# Dipole Emission to Surface Plasmon-Coupled Enhanced Transmission in Diamond Substrates with Nitrogen Vacancy Center- Near the Surface

## Abstract

**:**

## 1. Introduction

## 2. Resonance Energy Transfer

_{d}is undertaken, where z = 0 plane marks the diamond/air interface. The model is developed in 2D due to the limited computational resource available to the author. The analysis is then extended to a multilayer diamond/silver/air structure. Finite Element Method (FEM) simulations are carried out at the target wavelength of λ

_{0}= 700 nm that coincides with the spectral peak of the phonon sideband associated with the emission from nitrogen vacancies in nano-diamonds at room temperature [32,33]. The refractive index of the diamond substrate is set to n

_{sub}= 2.4 and the refractive index data for silver was taken from Palik [34]. Figure 1a depicts the scattered electric field intensity, |E|

^{2}, calculated for a diamond membrane with a NV- positioned at (x

_{d}, z

_{d}) = (0, −10) nm with its dipole moment aligned with the x-axis (i.e., horizontally oriented). The power ratio of P

_{air}/P

_{sub}≈ 0.5 is a clear indication that in the absence of any plasmonic structure most of the emitted power is scattered into the diamond. The power ratio ${P}_{\mathrm{air}}/{P}_{\mathrm{sub}}={\displaystyle \mathsf{\int}{\langle S\rangle}_{\mathrm{air}}\cdot ds}/{\displaystyle \mathsf{\int}\frac{{\langle S\rangle}_{\mathrm{sub}}}{{n}_{\mathrm{sub}}}\cdot ds}$ is calculated over a closed arc

**s**where <

**S**>

_{air,sub}are the time averaged Poynting vectors calculated in the air and inside the diamond substrate respectively. Introducing an optically thick silver film on top of the diamond substrate reduced the scattered power into the air to ~0, see Figure 1b. Electric field components, (E

_{x}, E

_{z}), calculated over the diamond/silver interface, Figure 1e, with their corresponding Fast Fourier Transforms (FFT), Figure 1g, revealed the presence of surface waves, having a wavelength λ

_{SPPd}= 230 nm. This agrees with the Surface Plasmon Polaritons (SPP) wavelength λ

_{SPPd}= 246 nm obtained analytically using ${k}_{\mathrm{SPP}}=\mathrm{Re}\left[\sqrt{\frac{{\epsilon}_{\mathrm{m}}{\epsilon}_{\mathrm{d}}}{{\epsilon}_{\mathrm{m}}+{\epsilon}_{\mathrm{d}}}}{k}_{0}\right]$ at λ

_{0}= 700 nm, where k

_{0}is the free space wavenumber, ε

_{m}and ε

_{d}are relative permittivities of the metal and dielectric respectively. It is apparent that some of the emanating power from the dipole is consumed in launching SPPs. Presence of a dipole-like activity associated with a metallic slit has been reported previously [35]. Therefore, by introducing a resonant aperture perforating the film just above the NV-, it is possible establish the dipole-dipole resonance energy transfer between the NV- and the aperture.

_{air}) in the air half-space vs. the film thickness, t, when an aperture having a width w

_{a}= 30 nm, perforates the silver film just above the emitter. The RDRE

_{air}was calculated for the air half-space using ${\mathrm{RDRE}}_{\mathrm{air}}={P}_{\mathrm{air}}/\left(0.5\times {P}_{0}\right)={\displaystyle \mathsf{\int}{\langle S\rangle}_{\mathrm{air}}\cdot ds}/0.5\times {\displaystyle \mathsf{\int}{\langle S\rangle}_{\mathrm{vac}}\cdot ds}$, where P

_{0}is the total power emitted by a nano-diamond in vacuum. The diameter of the isolated nano-diamond was also set to 30 nm in all calculations. This is a typical size for a nano-diamond with luminescence properties [36,37,38]. The factor of 0.5 in the denominator is due to P

_{0}being calculated along the arc length of a full circle, whereas P

_{air}was calculated over a semicircle in the air half-space. With z

_{d}= 221,210 nm, a maximum RDRE

_{air}≈ 140 was achieved for t = 110 nm that corresponds to the first Fabry-Pérot resonance of the aperture [39]. Figure 1d depicts the electric field intensity, |E|

^{2}, calculated for the diamond/silver/air multilayer with a 30 nm wide aperture perforating the 110 nm silver film just above the NV-. The power ratio P

_{air-Ag110nm+hole}/P

_{sub-Ag110nm+hole}= 10 is a clear indication that scattered power is preferential towards the air. Calculated energy transfer efficiency between the donor and the acceptor [21], E

_{T}= P

_{D→A}/(P

_{D}+ P

_{D→A}) = P

_{air-Ag110nm+hole}/(P

_{air-noFilm}+ P

_{air-Ag110nm+hole}) ≈ 1 was obtained for z

_{d}≤ −10 nm. Power ratios P

_{sub-Ag110nm+hole}/P

_{sub-noFilm}= 1.5 and P

_{air-Ag110nm+hole}/P

_{air-noFilm}= 33, however, suggest that the presence of the silver film with a resonant aperture enhances the reflected power inside the diamond as well as the transmitted power into the air, hinting at a change in the density of states. It is therefore intuitive to infer that RDRE

_{air}∝ P

_{D→A}. Radiation pattern of transmitted field, however, is dispersive with 1/e of its maximum intensity not reaching beyond z ≈ 1.7 μm along the optical axis, i.e., z-axis. Figure 1d also shows the presence of high intensity evanescent field along the silver/air interface. Electric field components, Figure 1f, and their corresponding FFTs, Figure 1h, calculated over the silver/air interface, confirmed the presence of surface waves with λ

_{SPPa}= 667 nm which is in agreement with λ

_{SPPa}= 682 nm obtained analytically. Although evanescent waves do not contribute to the far-field [21], it is possible to intercept and scatter them into freely propagating EM waves by introducing periodic surface gratings surrounding the aperture [40]. Note however, that for a horizontally oriented dipole moment, induced E

_{x}is an even function of x whereas E

_{z}is an odd function. Consequently, any contribution made by E

_{z}to the scattered fields leads to a destructive interference along the optical axis. Therefore, considering a 2D model, E

_{x}(with even parity) is the only contributing factor to the far-field intensity along the optical axis.

## 3. Design of Surface Gratings

_{h}emanating from the hole in the z-direction, ${\psi}_{n}={A}_{n}\frac{{e}^{i\left({k}_{0}{r}_{n}+{\varphi}_{n}\right)}}{{r}_{n}}\mathrm{sin}{\theta}_{n}$ is the scattered wave by the nth groove having an amplitude A

_{n}, ${r}_{n}=\sqrt{{\left(nP\right)}^{2}+{z}^{2}}$ is the distance between nth groove and an arbitrary point along the z-axis, ${\varphi}_{n}={\angle {\Psi}_{SPP}|}_{x={x}_{n}}$ is the phase of the superposed (not just forward propagating) SPP waves at the nth groove, see Figure 2a. The role of a corrugation (or a groove) positioned at x

_{n}= nP, is to utilize some of the power carried by the SPPs to excite LSPs inside the groove, Figure 2b, whose far-field radiation pattern is similar to that of a dipole antenna. The strength of the LSP inside the nth groove is denoted by A

_{n}. To simplify the argument consider Equation (1) with N = 1 in a symmetric configuration consisting of an aperture at x = 0 and a pair of grooves at x

_{1}and x

_{−1}with x

_{−1}= −x

_{1}, where the superposition of SPPs at x = x

_{1}may be defined as:

_{1}, the second term is the wave launched by the aperture towards x

_{−1}being reflected back towards x

_{1}, and the third term is the reflected wave from x

_{1}towards the aperture, being reflected back towards itself. The phase difference between the wave emanating from the aperture and the wave scattered by the groove calculated along the z-axis, ${\mathsf{\Phi}|}_{x=0}=\angle {\psi}_{\mathrm{h}}-\angle {\psi}_{1}$ vs. x

_{1}, showed that for x

_{1}= mλ

_{SPP}/2, Φ → 0 asymptotically only for z → ∞, Figure 2c, but z = ∞ is not a quantitative measure for the far-field. Although matching the period of the corrugations to the SPP wavelength leads to a collimated beam at z = ∞, simulations confirmed that it did not produce the highest possible intensity along the z-axis (results not shown here). Therefore, matching the period of the corrugations to λ

_{SPPa}is an oversight.

_{1}obtained at the silver/air interface using Equation (2). Note that the square of the amplitude is proportional to the total power available to excite LSPs inside the groove, hence ${A}_{1}\propto {\left|{\mathsf{\Psi}}_{\mathrm{SPPa}}\right|}_{x={x}_{1}}$. This simple configuration was also modelled in 2D using FEM. The numerically obtained magnetic field intensity, $\iint {\left|{H}_{y}\right|}_{x={x}_{1}}^{2}dxdz$, calculated inside the groove vs. x

_{1}agrees with the analytical solution. Total power available to the groove follows a Fabry-Pérot like resonance that depends on the groove’s distances from the aperture. The strongest LSP excitation was observed when the groove was positioned at x

_{1}= 505 nm. The situation becomes further complicated when considering the impact of Fabry-Pérot oscillations on the total power inside aperture, hence the enhancement/suppression of the RDR

_{air}. Figure 2d (inset) shows that $\iint {\left|{H}_{y}\right|}_{x=0}^{2}dxdz$ calculated inside the aperture has its maximum when x

_{1}= 475 nm and well away from x

_{1}= λ

_{SPPa}, marked by a star.

_{0}, high intensity radiation along the optical axis is guaranteed [42] (p. 399). But one must also maximize the radiating power available to both the aperture and the grooves. It is intuitive, therefore, to set x

_{1}to the average of the two peaks mentioned above, i.e., x

_{1}= 0.5 × (475 + 505). It is obvious that Equation (2) becomes convoluted for N > 1. Considering such complex interactions, the only viable approach in designing a BE structure in the presence of a quantum emitter, is to solve the Maxwell equations numerically. With t = 110 nm and w

_{a}= 30 nm being the optimum values for the film thickness and the aperture width, r

_{1}was set to x

_{1}+ w

_{1}/2 = 555 nm and a series of parametric sweeps were carried out over each of the remaining parameters w

_{1}, w

_{p}, h and P for z

_{d}= −10 nm. In each case, the spectrum for the scattered power into the air vs. the parameter was calculated to identify the peak position that corresponds to the optimum dimension. Note that this is the same approach used in determining the optimum film thickness, t, see Figure 1c. One possible configuration, C1, capable of collimating the emission as well as improving the RDR

_{air}was found to be w

_{a}= 30 nm, w

_{1}= 130 nm, w

_{p}= 210 nm, h = 70 nm, r

_{1}= x

_{1}+ w

_{1}/2 = 555 nm, P = 590 nm and t = 110 nm. Enhancement to the transmitted RDR

_{air}peaks for N = 5 with no significant changes to the radiation pattern for N > 5, see Figure 3. This implies that most (if not all) of the power carried by the SPPs are intercepted and scattered by the first five grooves. Therefore, for the rest of the calculations only 5 grooves are considered. Note that in this report, all 2D field plots are normalized to the same scale and comparable to one another.

## 4. Dipole’s Position and Orientation

_{air-C1}/(P

_{air-C1}+ P

_{sub-C1}) and RDRE

_{air}vs. z

_{d}is depicted in Figure 4a. Note that preliminary investigations with 1 nm steps, revealed the presence of a single peak in RDRE

_{air}at z

_{d}= −3 nm. Therefor the resolution for the final parametric sweep over z

_{d}was set to 1 nm in the range of −5 nm ≤ z

_{d}≤ 0 and 5nm for z

_{d}< −5 nm. An enhancement of RDRE

_{air}≥ 216 was achieved for z

_{d}≤ −5 nm with a maximum of 219 at z

_{d}= −3 nm and a drop to 1/e at z

_{d}= 17 nm. Rate of change in CE vs. z

_{d}was found to be much slower in comparison to that of RDRE

_{air}. A minimum of 80% efficiency in the range of z

_{d}≤ 25 nm with a maximum of 86% at z

_{d}= −5 nm was obtained. For of a horizontally oriented dipole, transmitted RDRE

_{air}showed a reciprocal relation to those corresponding to dipole emissions near flat metallic surfaces. Compare RDRE

_{air}spectrum in Figure 4a in this report to (P/P

_{0})

^{-1}spectrum in Figure 10.5(a) in “Principles of nano-optics” [21], for example. The slow drop of ~1/z

_{d}

^{0.8}in RDRE

_{air}is attributed to the energy transfer between the NV- and the resonant aperture that involves a mutli-channel mechanism. While the direct excitation of LSPs inside the resonant aperture, with its finite cross section, is dominated by short range interactions, the surface surrounding the aperture is able to capture NV-’s emission for longer distances, producing SPPs which consequently couple to the aperture’s LSPs. This slow rate of change with respect to the dipole’s distance from the surface is extremely beneficial. Beaming profiles for few arbitrary z

_{d}values are depicted in Figure 4b–e.

_{d}≤ −10 nm where the near total energy transfer occurs, the donor-acceptor relation between the NV- and the aperture implies that the performance of such a device is sensitive to the donor’s dipolar orientation with respect to that of the acceptor. Retaining only the near-field term, radiation pattern of the donor (i.e., NV-) may be written as [21]:

_{D}, see inset of Figure 5a. C1’s response vs. the direction of µ

_{D}in the x-z plane was calculated in the range of 0° ≤ Θ

_{xz}≤ 90°. Figure 5a shows that the power transfer between the donor (NV-) and the acceptor (aperture) does in fact comply with P

_{D→A}∝ |µ

_{A}·E

_{D}|

^{2}[21], where µ

_{A}is the dipole moment of the acceptor. Consequently, for a vertically oriented µ

_{D}, P

_{D→A}and RDRE

_{air}drop to zero. Unlike the simple donor-acceptor interaction observed in florescent molecules, the interaction between the NV- and C1 is influenced by plasmonic effects, therefore surface wave launched by the aperture must also be examined. Numerically obtained phase difference between E

_{x}oscillations on the corners of the aperture at the silver/diamond interface, ${{\mathsf{\Phi}}_{\mathrm{L}-\mathrm{R}}|}_{z=0}=\angle {E}_{x}(-{w}_{\mathrm{a}}/2,0)-\angle {E}_{x}({w}_{\mathrm{a}}/2,0)$, revealed that Φ

_{L-R}is a function of Θ

_{xz}, and may be defined as ${{\mathsf{\Phi}}_{\mathrm{L}-\mathrm{R}}|}_{z=0}\equiv f({\mathsf{\Theta}}_{\mathrm{xz}})\approx 180\xb0\left[1-\mathrm{cos}\left({\mathsf{\Theta}}_{\mathrm{xz}}\right)\right]$, Figure 5a. This implies that one can define the parity state of E

_{x}as $\mathrm{cos}\left(f({\mathsf{\Theta}}_{\mathrm{xz}})\right)|0\rangle +\mathrm{sin}\left(f({\mathsf{\Theta}}_{\mathrm{xz}})\right)|1\rangle $, where $|0\rangle $ and $|1\rangle $ are the states of pure even and pure odd parities with probabilities of cos

^{2}(f(Θ

_{xz})) and sin

^{2}(f(Θ

_{xz})) respectively. For pure $|1\rangle $ state of E

_{x}, however, one must not assume that NV’s emission is quenched. It can be demonstrated that for a vertically oriented µ

_{D}a small displacement in its horizontal position with respect to the center of the aperture, not only overcomes the suppressed P

_{D→A}, but also causes the transition from the odd to even state, $|1\rangle \to |0\rangle $, hence satisfying the two mandatory conditions for a high intensity constructive interference along the z-axis. Numerical results depicted in Figure 5b confirm that for 16 ≤ x

_{d}≤ 20 nm both conditions are satisfied. The trend observed in the RDRE

_{air}spectrum is due to the donor’s projected field intensity over the aperture, |E

_{D}|

^{2}

_{(x=0,z=0)}, varying with x

_{d}. Analytical values for |E

_{D}|

^{2}

_{(x=0,z=0)}vs. x

_{d}were calculated using Equation (3), with sin ϑ = |x

_{d}|/r, see Figure 5b (inset). Although, Equation (3) does not take into account the aperture’s dimension, reflection terms and interactions with surface waves, the trend in |E

_{D}|

^{2}spectrum agrees with that obtained numerically for RDRE

_{air}. Directional gain of the BE antenna, |E(θ)|

^{2}/|E

_{0}(θ)|

^{2}, vs. Θ

_{xz}is depicted in Figure 5c. The field intensity along the optical axis can be 800 times that of a free standing nano-diamond when Θ

_{xz}= 0°. Here |E(θ)|

^{2}and |E

_{0}(θ)|

^{2}are the electric field intensities as a function of angle, θ, from the x-axis calculated for C1 and a nano-diamond particle in vacuum respectively. FWHM of the beam remains approximately 14° for all Θ

_{xz}values. Corresponding directional gains vs. x

_{d}= {10, 16, 21, 33 nm when Θ

_{xz}= 90° are shown in Figure 5d. The angle θ, may be expressed in terms of Θ

_{xz}, x

_{d}and z

_{d}as: $\vartheta =90\xb0-\left|90\xb0-{\mathsf{\Theta}}_{\mathrm{xz}}-{\mathrm{tan}}^{-1}\left(\frac{\left|z\mathrm{d}\right|}{x\mathrm{d}}\right)\right|$, valid in the range of 0° ≤ Θ

_{xz}≤ 90° and x

_{d}≥ 0.

**µ**

_{D}in the xy plane, C1 was first modelled in 3D as a BE structure with a 250 nm long slit and later with a symmetric cross-shaped aperture having the same arm-lengths. The spherical simulation domain was divided by the planar structure into two hemispheres, with the bottom being the diamond substrate and the upper being the air. The first Fabry-Pérot resonance of the aperture in both cases occurs at t = 130 nm. To reduce the required computational resources, number of corrugations were limited to N = 3. All other dimensions were kept in accordance to the 2D model. A parametric sweep over the orientation of

**µ**

_{D}in the xy plane, Θ

_{xy}, showed that RDRE

_{air}∝ |

**µ**

_{A}·E

_{D}|

^{2}for the slit, as anticipated. RDRE

_{air}in 3D was calculated using ${P}_{\mathrm{air}}/{P}_{\mathrm{sub}}={\displaystyle \mathsf{\int}{\langle S\rangle}_{\mathrm{air}}\cdot ds}/{\displaystyle \mathsf{\int}\frac{{\langle S\rangle}_{\mathrm{sub}}}{{n}_{\mathrm{sub}}}\cdot ds}$ with integrations being carried over the surface

**s**enclosing the hemisphere. The maximum RDRE

_{air}obtained from the 3D model was calculated to be ~100 (results not shown here). Far-field intensities, |E

_{far}|

^{2}vs. Θ

_{xy}also followed a similar trend with collimated beams being achieved for Θ

_{xy}< 90°, Figure 6a–c. Far-field intensity emanating from the symmetric cross-shaped aperture, on the other hand, was impervious to the change in Θ

_{xy}but the maximum far-field intensity in this case was almost half of that achieved by the slit, i.e., |E

_{far-cross}|

^{2}/|E

_{far-slit}|

^{2}= 0.45 at their respective maxima. Presence of an additional arm in the cross-shape aperture reduces the nearby metallic surface area over which |E

_{D}|

^{2}is projected on to, hence the reduction in absorption cross section of the acceptor.

**µ**

_{D}, field amplitude of the donor projected over the aperture vs. x

_{d}was calculated analytically using Equation (3) with sin ϑ = |z

_{d}|/r, see Figure 7a. The trend observed in |E

_{D}|

^{2}

_{(x=0,z=0)}spectrum agrees with that of the RDRE

_{air}obtained numerically, Figure 7b. Numerical results revealed that the device tolerates a lateral displacement of Δx

_{d}≈ ±15 nm beyond which the RDRE

_{air}drops below 1/e of its maximum. The phase difference ΔΦ

_{L−R}in this case rises at x

_{d}≥ 7 nm, reaching a maximum of ~105° at x

_{d}= 23 nm but never attaining a 180° difference to qualify E

_{x}as a pure odd function. RDRE

_{air}, on the other hand, drops to 1/e of it maximum at x

_{d}≈ 15 nm where ΔΦ

_{L−R}is 70°. Defining the state of even function-ness as cos

^{2}(ΔΦ

_{L−R}/2), a 70° phase difference must drop the RDRE

_{air}only to 0.67 (not 1/e) of its maximum. The 55% excess reduction, therefore, is partially due to the E

_{x}not being a pure even function and partially due the reduction in |E

_{D}|

^{2}

_{(x=0,z=0)}with increase in x

_{d}.

## 5. Crystallographic Directions

**µ**

_{D}with a plasmonic slit offers the best outcome with the highest RDRE

_{air}and tolerance. Descriptions on triplet notations and Miler indices used in crystallography, may be found in undergraduate solid state texts [45,46]. With NV- symmetry axis being one of the <111> crystallography directions [26,27], one may narrow the search for a horizontally oriented

**µ**

_{D}. Assuming that the planar design lies on an arbitrary plane, z, defined by its normal vector $\widehat{z}$, possible scenarios are (a) z plane parallel to diamond’s $\left(1\overline{1}0\right)$ plane which implies that

**µ**

_{D}is parallel to the $\left[\overline{1}\overline{1}2\right]$ direction, hence the slit must be fabricated such that

**µ**

_{A}‖

**µ**

_{D}. To maximize the emission from the NV- in this case, the excitation field,

**E**

_{ex}, must also be polarized in the $\left[\overline{1}\overline{1}2\right]$ direction; (b) z plane ‖ $\left(\overline{1}\overline{1}2\right)$ with

**E**

_{ex}‖

**µ**

_{A}‖

**µ**

_{D}‖ $\left[1\overline{1}0\right]$; (c) z plane ‖ 111) (or $\widehat{z}$ being aligned with the symmetry axis of the NV-) with no preferential direction for neither

**E**

_{ex}nor

**µ**

_{D}since

**µ**

_{D}= α

^{2}

**µ**

_{D1}+ β

^{2}

**µ**

_{D2}, where α

^{2}+ β

^{2}= 1, see Figure 8. Depending on the application, one can identify other diamond planes over which the planar structure to be set. For examples: if the emphasis is on the extraction of single photons, then the diamond plane must be chosen such that (

**µ**

_{A}‖

**µ**

_{D1}AND

**µ**

_{A}⊥

**µ**

_{D2}) OR (

**µ**

_{A}‖

**µ**

_{D2}AND

**µ**

_{A}⊥

**µ**

_{D1}). On the other hand, if one is to investigate the two-entangled photon arising from the mutual excitations of

**µ**

_{D1}and

**µ**

_{D2}, then it is best to implement the planar structure using a symmetric cross aperture on the (111) plane with the arms of the cross being aligned with $\left[\overline{1}\overline{1}2\right]$ and $\left[1\overline{1}0\right]$.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**|E|

^{2}calculated for a diamond substrate with Nitrogen Vacancy center (NV-) with horizontally oriented dipole moment, positioned at (x

_{d}, z

_{d}) = (0, −10) nm (

**a**) with no metallic film and (

**b**) with an optically thick silver film over the diamond substrate; (

**c**) Radiative Decay Rate (RDRE

_{air}) vs. the film thickness with a 30 nm wide aperture perforating the film above the NV-; (

**d**) |E|

^{2}calculated for diamond/silver/air with a 30 nm wide aperture perforating the 110 nm silver film above the NV-. E

_{x}and E

_{y}calculated at (

**e**) diamond/silver with no aperture and (

**f**) silver/air interfaces with the aperture. Fast Fourier transforms of E

_{x}and E

_{y}calculated at (

**g**) diamond/silver and (

**h**) silver/air interfaces.

**Figure 2.**(

**a**) Wave components governing the interference mechanism along the optical axis of a plasmonic bullseye device; (

**b**) Schematic of a bullseye device and its components; (

**c**) The phase difference between the wave emanating from an aperture and the wave scattered by a groove, ${\mathsf{\Phi}|}_{x=0}=\angle {\psi}_{\mathrm{h}}-\angle {\psi}_{1}$, vs. x

_{1}calculated along the z-axis bases on superposed SPPs; (

**d**) Square of the amplitude of superposed surface waves, ${\left|{\mathsf{\Psi}}_{\mathrm{SPPa}}\right|}_{x=x1}^{2}$, obtained analytically and $\iint {\left|{H}_{y}\right|}_{x={x}_{1}}^{2}dxdz$, obtained numerically. (

**Inset**) $\iint {\left|{H}_{y}\right|}_{x=0}^{2}dxdz$ calculated numerically inside the aperture.

**Figure 3.**(

**a**) RDRE

_{air}and (

**b**–

**f**) |E|

^{2}vs. the number of grooves N for a BE with r

_{1}= 555 nm, P = 590 nm, w

_{1}= 130 nm and w

_{p}= 210 nm, w

_{a}= 30 nm and t = 110 nm, on a diamond substrate with NV- at (x

_{d}, z

_{d}) = (0, −10) nm, emitting at λ = 700 nm.

**Figure 4.**(

**a**) RDRE

_{air}and collection efficiency, CE = P

_{air-C1}/(P

_{sub-C1}+ P

_{air-C1}) vs. z

_{d}. (

**b**–

**e**) Radiation patterns, |E|

^{2}, for various dipole distances, z

_{d}.

**Figure 5.**(

**a**) Numerically obtained RDRE

_{air}and the phase difference between E

_{x}oscillations on the corners of the aperture at the silver/diamond interface, ${{\mathsf{\Phi}}_{\mathrm{L}-\mathrm{R}}|}_{z=0}=\angle {E}_{x}(-{w}_{\mathrm{a}}/2,0)-\angle {E}_{x}({w}_{\mathrm{a}}/2,0)$ vs. the orientation of the µ

_{D}in the x-z plane, Θ

_{xz}; (

**b**) Numerically obtained RDRE

_{air}and Φ

_{L-R}vs. x

_{d}when Θ

_{xz}= 90°; (

**b**)-(inset) Analytical values for donor’s field intensity, |E

_{D}|

^{2}, projected over the aperture at (0,0) vs. x

_{d}when Θ

_{xz}= 90°; (

**c**) Directional gains of the BE antenna, |E(θ)|

^{2}/|E

_{0}(θ)|

^{2}, for 0° ≤ Θ

_{xz}≤ 90° when x

_{d}= 0; (

**d**) Directional gains for x

_{d}= {10, 16, 21, 33} nm when Θ

_{xz}= 90°.

**Figure 6.**3D realization of the model with a 30 nm wide and 250 nm long slit and cross-shaped apertures in a 130 nm silver film. Far-field intensities (V/m)

^{2}vs. the orientation of µ

_{D}in the xy plane, Θ

_{xy}, for (

**a**–

**c**) slit with Θ

_{xy}= {0°, 45°, 90°} and (

**d**) cross with Θ

_{xy}= 45°.

**Figure 7.**(

**a**) Analytical values for donor’s field intensity, |E

_{D}|

^{2}, projected over the aperture at (0,0) vs. x

_{d}when Θ

_{xz}= 0°. (inset) schematic showing for the donor-acceptor relation; (

**b**) Numerically obtained RDRE

_{air}and Φ

_{L−R}vs. x

_{d}when Θ

_{xz}= 0°; (

**c**–

**f**) Radiation profiles, |E|

^{2}, for various values of x

_{d}.

**Figure 8.**Best case scenario for implementing

**µ**

_{A}with respect to

**µ**

_{D}with z plane ‖ (111) (or z-axis being aligned with the symmetry axis of the NV-). There is no preferential direction for either

**E**

_{ex}or

**µ**

_{D}since

**µ**

_{D}= α

^{2}

**µ**

_{D1}+ β

^{2}

**µ**

_{D2}, where α

^{2}+ β

^{2}= 1.

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Djalalian-Assl, A.
Dipole Emission to Surface Plasmon-Coupled Enhanced Transmission in Diamond Substrates with Nitrogen Vacancy Center- Near the Surface. *Photonics* **2017**, *4*, 10.
https://doi.org/10.3390/photonics4010010

**AMA Style**

Djalalian-Assl A.
Dipole Emission to Surface Plasmon-Coupled Enhanced Transmission in Diamond Substrates with Nitrogen Vacancy Center- Near the Surface. *Photonics*. 2017; 4(1):10.
https://doi.org/10.3390/photonics4010010

**Chicago/Turabian Style**

Djalalian-Assl, Amir.
2017. "Dipole Emission to Surface Plasmon-Coupled Enhanced Transmission in Diamond Substrates with Nitrogen Vacancy Center- Near the Surface" *Photonics* 4, no. 1: 10.
https://doi.org/10.3390/photonics4010010