1. Introduction

Optical waveguides can be used to transfer light much more efficiently than free space optics [1] thus representing a desirable practical method for nano-particle detection, fluorescent sensing and single-photon sources [2–6]. However, coupling the emitted light into a waveguide can be inefficient due to the dipolar radiation pattern of nano-scale sources and for this reason it is still a challenging research task [7–9].

One method that has been extensively explored is direct fiber coupling, where the dipole source is located in the vicinity of a single-mode fiber tip. Fiber-based micro-cavities [10], fibers with a nano-scale tapered tip [11], lensed fibers [12] and etched single mode fibers [13] are a few examples of this approach. On the other hand, multimode fibers, which have a larger core size ($\sim$50 $\mu m$), suffer from mode mismatch with the near field of the dipole.

Another method is embedding the molecules inside the waveguide [14,15]. The coupling efficiency could reach up to 70% in a tapered fiber, but the approach requires immobilization of the molecules in the fiber, which is not always desirable. Moreover, researchers have investigated the evanescent coupling to a tapered fiber [16–19]. This method is particularly useful for single-molecule detection, but it is quite challenging from the practical point of view. The reported coupling efficiency for this technique varies from 10% to 80% in the simulations depending on the fiber size and the material.

Recently, a planar antenna structure that beams the emission of single molecules has been proposed [20,21]. This relies on the concept of an optical Yagi-Uda antenna [22], where the reflector and the director elements are thin metal films. Hence, the antenna strongly modifies the emission pattern leading to a half-width at half maximum of less than 20 degrees. This modification can be applied to the concept of fiber collection.

Here, we investigate the coupling of a dipolar emitter in a planar antenna with an optical fiber, which has a core radius larger than the source emission wavelength $\lambda$ (hence excluding the case of nano-fibers [11]) and a facet coated with a thin gold layer, acting as director element. This configuration improves the overlap of the radiated field with the guided modes in the fiber. As a result, the coupling efficiency of the dipole into the fiber increases.

The proposed detection scheme is targeting, especially, bioassays based on surface chemistry, in which the vertical position of the molecule can be defined, for instance as in surface-plasmon resonance biosensors [23].

Furthermore, we study the relationship between mode coupling efficiency and numerical aperture (NA) of the fiber. The critical numerical aperture, for which the fiber has the maximum coupling efficiency, has been obtained. The coupling efficiency of a dipole embedded in different transparent materials has also been calculated. This is relevant for the efficient coupling of organic molecules in media such as water. Finally, the effect of tilting and displacement of the fiber in the setup is investigated.

2. Methodology and layout of the problem

In order to optimize the coupling efficiency of a dipole into a fiber, the geometry, material and position of the fiber must be specified. This includes many variables such as core radius, refractive indices of core and cladding, the reflector-director distance, the active medium in which the dipole source is embedded and so on. In this work the effect of these variables for the optimization of the fiber coupling is studied using the finite difference time-domain (FDTD) method [24]. The simulations have been performed with a commercial software package (FDTD Solutions, Lumerical Inc. [25]). To reduce the computation time and memory we took advantage, whenever possible, of the symmetry of the layout. Symmetric boundary conditions are implemented by forcing the appropriate field components to zero. These boundaries can not be used for tilting and radial displacement of the dipole from the center.

We consider a gold substrate as the reflector, in which a dipole emitter is located at a distance $d_1$, creating an image dipole [26]. The distance between dipole and fiber tip in the radiation direction is denoted as $d_2$, shown in Fig. 1. When two dipoles radiate with an appropriate phase delay, they can give rise to a beaming effect in a certain direction. This occurs when $d_1$ and $d_2$ fall within the range of $\lambda /(6n) < d_{1,2} < \lambda /(4n)$, where $\lambda$ is the emission wavelength and $n$ is the refractive index of the active medium [20,21].

 

Fig. 1. Simulations layout. A single dipole is situated 130 nm above a gold coated substrate (reflector). The dipole emission is collected by the fiber and monitored with a frequency-domain field profile. The monitor is placed in the far field, approximately $4\lambda$ from the dipole. The cladding radius, fiber length and substrate (glass) are semi-infinite. The values of the fixed parameters are given in the figure. The distance $d$ between director and reflector is $d = d_1 + d_2 - 20$ nm.

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The radiated power ($P_r$) is calculated as the power passing through a frequency-domain field monitor across the fiber (see Fig. 1) at a distance 4$\lambda$ in the radiation direction and it corresponds to the power out-coupled from the planar antenna [20].

The power transmitted into the mode $m$ of the fiber is determined by the overlap integral ($a_m$) of the guided mode across the fiber.

The dipole has a fixed amount of power in free space, known as the source power ($P_s$). The ratio of the total emitted power of the dipole inside the system ($P_t$) over the source power gives the Purcell factor ($F$):

One way to enhance the overlap integral is to change the fiber geometry and material. Further enhancements can be obtained by coating a thin gold layer on the tip of the fiber, also known as a director, which provides directional beaming.

In all simulations, the dipole source has a horizontal orientation with respect to the reflector in order to minimize the coupling of energy to surface plasmon-polariton (SPP) modes present in both the reflector and the director interfaces. The emission of vertically oriented dipoles is quenched by the antenna. Therefore, in the case of molecules with a tilted dipole moment, the detection efficiency would scale as $\sin ^2\phi$, where $\phi$ is the angle between the dipole orientation and the antenna axis. For randomly-oriented molecules, the detection efficiency would thus be reduced by a factor of 2/3, which corresponds to the average value of $\sin ^2\phi$ over the solid angle.

For simplicity, some of the following geometrical parameters are considered constant (see Fig. 1). The cladding radius is infinitely large. The distance between reflector and dipole is $d_1 = 130$ nm (except Fig. 8). The frequency-domain field monitor for the overlap integral calculation is located far inside the fiber, but the exact position is not critical. The thicknesses of reflector and director are assumed to be 100 nm and 20 nm, respectively. Changing the thickness of the director could also affect the coupling efficiency. For instance, when the director is 20 nm thick the coupling efficiency into the fundamental mode of the fiber has a maximum value of 50%. However, this value reduces to 44% for 30 nm and to 43% for 10 nm director thickness at a constant reflector-director distance, $d = 295$ nm. Further simulations show that having a thicker director increases the directionality, but the radiated power decreases [20].

The adhesion of the gold director on the tip of the fiber can be improved by adding a thin layer of titanium. Calculations show that the inclusion of a 2 nm-thick titanium film does not substantially increase losses. We find that, in general, for the designs of Fig. 1 $P_{\textrm {r}}$ drops by about 2%, when 2 nm of gold are replaced by 2 nm of titanium [27]. Adding an adhesion layer to the reflector does not affect the antenna’s properties, since the gold layer is thick and light cannot pass through it.

3. Results and discussion

3.1 Coupling enhancement with a designed fiber

Coupling light radiated by a single dipole emitter into a commercial fiber (single mode, e.g. SM600) without any objective lens is challenging. This is due to the mismatch of fiber mode (see Fig. 2(a)) and a dipole field. The coupling efficiency can be increased by placing a reflecting layer behind the emitter (see in Fig. 2(c) the radiated field profile into the fiber) and it can be further improved by adding a director layer on the fiber tip (see Fig. 2(d)). The dipole is centered with respect to the fiber and its emission has been set at $\lambda = 700$ nm. This represents the wavelength range of well-known near-infrared dyes and solid-state quantum emitters [28–30].

 

Fig. 2. Propagation of fundamental mode in (a) reference fiber ($R_{\textrm {core}}$ = 1.6 $\mu$m, $n_{\textrm {core}} = 1.4949$ and $n_{\textrm {cladd}} = 1.4533$) and (b) custom fiber ($R_{\textrm {core}}$ = 0.8 $\mu$m, $n_{\textrm {core}} = 1.45$ and $n_{\textrm {cladd}} = 1.3$) simulated by the commercial software Mode Analysis, Lumerical Inc. [25]. In (a) and (b) the white line shows the fiber core and they have the same colour bar. The radiated field of the dipole at $4 \lambda$ away from the source for (c) reference fiber with reflector and distance $d_1 + d_2 = 295 + 20$ nm, (d) reference fiber with reflector-director and distance $d = 295$ nm, (e) custom fiber with reflector and $d_1 + d_2 = 295 + 20$ nm, and (f) custom fiber with reflector-director with distance $d = 295$ nm has been shown. (c)–(f) are normalized with respect to maximum value in (f) and have the same colour bar. In each case the geometry of the simulation is depicted below the field image. (g) Plots of the radiated field into the fiber for the reference fiber (c), (d) and the custom (e), (f) configurations with respect to the reflector-director distances, normalized by the total emitted power $P_t$ of the dipole. The right axis indicates the Purcell factor of the dipole, which can be multiplied by the coupling efficiency to determine the power coupled to the fiber. This indicates that, however, the mode reaches the maximum coupling (59%) at 285 nm, but a larger collected power is obtained at 295 nm due to the Purcell factor.

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Alternatively, shrinking the core radius to 0.8 $\mu$m and using materials with higher refractive index contrast for the core and the cladding could also enhance the efficiency. Figure 2(b) shows the fundamental guided mode of such fiber, while Fig. 2(e) and 2(f) show the radiated field into the fiber after adding a gold reflector and a director, respectively. In Fig. 2 the reference fiber is an solid core optical fiber with the following parameters: core radius ($R_{\textrm {core}}$) = 1.6 $\mu$m, core refractive index ($n_{\textrm {core}}$) = 1.4949, cladding radius ($R_{\textrm {cladding}}$) = 50 $\mu$m and cladding refractive index ($n_{\textrm {cladd}}$) = 1.4533 which are approximate to the commercial available fiber from Thorlabs “UHNA3”, high NA single mode fiber. The common single mode fibers such as Thorlabs SM600 have a coupling efficiency of less than 1% due to the low NA. For the customized fiber the parameters are: $R_{\textrm {core}}$ = 0.8 $\mu$m, $n_{\textrm {core}}= 1.45$, $R_{\textrm {cladding}}$ = 50 $\mu$m and $n_{\textrm {cladd}}=1.3$.

Since the radiated field from the dipole in the planar Yagi-Uda antenna exhibits a small emission angle, the core size of the fiber should be smaller than that of commercial fibers. In addition, a high-NA allows to collect more light into the fiber. These conditions can be achieved by tapering the core of a commercial fiber adiabatically [31–33] or making a photonic-crystal cladding fiber [34]. Figure 2(g) compares the coupling efficiency of reference fiber and custom fiber at different distance ($d$) between the reflector and the director.

In Fig. 2 the coupling efficiency is simulated for different distances between 200 nm to 600 nm. For the custom-designed fiber with director, despite the cavity effect at 350 nm and 700 nm, the maximum coupling is at 295 nm due to the antenna effect [20]. More interestingly, even a dipole at the center of the active medium ($d_1 = 350$ nm) when $d = 700$ nm does not give a higher coupling efficiency for both custom-designed and reference fiber.

We would like to point out that the custom fiber could collect more photons than those emitted by a $P_s$ dipole in vacuum. This is due to the Purcell factor, which reaches a value of 2.1 at $d = 295$ nm. By multiplying the coupling efficiency and the Purcell factor, one can find the coupling efficiency normalized with respect to the fixed value $P_s$. Consequently, the ratio between the power coupled into the fundamental mode (see Eq. (1) and (3)) and the source power ($P_s$) is $T_1/P_s = 1.06$.

3.2 Critical numerical aperture

Next, we investigate the relationship between the collection efficiency and the fiber NA. The numerical aperture of a fiber is calculated by

A fiber with high-NA could enhance the collected light from a single dipole, but this enhancement does not increase with the NA. For a fixed value of $n_{\textrm {core}}$, reducing the refractive index of the cladding below a certain value will not change the coupling efficiency as shown in Fig. 3. Using the $n_{\textrm {cladd}}$ values depicted in Fig. 3, one can find the critical NA using Eq. (5).

 

Fig. 3. NA of the fiber, represented by varying cladding refractive index $n_{\textrm {cladd}}$. The core radius here is 0.8 $\mu$m and it has a refractive index of 1.45 (custom fiber configuration). This figure indicates that for high-NA fibers there is more than one propagating mode. The curves show that for $n_{\textrm {cladd}}$ smaller than 1.3 the coupling efficiency is nearly constant for the modes. By using Eq. (5) the critical NA is equal to 0.64 (black dashed line). The radiated power curve indicates the total out-coupled light from the reflector-director configuration in the propagation direction normalized by total emitted power $P_t$.

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In Fig. 3, $n_{\textrm {core}} = 1.45$ and $n_{\textrm {cladd}}$ varies from 1 (air) to 1.44 (close to the refractive index of the core). The graph indicates that below $n_{\textrm {cladd}}=1.3$ the coupling efficiency is almost constant. This is called critical NA and for such fiber it is equal to 0.64. Furthermore, the critical NA depends on the core size and its refractive index. This value is in the range between 0.45 to 0.65 for different sizes or refractive indices of the core in the free space medium. This shows that having a high-NA fiber is essential for the coupling, but fibers with very large NAs are not necessary.

3.3 Fiber core radius and higher order mode coupling

As shown in Fig. 2 the coupling efficiency raises when the fiber is coated with a thin gold layer. This happens not only for the fundamental mode, but also for some higher order modes. The number of modes in a fiber depends on the core radius and the NA of the fiber. The high-NA fibers normally have a higher number of modes compared to the low-NA ones for the same geometry. Moreover, the number of modes increases with the size of the core radius.

Figure 4 indicates the coupling efficiency for the $1^{\textrm {st}}$, $4^{\textrm {th}}$ and $9^{\textrm {th}}$ modes of the fiber with variable core radius. The other modes would just slightly change the coupling strength therefore we can neglect them. The refractive indices of core and cladding are 1.45 and 1.3, respectively. By increasing the core size from 0.6 to 2.1 $\mu$m, the coupling to a fundamental mode decreases, but the coupling to higher-order modes increases the density of states. This effect has been shown also for a single dipole evanescently coupled to a multimode fiber [7].

 

Fig. 4. Coupling efficiency of different modes in the custom gold coated (director) fiber. The distance is fixed at $d = 295$ nm. The $1^{\textrm {st}}$, $4^{\textrm {th}}$ and $9^{\textrm {th}}$ modes with dipole emission are demonstrated. The red curve points out the Purcell factor (right axis). The radiated power shows that the out-coupled amount of light from the antenna structure does not depend on the size of the core or the cladding. For a core radius less than 1 $\mu$m there are only four modes propagating through the fiber and the fiber with less than 0.4 $\mu$m core radius is a single mode fiber.

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Figure 4 shows that the fiber with core radius of 0.4 $\mu$m can be used as a single-mode fiber for coupling with more than 50% efficiency. However, the fiber with slightly larger core size contains few higher order modes with reasonable amount of coupling efficiency. Therefore, in Fig. 4 the fiber with 1.2 $\mu$m core size has a total coupling efficiency of 70% (i.e. $(T_1+T_4+T_9)/P_s = 1.55$) due to the contribution of $1^{\textrm {st}}$, $4^{\textrm {th}}$ and $9^{\textrm {th}}$ order modes.

3.4 Distance between the reflector and the director

We have shown that the coupling efficiency is maximal at a specific distance between reflector and director (see Fig. 2), but actually the coupling efficiency is relatively high also for determined larger distances. This can be exploited for detecting larger objects, like particles, or even cells in fluids.

The Purcell effect and the coupling efficiency of the fiber vary when the distance between director and reflector changes. Figure 5 reveals this for the custom fiber as mentioned in Fig. 2. By increasing the distance between reflector and director, the coupling efficiency grows and then it gradually decreases. This happens due to transition of the antenna effect with the buildup of the cavity mode in the reflector-director configuration [20]. Moreover, the maximum coupling efficiency is not at the same distance for each mode.

 

Fig. 5. Coupling efficiency versus reflector-director distance. The coupling raises fast, but it gradually decreases by increasing the distance. The fluctuation of radiated power and mode coupling efficiency are like the Purcell factor. The distance between each two peaks is 350 nm, which corresponds to $\lambda /2$.

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As shown in Fig. 5, the coupling efficiency at larger distances is still higher than some other methods for direct fiber coupling. For example, here at $d = 990$ nm the coupling efficiency of the fundamental mode is 27%, which is much higher than the other methods for direct fiber coupling [36,37].

3.5 Wavelength dependence of dipole-fiber coupling

In all previous simulations the dipole emitted at a fixed wavelength ($\lambda =700$ nm). Here, the dipole wavelength is swept from 600 nm to 1200 nm. Figure 6 shows the wavelength dependency of the coupling for three different reflector-director distances. The fiber parameter is the same as the custom fiber in Fig. 2. By increasing the distance between reflector and director, the maximum coupling occurs at larger wavelengths. This method can be helpful for spectroscopy measurements [38]. Moreover, it could be exploited to distinguish different fluorescent markers without colour cameras nor optical filters.

 

Fig. 6. Maximum coupling efficiency as a function of wavelength for the reflector-director distance of 260, 295 and 330 nm. It is possible to tune the coupling for the desired wavelength by changing the reflector-director distance.

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3.6 Coupling efficiency in different active media

Positioning a dipole with nano-meter accuracy in air between reflector and director is experimentally a challenging task. In addition, some organic molecules would need a liquid or a polymer matrix rather than air. As a result, varying the active medium between the director and the reflector provides an opportunity to explore different types of organic or inorganic emitters.

Figure 7 determines the coupling efficiency for three different materials as active medium: water ($n_{\textrm {water}} \approx 1.33$), polyvinyl alcohol (PVA) ($n_{\textrm {PVA}} \approx 1.47$) and diamond ($n_{\textrm {dia.}} \approx 2.41$). However, the coupling efficiency is around 20% for these active media, but by changing the core size and material one could improve this efficiency for each case. As an example, the optimal coupling efficiency for PVA is simulated in Fig. 8. Here, the distance between reflector and dipole is 85 nm and the PVA has a thickness of 180 nm. The coupling efficiency is simulated for two different fibers, “custom” which is the same as the custom fiber in previous simulations ($R_{\textrm {core}}$ = 0.8 $\mu$m, $n_{\textrm {core}} = 1.45$ and $n_{\textrm {cladd}} = 1.3$) and “optimal” with $R_{\textrm {core}}$ = 0.6 $\mu$m, $n_{\textrm {core}} = 1.77$ and $n_{\textrm {cladd}} = 1.3$.

 

Fig. 7. Maximum coupling efficiency as a function of wavelength for three different active media, with various refractive indices: $n_{\textrm {water}} \approx 1.33$, $n_{\textrm {PVA}} \approx 1.47$ and $n_{\textrm {dia.}} \approx 2.41$. The media have 295 nm thickness.

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Fig. 8. Maximum coupling efficiency as a function of wavelength for PVA active medium, with custom fiber ($R_{\textrm {core}}$ = 0.8 $\mu$m and $n_{\textrm {core}} = 1.45$) and so-called optimal fiber ($R_{\textrm {core}}$ = 0.6 $\mu$m and $n_{\textrm {core}} = 1.77$). The inset, indicates the reflector-director distance and the dipole position. By adding a higher refractive index active medium the reflector-director distance and the core radius should be decreased.

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By increasing the refractive index of the fiber core the coupling efficiency increases. Moreover, for a large core refractive index, reducing size of the core could also improve the coupling efficiency. The minimum core size depends on the core refractive index. In Fig. 8 the coupling efficiency of $1^{\textrm {st}}$ and $4^{\textrm {th}}$ modes of the “optimal” fiber at 700 nm is $\simeq 70\%$ and it can be increased for a higher core refractive index and smaller core size.

For active media with a larger refractive indices the coupling efficiency is not as high as in air due to the excitation of SPP modes at the interface between the gold layers with the dielectric material. This effect can be reduced by introducing intermediate layers with a lower refractive index with respect to active medium [21].

3.7 Dipole positioning and fiber tilting

The coupling efficiency of the fiber depends on the position of the fiber with respect to dipole and reflector. By lateral displacement of the dipole from the center of the fiber the coupling efficiency is reduced. Figure 9 shows the radial displacement of the dipole from the center along the $x$-axis for custom design fiber. The coupling-efficiency curve exhibits a Gaussian profile with a full-width at half-maximum of 800 nm. The sensitivity to the dipole position could thus add challenges to the experiment, but it could also turn out to be useful as a method for microscopy with low background noise.

 

Fig. 9. Radial distance of the dipole from the optical axis of custom design fiber. The signal has a Gaussian shape with around 800 nm full-width at half-maximum. The coupling efficiency for the higher-order modes is highly dependent on the position of the dipole (not shown).

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Moreover, the tilting of the fiber could reduce the coupling efficiency. The coupling efficiency for the fiber without director layer is almost constant with tilting. Here, the tilting angle ($\theta$) is the angle between reflector and director. As shown in Fig. 10 the coupling efficiency into the fundamental mode of custom and reference fibers with director drops nearly by 4% for one degree tilt of the fiber. As the core radius of the fiber reduces, the tilting has less influence on the coupling efficiency. For a fiber with 0.8 $\mu$m core radius (custom fiber), 2 degrees tilting would reduce the coupling efficiency by only $\approx$ 10%.

 

Fig. 10. The coupling efficiency for the tilted fiber. The custom and reference fiber (Thorlabs “UHNA3”) are tilted by 1 degree and afterwards 2 degrees, showing that the coupling is slightly reduced by small tilting. The inset, shows the geometry of tilted fiber. Comparison of custom and reference fiber illustrates that the system is more sensitive to the tiling for larger fiber cores.

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4. Conclusions

Our study proposes novel optical fiber designs in combination with planar Yagi-Uda antennas for the efficient coupling of light with single emitters and nano-objects.

A gold substrate and a thin gold layer (director) on the tip of the fiber increases the power coupled into the fiber up to 1.5 fold more than the total emitted power of the dipole in free space. This is possible due to the combination of Purcell enhancement and antenna effect. Also, the choice of gold should not be seen as a limiting factor. Other materials like silver or aluminum would lead to similar findings. Such large coupling conditions could be obtained by fiber tapering [11], photonic-crystal cladding [34] or by soft glass material [39] to fulfill the critical NA condition. Moreover, a higher fiber core refractive index would increase the coupling efficiency, when it is above the critical NA condition. Although the results have been presented for a specific wavelength, they are general given the scale invariance of Maxwell’s equations.

The proposed approach would be particularly attractive for the detection of quantum emitters at interfaces, such as fluorophores in bioassays based on surface chemistry, semiconductor nanocrystals or fluorescence beads on thin films, and also for the design of efficient single-photon sources. Moreover, the significant coupling efficiency obtained even for micron-scale distances could be exploited for detecting entities like large viruses or bacteria, finding further applications in sensing, diagnostics, and light microscopy.