1. Introduction

The negatively charged nitrogen-vacancy (NV) color center in diamond has proven itself to be a highly sensitive sub-micrometer-scale sensor for magnetic fields [1, 2], electric fields [3], temperature [4, 5] and so on. The sensing mechanism is based on the coherent control and readout, which is effective even under ambient conditions, of the ground-state spin. Since the projected spin state is inferred from photoluminescence, however, poor collection efficiency of photons from a bulk diamond substrate represents one of the major limitations to obtaining the best sensitivity of the sensor. Moreover the small absorption cross section of NV center hinders its ensemble from being efficiently pumped [6], and often leads to the use of large optical power which may form an obstacle to miniaturized devices [7] and cause optical damage to the sample to be measured.

A variety of monolithic photonic structures such as immersion lens [8], photonic crystal cavity [9, 10], microring resonator [11], vertical waveguide pillar [12, 13], plasmonic resonator [14], circular grating [15] has been demonstrated to improve detection efficiency of photoluminescence. None of them, however, is optimal for themselves if we consider their use for a sensitive wide-field metrology [16–21] with some desirable features such as NV centers placed in the shallow region below the surface, robust detection of the broadband emission at room temperature, ability to form an array with a sub-micron separation, and a structure preferably of low aspect ratio. We also note that photonic means for efficient pumping of a dense array of NV center is rarely found in the literature. The light trapping geometry [6] was reported to achieve a breakthrough in efficient excitation, but does not provide a uniform illumination of the pump field over the nanometric shallow layer near the surface.

Mie resonances, which indicate resonantly enhanced effective polarizability produced by displacement currents in a dielectric particle, are also known to have attractive features such as high field-concentration and directional waveguiding. In particular, those supported in a sub-wavelength resonator of high refractive index material e.g. Si have been extensively studied during the last decades in the context of low-loss all-dielectric metamaterial [22–27]. We also note that materials with moderate refractive indices, between 2 and 3 as in diamond, have been considered to obtain similar optical functions [28, 29]. It has not been recognized, however, that concomitant reduction in the quality factor and coupling efficiency may lead to further modification of the field distribution near the top surface and provide a mechanism to enhance the field concentration.

Here we present a numerical simulation on the use of a few Mie resonance modes for enhanced metrology using a shallow NV center ensemble. Remarkably it turns out that we can assign a set of adjacent high-order modes to each of the absorption and emission band of typical NV centers at room temperature. The field-concentration in the shallow layer below the top surface becomes even higher due to the interference between the waves coupled and uncoupled to the resonator. Consequently we obtain about ten-fold enhancement both in the pumping and detection efficiencies of NV centers located at the center of the transverse cross section. Even after averaging over a 20 nm thick top layer the pumping efficiency is kept above several times larger than that of the bulk substrate case. The detection efficiency is also robust over spatial distribution of NV center within the region in which the pump field is strong enough to make the NV centers active. Further analysis on the inter-resonator coupling for a periodic array of resonators suggests a minimum possible separation between them should be around 700 nm.

By combining its high sensitivity, low pumping power, and sub-micrometer spatial resolution this technique can be used to build a compact sensor system for imaging and sensing problems in biology, chemistry, materials science, etc.

2. Problem setting

Since we are interested in a sensor configuration which allows the shortest distance possible between a target sample and NV centers, we suppose NV centers created in a shallow layer near the top surface of a diamond substrate. The target sample, preferably transparent, is placed above the diamond top surface with a minimum stand-off distance as far as the optical mode supported in the diamond resonator is not disturbed.

The measurement setup considered in this paper is basically an epifluorescence microscope. We apply the pump field from the bottom side of the substrate (see Fig. 1(a)) through an objective lens and collect the photoluminescence signal produced from the color center via the same lens. To illuminate a wide range of the diamond substrate the incident pump laser is supposed to be focused on the back focal plane of the objective lens [16]. The polarization of the pump is chosen to be linear and coincides with one of the two possible directions of the dipole axis of NV centers. Specifically we assume the most common situation in which NV axis lies along [111] direction in a [100]-oriented substrate [16] in which the corresponding directions of possible dipole are [1̄1̄2] and [11̄0]. Since the pump beam is along [001] direction, one can prepare the input polarization to be linear along [11̄0] to induce the largest possible dipole transition [30]. Of course the crystal symmetry providing four possible NV axes constitutes a common problem in an ensemble experiment in which only a fraction of NV centers are used for sensing given the fixed quantization axis.

 

Fig. 1 (a) Schematic diagram of optical pumping and detection in the form of a wide field microscope. The subwavelength resonator is patterned on the top side of the diamond plate over which a sample to be measured are placed. NV centers are assumed to be created near the top surface. The pump beam is injected from the bottom through an objective lens which also collects the photoluminescence. A dichroic mirror and a set of spectral filters (not shown) are used to separate the signal from the pump. (b) Location of the NV center used to extract a set of pumping efficiency. The first is the center of the transverse cross section at a vertical depth of 10 nm, and the next considered is halfway moved towards the edge of the resonator at the same depth. To extract averaged quantities one region of interest is chosen to be a 20 nm thick layer at the top of the resonator, and the other includes the entire resonator. The polarization of the pump beam is along x direction.

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We take the refractive index of diamond as 2.43 irrespective of considered wavelength, i.e. from 300 nm to 900 nm, and the optical loss is neglected. It is worth mentioning here that typical absorption and emission spectrum of NV centers in its negative charge state is centered around the wavelengths of 570 nm and 680 nm respectively with the bandwidths of both spectra spanning more than 100 nm at room temperature. To determine the central wavelength of Mie resonances we further take account of some experimental preferences. For example the wavelength of the pump is often chosen as 532 nm for practical reasons such as the availability of solid-state lasers. In regard to detection we target 700 nm to reflect the fact that one often filters out the signal of shorter wavelength to reject the emission from NV centers in a neutral charge state.

3. Pumping efficiency

To extract the pumping efficiency we perform a series of FDTD (Finite-Difference Time-Domain) simulations [32] in which a plane wave source is injected normally from the bottom of the resonators. The enhancement factor is defined as the ratio of the electric field intensity evaluated at the region of interest (Fig. 1(b)), that is, |E_Mie|²/|E_ref|² where E_Mie is the electric field in the presence of the resonator, and E_ref is that without the resonator. Note that due to partial reflection at the diamond-air interface there exists a standing wave pattern with its antinode near the surface. Thus even in the absence of the resonator the field intensity is already stronger than that of free space, and we seek for further enhancement by introducing the resonator.

In the literature reporting the creation of shallow NV centers the most frequently found value of implanted depth is around 10 nm with a typical spread less than 10 nm [13, 16–19, 21, 31]. Here we consider efficiencies evaluated at four different regions of interest as shown in Fig. 1(b) – the first is the center of the resonator on the transverse plane at a depth of 10 nm, the second is the point halfway moved towards the edge of the resonator in x direction. The third one is the averaged value over a 20 nm thick top layer of the resonator to fully cover possible region of color centers, and the fourth is also given as an averaged one over the entire resonator to identify confined modes themselves.

3.1. Vertical cylinder resonator

We start by investigating a diamond resonator of circular cross section with a straight side wall. By fixing the radius as 125 nm and varying the height from 150 nm to 900 nm, we notice large field-concentration associated with the well-known fundamental mode, often referred to as the magnetic dipole mode, from 600 nm to 800 nm range for the height larger than 400 nm as in Fig. 2(d). All of the four spectra in Fig. 2 mark the same spectral position of Mie resonance modes although the degree of enhancement may vary depending on the region of evaluation.

 

Fig. 2 The pumping enhancement as a function of wavelength of the light source and the height of the resonator for a vertical cylinder with its radius 125 nm. Above each panel the region of interest used for calculating the enhancement factor is described. The off-center position considered in (b) is 62.5 nm in x direction. The absorption band of the NV center is designated as a white dashed box for comparison.

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To match the fundamental mode to the absorption band of NV centers we should reduce the radius of the resonator which leads to an unpractical design with a very high aspect ratio as well as a challenging lateral dimension for the conventional lithographic technique. Rather we increase the radius to introduce a series of high-order modes which turns out to be even better to be assigned to each of the absorption and emission band of NV centers.

In Fig. 3(d) we notice two adjacent modes from 400 nm to 800 nm wavelength range for the low aspect-ratio condition, i.e. the height from 200 nm to 400 nm. In the spirit to maximize the averaged efficiency in Fig. 3(c) we choose the height of 300 nm as our optimal design. The peak enhancement factor at the center then exceeds 10 while the averaged one is around 3 ∼ 4.

 

Fig. 3 The same plot as Fig. 2 with the radius changed to 195 nm. The off-center position is 97.5 nm away from the center in x direction.

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The amplitude profile of the electric field |E⃗| on the xz-cross section is presented in Fig. 4 for a set of representative wavelengths. We first identify the circulating electric field in the longest wavelength which corresponds to the magnetic dipole mode (or HE_11δ mode, see Fig. 4(a)).

 

Fig. 4 The xz-cross sectional view of the electric field amplitude |E⃗| for selected wavelengths λ of the pump beam, that is, 530 nm (absorption band), 700 nm (emission band), and 900 nm (for identifying the fundamental resonance). The direction of the pump injection is (a)–(c) from the diamond substrate and (d)–(f) from the air side above the resonator. The arrow plot indicate the direction of the field in xz-plane cut in the center of the cylinder (y = 0). The color bar scale is applied commonly to all six plots and given in the unit of V/m.

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At the targeted emission band of the NV center (700 nm) the mode is attributed to the magnetic quadrupole mode (or HE_111+δ mode, Fig. 4(b)) in which two magnetic dipoles are vertically arranged with opposite phases [27]. Due to the low-Q nature, however, the field profile contains a significant portion of uncoupled background field which forms a moderately focused wave in the direction of propagation [33, 34]. We note that in this particular arrangement of measurement the mode-formation and the single-pass near-field modulation are competing each other. The resulting hybrid interference pattern gives rise to overall increase of the field towards the top surface. If we flip the direction of the pump beam injection for the same resonator design, the focusing of the background field is reversed as shown in the corresponding plots in Fig. 4(d)–(f). Although this wavelength is irrelevant to the pumping role discussed in this section, this mode will be identified as a detection channel in the later section.

The next order resonance characterized as HE_12δ mode (Fig. 4(c)), which contains three magnetic dipoles horizontally placed with alternating phases [27], is exactly tuned to the targeted absorption spectrum of NV center (530 nm). Again we note the dependence of field-concentration on the excitation direction. A close look at the mode profile near the shallow layer reveals the maximum of the field coincides with the central spot of the NV center at a 10 nm depth (Fig. 5(a)). We also note that the direction of the field stays parallel to the pump field for the region of strong intensity. The portion of x–component of the electric field |E_x|/|E⃗| is depicted in Fig. 5(b).

 

Fig. 5 Zoom-in view of (a) the electric field intensity |E⃗|² and (b) the portion of the x–polarized field |E_x| to |E⃗| near the top layer of 20 nm thickness (white dashed line) for the wavelength of 530 nm.

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3.2. Tapered resonator

We present another design of the resonator which is built upon the same principle but further increases the averaged enhancement factor by tapering the vertical post. The tapering effectively augments the field density by cutting off the corner of the cylinder in which the field concentration is not very strong. Thus it does not affect the point enhancement at the center very much while the averaged enhancement is doubled as shown in Fig. 6. Here the radii of the top and the bottom cross section are 100 nm and 350 nm respectively. We observe the spectral linewidth of each mode is even broader to result in almost flat response over the absorption spectrum of NV centers.

 

Fig. 6 The same plot as Fig. 2 for the tapered design with the upper and lower radius 100 nm and 350 nm respectively. The off-center position is moved by 50 nm in this case.

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The mechanism of enhancement and the relevant mode profile are basically similar to that of the straight cylinder as seen in Fig. 7 for the height of 400 nm. Each design provides its own merit in the sense that, despite the increase in the average enhancement factor, the absolute area available to shallow NV centers is smaller than that given to the straight cylinder.

 

Fig. 7 The xz-cross sectional view of the electric field amplitude as in Fig. 4 for the tapered design. The direction of the pump injection for (a)–(c) and (d)–(f) are opposite to each other as before.

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It is possible to consider even higher oder modes which have similar field-confinement effect near the top surface, but we observe the radial confinement on the transverse plane becomes tighter for those modes (not shown) thereby reduces the averaged efficiency. Thus we restrict our discussion to a few modes considered so far.

3.3. Periodic array

In practice the size of an entire target sample to be measured may well exceed that of the resonator, and a typical field of view of the wide field imaging is larger than 100μm [20, 21]. Thus it is natural to consider an array of the resonator. The key parameter in this configuration is the separation between resonators which determines the strength of inter-resonator coupling effect and resulting deviation from the single element characteristics.

Since the incident field is normal to the substrate, the scattered field having a small angle of diffraction does not smear into the nearest site of the resonator. However the diffraction angle reaching 90° is significantly coupled to the neighboring site in the horizontal direction. By mth-order diffraction occurring for a sin θ_m = mλ where a corresponds the period of the array, the diffraction in the direction of θ_m ∼ 90° is expected at the wavelength of the pump λ ≃ a/m.

The array effect in the straight cylinder design for a square lattice configuration is summarized in Fig. 8. As predicted from the above argument we notice strong modulation in the efficiency spectrum whenever the diffraction condition is met. For example, the period of 1000 nm does not induce the first-order diffraction in the spectral range shown in Fig. 8(a)–(b) whereas the second-order diffraction appears near λ ∼ 1000 nm÷2 = 500 nm as a dip. For a = 700 nm, both first- and second-order diffraction effects are visible. The smaller separation of a =500 nm involves only the first-order diffraction in NV absorption range. The horizontal coupling is evident in the field profiles presented in Fig. 8(c) and (d) which are obtained for wavelengths showing large deviation from the single element efficiency. The coupling does not necessarily reduce the efficiency as the peak near λ = 740 nm suggests, but the underlying mechanism is not clearly understood at this stage.

 

Fig. 8 The pumping enhancement factor for the straight resonator design in the square lattice configuration estimated at (a) the center of the resonator and (b) the shallow layer for a number of period a=1000, 700, 500 nm. The reference result for the single element case is overlaid by a black dotted curve. The absorption band of the NV center is marked by a pink rectangle. We also depict the electric field profile produced in the space between two adjacent resonators on xz-plane for a =700 nm with (c) λ =500 nm and (d) λ =740 nm.

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Therefore the pumping efficiency is not degraded for a period larger than 500 nm by the first-order diffraction, and it is also desirable to exclude the value a ≃1000 nm to suppress the second-order effect. The choice of a =700 nm seems reasonable here, and will also be commensurate with the result of detection analysis given in the later section.

4. Detection efficiency

While it is straightforward from the field-concentration spectrum that the tuned Mie resonance leads to enhancement of the pumping efficiency, we should perform another series of simulation to verify that the detection efficiency is indeed enhanced by relevant Mie resonances. For convenience, the detection efficiency is calculated for a point NV, which can be considered as a dipole radiator. Since the pump field is concentrated near the central region on the transverse plane of the resonator, we assume that a few points at a 10 nm depth in the central region of the cross section well represents the effective distribution of NV ensemble sufficiently pumped inside the resonator. The polarization of the dipole emission is taken to be horizontal as discussed in the previous section (see Fig. 5(b)).

In detail, the detection process is simulated as follows. From FDTD simulation we first obtain the near field distribution on a plane beneath the resonator (100 nm below the bottom level), and then acquire the far-field distribution by the plane wave decomposition and propagation algorithm. Next we calculate the Fresnel coefficient at the diamond-air interface for each propagation direction of the far-field, multiply the coefficient to the corresponding component of the field to obtain the final field distribution incident on imaging optics. For a given collection angle of the optics, e.g. an objective lens, we integrate the collected power and normalize it by the input dipole power.

4.1. Vertical cylinder resonator

The detection efficiency for the straight circular post design is presented in Fig. 9(c). We consider the collection angle of 10°, 30°, 50° which corresponds to the numerical aperture (NA) of 0.17, 0.5, 0.77 respectively. The Mie resonances are clearly identified near wavelength ranges near 530 nm and 700 nm as in the field concentration spectrum. The spectrum of detection efficiency is also consistent with the scattering cross section spectrum calculated for an isolated cylindrical resonator without a substrate (Fig. 9(a)), but we notice a small amount of shift due to the substrate effect [25]. Around the targeted wavelength the efficiency reaches 50 % for NA=0.77 case. Also shown for reference is the efficiency obtained without any resonator, i.e. bulk diamond for the collection angle of 50°. It turns out that the efficiency remains less than 5 % even for a large collection angle due to angular spread of the dipole emission as well as total internal reflection at the diamond-air interface. Consequently the Mie resonator gives rise to an order of magnitude improvement given the same imaging volume. In addition the reduction of efficiency introduced by placing NV centers off the center is not very serious even at the midpoint between the center and the edge in x direction (97.5 nm and 50 nm for each design) where the pump intensity is reduced to about half of its maximum. The intensity profile of the photoluminescence from a point-NV center is plotted on a logarithmic scale in Fig. 9(e) where the directional emission is clearly demonstrated.

 

Fig. 9 In the upper panels the scattering cross section normalized by the geometric cross section is presented. In the middle panels we depict the detection efficiencies for several collection angles of the lens in the presence of the resonator– at the center (solid lines) and off the center (dashed line) on the transverse plane. For comparison the portion of energy captured as the near field at the bottom of the resonator (dash-dot line) and the efficiency given without any resonator (dotted line). The emission band of the NV center is highlighted by a pink rectangle. In the lower panels the electric field intensity |E⃗|² is shown on a log-scale at the wavelength of 700 nm. The resonator design is the cylindrical resonator for (a),(c),(e) and the tapered design for (b), (d), (f).

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4.2. Tapered resonator

We also provide a result for the tapered pillar design in the right column of Fig. 9. The scattering cross section reveals the existence of more closed spaced modes compared to the straight post [13]. Those peaks tend to be broadened in the efficiency spectrum as in the pumping explained in the previous section. Consequently the tapered design guides about 50 % of the photoluminescence into the imaging optics of NA=0.77 over a broad range of wavelength from 350 nm to 800 nm. The guided signal is also visualized in Fig. 9(f).

4.3. Periodic array

Here we consider again the coupling effect taking place in a periodic array configuration. As opposed to the pumping case, the field produced from each site does not produce coherent interference because each NV center is an independent light source. Thus, instead of coherently summing all of the contribution from each site, we inspect emission characteristics of the single site and pay attention to the difference introduced by approaching the other sites towards it. To do so we set up a 3 × 3 square lattice array, in which only the central site possesses an active dipole emitter located 10 nm below the top surface as in the single element calculation. In Fig. 10 we compare the detection efficiencies for arrays of different periods with the collection angle θ = 50° fixed. Both the vertical and the tapered design start to lose the photoluminescence as the period is reduced down to 500 nm. This criterion is consistent with the result obtained from the pump analysis.

 

Fig. 10 The detection efficiency obtained from the resonator array compared with the case of single element case. Two different values of the separation a are used for the design of (a) the vertical resonator and (b) the tapered one. The emission band of the NV center is highlighted by a pink rectangle. The reference efficiency obtained from a bulk diamond is depicted as a dotted curve.

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To understand reduced efficiency for a closely packed array, we inspect the far-field image of NV emission constructed by a lens of NA=0.77. The chirped z-transformation is employed to construct the image pattern from the near field information. For a = 700 nm the signal intensity measured at a neighboring site is less than 5% regardless of the position of the emitter (Fig. 11(a)). For a smaller period of array, however, we notice significant amount of cross talk into the nearest site, in particular, along the perpendicular direction to the polarization of the dipole radiation. And the coupling along the parallel direction to the polarization is induced if the NV center is positioned off the center horizontally (Fig. 11(b)). The cross-talk amounts ∼ 12% in the case of a = 500 nm.

 

Fig. 11 The array effect manifested in the image for two different values of site separation – (a) a =700 nm and (b) a =500 nm. The upper row corresponds to the near field intensity pattern 100 nm below the bottom of the resonator for the NV center placed at the center and off the center respectively. The lower row represents the corresponding images obtained by a lens system of NA=0.77. The emission wavelength is 700 nm and the resonator design is the straight cylinder of 195 nm radius. Note the position of the 3 × 3 array is marked by white circles. The intensity distribution is normalized by the maximum value found in each plot. The limit of color scale is set to correspond to 30% of the maximum intensity to discern small difference induced by the coupling effect.

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5. Conclusion

We studied Mie resonances supported in a diamond nano-resonator as a useful tool to increase pumping efficiency as well as detection efficiency of sensors using shallow NV centers. Although it turned out that the fundamental mode, which has been extensively studied in the context of silicon metasurface, was not favorable to use for this purpose, we identified that a series of higher order modes could provide localization of the pump field in the vicinity of the top surface of the resonator while, at the same time, another higher order mode played a role to guide photoluminescence from shallow NV centers. The periodic effect did not significantly deteriorate the above-mentioned features as long as we chose the separation between resonators larger than the wavelength of NV emission.

This configuration is particularly useful for application in which efficient pumping of a shallow two-dimensional sheet of many color centers is required along with the sub-micrometer spatial resolution. The most relevant example is wide-field magnetometry and thermometry with ensemble NV centers. The increased efficiency of photon collection leads to enhanced local sensitivity for the given imaging volume which is typically of diffraction-limited size.

One of the drawbacks of this approach is the loss of color centers by etching out more than 70 % of the whole area. This leads to inevitable spatial undersampling and reduction of the overall sensitivity as a sensor of the macroscopic signal influencing the whole diamond plate. The latter issue is more relevant to the detection of a spatially uniform signal, and not of much concern to the imaging experiment for resolving spatial variation of the target signal. The undersampling issue may not be significant for image acquisition since the dimension of empty space between the resonators can be made smaller than the diffraction limited resolution, and can also be partially overcome by compressive sensing [35] depending on the nature of the signal.