.803
	22	z	1.352	2.069	2.797
50	5	x	0.999	1.595	1.593
	5	z	1.001	1.591	1.593
	45	x	0.530	1.847	0.979
	45	z	1.296	1.792	2.322

For understanding the nanoscale-cavity effect on the SP coupling behavior inside a nano-cavity, we consider structure C illustrated in Fig. 1(c) in which a spherical Ag NP of b in radius is located at the nano-cavity center (the coordinate origin) for inducing localized surface plasmon (LSP) resonance to SP-couple with a radiating dipole situated at the coordinate of (0, 0, s). To separate the nanoscale-cavity effect from the SP coupling effect on radiated power, structure D (E) illustrated in Fig. 1(d) [1(e)] are considered for comparing with structure C when the Ag NP is removed (the cavity radius becomes infinitely large). Although analytic solutions can also be derived for the radiated powers in structures C-E, they are complicated. A numerical simulation method is chosen based on the commercial software of COMSOL [23, 24]. By setting R = 60 nm, b = 20 nm, and s = 50 nm, we obtain the numerical results of normalized radiated powers (normalized with respect to the corresponding results in other structures) in Figs. 9(a) and 9(b) for the x- and z-dipole, respectively. In numerical computations, we use the experimental data for the dielectric constant of Ag [34]. In either Fig. 9(a) or 9(b) the curve labeled by D/R (E/R) describes the effect of nano-cavity without SP coupling (the effect of SP coupling in a homogeneous medium of n₁ in refractive index) on radiated power. Also, the curve labeled by C/E (C/D) shows the effect of nano-cavity (SP coupling) under the condition of SP coupling (in the nano-cavity structure). Then, the curve labeled by C/R demonstrates the combined effect of nano-cavity and SP coupling. The behaviors of normalized radiated power with a dipole inside a nano-cavity (D/R) has been discussed in Figs. 7(b) and 8(b). In either case of x- or z-dipole near the Ag NP in a homogeneous medium (E/R) the SP coupling leads to a Fano oscillation in radiated power around 2.8 eV. However, their relative phases are opposite. In this situation, the introduction of a spherical nano-cavity structure to the SP-coupling system (C/D) also results in Fano oscillations in radiated power. However, the oscillation amplitudes become smaller, particularly in the case of x-dipole. The introduction of SP coupling to a nano-cavity structure (C/E) reverses the phase of the Fano oscillation from the result of C/D in either dipole orientation case. In this situation (C/E) for an x- (z-) dipole, the oscillation amplitude is enhanced (reduced). Finally, by normalizing the radiated power with respect to that of the reference structure in combining the nanoscale-cavity and SP coupling effects (C/R) the Fano oscillation amplitude in the case z-dipole is significantly larger than that in the case of x-dipole. As illustrated in curves C/E, in either case of dipole orientation, the nanoscale-cavity effect can increase the radiated power of an SP coupling system in most concerned spectral range. An x-dipole can lead to a relatively larger maximum enhancement factor even though its absolute radiated power can be smaller, when compared with a z-dipole.

The structure of a closed spherical nano-cavity with radiating dipoles inside used for the theoretical/numerical studies above may be impractical for experimental implementation. However, a closed nano-cavity can be implemented with a surface nano-hole filled with a photoresist solution of colloidal QDs and covered by a deposited metal or dielectric over-layer after the photoresist is solidified [15, 22]. Although it is reasonable to use a two-level system for describing the electron transition and radiation behaviors of a colloidal QD in evaluating the feedback or Purcell effect, the resultant saturation effect can be overestimated. Therefore, the feedback effect and hence the overall nanoscale-cavity effect can be underestimated when a source dipole is close to the cavity boundary. The nanoscale-cavity effect can be observed not only for a radiating dipole inside a nano-cavity, but also for a dipole outside but close to a nano-cavity. A practical implementation of such a structure is to fabricate a nano-cavity near a QW [7, 8, 15, 22]. The nanoscale-cavity effect in such a structure can enhance the color conversion efficiency from the QW (a radiating dipole outside the nano-cavity) into a QD inserted into the nano-cavity.

The far-field cavity resonance behavior is quite sensitive to the cavity geometry. However, the nanoscale-cavity effect is not so sensitive to the cavity geometry. As illustrated in the inset of Fig. 10, we consider a cubic nano-cavity with dimension L for comparing its radiated power with that of a spherical nano-cavity. For a reasonable comparison, we choose the L value such that the volume of the cubic cavity is the same as that of a spherical cavity of R in radius, i.e., L³ = 4πR³/3. Figure 10 shows the results of the normalized radiated power, P_n, of an x-dipole located at the center of the cubic-cavity structure at the five labeled wavelengths as functions of L (the lower abscissa) or corresponding R (the upper abscissa). One can see that when L is smaller than ~ 100 nm, the nanoscale-cavity effect leads to the P_n behaviors close to those in the spherical-cavity structure, indicating that the nanoscale-cavity effect is quite insensitive to the cavity geometry.

In summary, based on a spherical nano-cavity structure, we have formulated the electric fields and radiated powers of a radiating dipole located inside the nano-cavity. Such formulations revealed that the nano-cavity structure or nanoscale-cavity effect could enhance the near-field intensity inside the cavity and far-field radiated power of the dipole. The enhancements caused by the nanoscale-cavity effect had two contributions, including the classical electromagnetic scattering as formulated and the Purcell or feedback effect. The enhancement of field intensity could increase the FRET efficiency when both donor and acceptor were located inside the nano-cavity. By combining the enhancements of field intensity and radiated power, we showed the increase of the color conversion efficiency through the nanoscale-cavity effect. Besides, through numerical simulations, we demonstrated the enhancement of the SP coupling effect for further increasing the radiated power of a dipole located nearby an Ag NP inside a nano-cavity.

Funding

This research was funded by the Ministry of Science and Technology, Taiwan, the Republic of China, under the grant of MOST 111-2221-E-002-073 and MOST 110-2221-E-002-131.

Author Contributions

Yang Kuo: Formulation and numerical algorithm preparation, numerical computations; C. C. (Chih-Chung) Yang: Concept proposing, data interpretations, writing.

Data Availability

All data included in this paper are available upon request by contacting with the corresponding author.

Conflicts of interest/Competing interests Availability of data and material The authors declare that they have no conflict of interest.

Ethical Approval This is a theoretical/numerical study on the nanoscale-cavity effects.

Consent to Participate This is a theoretical/numerical study of the nanoscale-cavity effects on dipole emission, Förster resonance energy transfer, and surface plasmon coupling.

Consent for Publication All authors of this paper agree to publish our theoretical/numerical research.

Purcell EM (1946) Spontaneous emission probabilities at radio frequencies. Phys Rev 69:681–681
Chen D, Xiao H, Han J (2012) Nanopores in GaN by electrochemical anodization in hydrofluoric acid formation and mechanism. J Appl Phys 112(6):064303
Griffin PH, Oliver RA (2020) Porous nitride semiconductors reviewed. J Phys D: Appl Phys 53(38):383002
Schwab MJ, Chen D, Han J, Pfefferle LD (2013) Aligned mesopore arrays in GaN by anodic etching and photoelectrochemical surface etching. J Phys Chem C 117(33):16890–16895
Schwab MJ, Han J, Pfefferle LD (2015) Neutral anodic etching of GaN for vertical or crystallographic alignment. Appl Phys Lett 106(24):241603
Tseng WJ, van Dorp DH, Lieten RR, Vereecken PM, Borghs G (2014) Anodic etching of n-GaN epilayer into porous GaN and its photoelectrochemical properties. J Phys Chem C 118(51):29492–29498
Chen CH, Kuo SY, Feng HY, Li ZH, Yang S, Wu SH, Hsieh HY, Lin YS, Lee YC, Chen WC, Wu PH, Chen JC, Huang YY, Lu YJ, Kuo Y, Lin CF, Yang CC (2023) Photon color conversion enhancement of colloidal quantum dots inserted into a subsurface laterally-extended GaN nano-porous structure in an InGaN/GaN quantum-well template. Opt Express 31(4):6327–6341
Yang S, Feng HY, Lin YS, Chen WC, Kuo Y, Yang CC (2023) Effects of surface plasmon coupling on the color conversion from an InGaN/GaN quantum-well structure into colloidal quantum dots inserted into a nearby porous structure. Nanomaterials 13(2):328
Chanyawadee S, Lagoudakis PG, Harley RT, Charlton MDB, Talapin DV, Huang HW, Lin CH (2010) Increased color-conversion efficiency in hybrid light-emitting diodes utilizing non-radiative energy transfer. Adv Mater 22(5):602–606
Krishman C, Brossard M, Lee KY, Huang JK, Lin CH, Kuo HC, Charlton MDB, Lagoudakis PG (2016) Hybrid photonic crystal light-emitting diode renders 123% color conversion effective quantum yield. Optica 3(5):503–509
Zhuang Z, Guo X, Liu B, Hu F, Li Y, Tao T, Dai J, Zhi T, Xie Z, Chen P, Chen D, Ge H, Wang X, Xiao M, Shi Y, Zheng Y, Zhang R (2016) High color rendering index hybrid III-nitride/nanocrystals white light-emitting diodes. Adv Funct Mater 26(1):36–43
Athanasiou M, Papagiorgis P, Manoli A, Bernasconi C, Poyiatzis N, Coulon PM, Shields P, Bodnarchuk MI, Kovalenko MV, Wang T, Itskos G (2020) InGaN nanohole arrays coated by lead halide perovskite nanocrystals for solid-state lighting. ACS Appl Nano Mater 3(3):2167–2175
Wan R, Li G, Gao X, Liu Z, Li J, Yi X, Chi N, Wang L (2021) Nanohole array structured GaN-based white LEDs with improved modulation bandwidth via plasmon resonance and non-radiative energy transfer. Photon Res 9(7):1213–1217
Du Z, Li D, Guo W, Xiong F, Tang P, Zhou X, Zhang Y, Guo T, Yan Q, Sun J (2021) Quantum dot color conversion efficiency enhancement in micro-light-emitting diodes by non-radiative energy transfer. IEEE Electron Dev Lett 42(8):1184–1187
Huang YY, Li ZH, Lai YC, Chen JC, Wu SH, Yang S, Kuo Y, Yang CC, Hsu TC, Lee CL (2022) Nanoscale-cavity enhancement of color conversion with colloidal quantum dots embedded in the surface nano-holes of a blue-emitting light-emitting diode. Opt Express 30(17):31322–31335
Förster T (1946) Energy transport and fluorescence. Naturwissenschaften 33:166–175
Chanyawadee S, Lagoudakis PG, Harley RT, Charlton MDB, Talapin DV, Huang HW, Lin CH (2010) Increased color-conversion efficiency in hybrid light-emitting diodes utilizing non-radiative energy transfer. Adv Mater 22(5):602–606
Demira HV, Nizamoglub S, Erdemb T, Mutluguna E, Gaponikc N, Eychmuller A (2011) Quantum dot integrated LEDs using photonic and excitonic color conversion. Nano Today 6(6):632–647
Zhang F, Liu J, You G, Zhang C, Mohney SE, Park MJ, Kwak JS, Wang Y, Koleske DD, Xu J (2012) Nonradiative energy transfer between colloidal quantum dot-phosphors and nanopillar nitride LEDs. Opt Express 20(S2):A333–A339
Ni CC, Kuo SY, Li ZH, Wu SH, Wu RN, Chen CY, Yang CC (2021) Förster resonance energy transfer in surface plasmon coupled color conversion processes of colloidal quantum dots. Opt Express 29(3):4067–4081
Chen YP, Ni CC, Wu RN, Kuo SY, Su YC, Huang YY, Chen JW, Hsu YC, Wu SH, Chen CY, Wu PH, Kiang YW, Yang CC (2021) Combined effects of surface plasmon coupling and Förster resonance energy transfer on the light color conversion behaviors of colloidal quantum dots on an InGaN/GaN quantum-well nanodisk structure. Nanotechnology 32(13):135206
Yang S, Lai YC, Feng HY, Lee YC, Li ZH, Wu SH, Lin YS, Hsieh HY, Chu CJ, Chen WC, Kuo Y, Yang CC (2023) Enhanced color conversion of quantum dots located in the hot spot of surface plasmon coupling. IEEE Photon Technol Lett 35(5):273–276
Kuo Y, Ting SY, Liao CH, Huang JJ, Chen CY, Hsieh C, Lu YC, Chen CY, Shen KC, Lu CF, Yeh DM, Wang JY, Chuang WH, Kiang YW, Yang CC (2011) Surface plasmon coupling with radiating dipole for enhancing the emission efficiency of a light-emitting diode. Opt Express 19(14):A914–A929
Kuo Y, Chang WY, Lin CH, Yang CC, Kiang YW (2015) Evaluating the blue-shift behaviors of the surface plasmon coupling of an embedded light emitter with a surface Ag nanoparticle by adding a dielectric interlayer or coating. Opt Express 23(24):30709–30720
Cai CJ, Wang YT, Ni CC, Wu RN, Chen CY, Kiang YW, Yang CC (2020) Emission behaviors of colloidal quantum dots linked onto synthesized metal nanoparticles. Nanotechnology 31(9):095201
Wang YT, Liu CW, Chen PY, Wu RN, Ni CC, Cai CJ, Kiang YW, Yang CC (2019) Color conversion efficiency enhancement of colloidal quantum dot through its linkage with synthesized metal nanoparticle on a blue light-emitting diode. Opt Lett 44(23):5691–5694
Chang WY, Kuo Y, Kiang YW, Yang CC (2019) Simulation study on light color conversion enhancement through surface plasmon coupling. Opt Express 27(12):A629–A642
Lin CH, Chiang HC, Wang YT, Yao YF, Chen CC, Tse WF, Wu RN, Chang WY, Kuo Y, Kiang YW, Yang CC (2018) Efficiency enhancement of light color conversion through surface plasmon coupling. Opt Express 26(18):23629–23640
Chen CY, Ni CC, Wu RN, Kuo SY, Li CH, Kiang YW, Yang CC (2021) Surface plasmon coupling effects on the Förster resonance energy transfer from quantum dot into rhodamine 6G. Nanotechnology 32(29):295202
Boyd RW (2008) Nonlinear Optics, 3rd ed.:278–293. Academic press
Loudon R (1991) The Quantum Theory of Light, 2nd ed.:78–79. Oxford university press
Meystre P, Sargent M III (1999) Elements of Quantum Optics:344. Springer
Jackson JD (1975) Classical Electrodynamics, 2nd ed.:395. Wiley
Palik ED (1991) Handbook of Optical Constants of Solids. Academic press

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Theoretical/Numerical Studies of the Nanoscale-cavity Effects on Dipole Emission, Förster Resonance Energy Transfer and Surface Plasmon Coupling

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