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Abstract
We present a combined experimental and numerical study of the far-field emission properties of optical travelling wave antennas made from low-loss dielectric materials. The antennas considered here are composed of two simple building blocks, a director and a reflector, deposited on a glass substrate. Colloidal quantum dots placed in the feed gap between the two elements serve as internal light source. The emission profile of the antenna is mainly formed by the director while the reflector suppresses backward emission. Systematic studies of the director dimensions as well as variation of antenna material show that the effective refractive index of the director primarily governs the far-field emission pattern. Below cut off, i.e., if the director’s effective refractive index is smaller than the refractive index of the substrate, the main lobe results from leaky wave emission along the director. In contrast, if the director supports a guided mode, the emission predominately originates from the end facet of the director.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical antennas enable the efficient conversion of propagating light waves into localized modes and vice versa [1,2]. This capability opens up a multitude of possibilities to engineer light fields and light-matter interactions at the nanoscale. Local field enhancement in the vicinity of an optical antenna can elevate the optical pumping rate of a nano-emitter and lead to an increase of the spontaneous emission rate via the Purcell effect [3,4]. In addition, the optical antennas can modify the radiation characteristics of the nano-emitter leading to a better collection efficiency [5,6].
So far, the majority of optical antenna designs have been based on plasmonic modes supported by metallic nanostructures. These plasmonic nanoantennas are of resonant nature and often considerably smaller than the respective design wavelength [7–9]. Despite of the typically low quality factors of the modes, plasmonic nanoantennas can give rise to strong spontaneous emission rate enhancements due to the strong field concentration into deep sub-wavelength volumes [10,11]. For instance, plasmonic patch antennas can give rise to Purcell factors of several hundred [12,13]. Plasmonic nanoantennas also allow to manipulate the emission properties of nearby emitters. As an example, unidirectional emission of a quantum dot coupled to a plasmonic Yagi-Uda antenna has been demonstrated [14,15].
An inherent disadvantage of plasmonic nanoantennas is absorption losses caused by ohmic dissipation in metals. For this reason, optical antennas made from transparent dielectrics are an attractive alternative. In particular, optical antennas comprising high refractive index materials such as silicon, germanium or gallium phosphide have attracted considerable interest [16–26]. These high-index dielectric optical antennas can support both electric and magnetic Mie type resonances in the visible spectral domain. By balancing the electric and magnetic dipole moments of the optical antenna one can satisfy the first Kerker condition and hence achieve highly directional light scattering in the forward direction [27]. This property makes high-index dielectric optical antennas attractive for metasurfaces [28–31]. Moreover, the significant field enhancements provided by the Mie type resonances of high-index dielectric optical antennas are also interesting for nonlinear optics [32–34], spectroscopy [35–38] and sensing applications [39–42].
Dielectric materials can also serve as an attractive platform for travelling wave optical antennas [43,44]. For instance, we recently demonstrated a travelling wave optical antenna made from Hafnium dioxide that features highly directional emission [45]. Because of their non-resonant nature, these optical antennas are characterized by a large bandwidth and robustness against fabrication imperfections. In comparison to resonant optical antennas, travelling wave optical antennas also have lower requirements for the magnitude of the refractive index of the respective dielectric material. This significantly increases the list of dielectric materials suitable for the realization of optical antennas.
Here, we present an experimental and numerical parameter study on the emission properties of optical dielectric travelling wave antennas made from materials with moderate refractive index value in the range between $1.79$ and $2.4$. The antennas considered in this work are composed of two dielectric elements, namely a director and a reflector. Semiconductor quantum dots deposited in the gap between the two elements serve as integrated light source. Systematic variations of the antenna dimensions as well as use of different dielectric materials provide a deeper understanding of these antennas. Our experiments and numerical calculations show that the effective refractive index of the director primarily governs the far-field emission patterns of the antennas.
2. Experimental methods
In our experiments, we investigate travelling wave optical antennas either made from Hafnium dioxide (HfO$_2$), Indium tin oxide (ITO), or Titanium dioxide (TiO$_2$) with refractive indices $n = 1.9, 1.79$ and $2.4$, respectively. Colloidal CdSeTe quantum dots with ZnS shell (Qdot 800 Carboxyl Quantum Dots, Thermo Fisher Scientific) with an emission wavelength of 780 nm serve as internal light source. The fabrication of these structures is based on the procedure described in [45]. In the first step, we fabricate the dielectric optical antennas. For this purpose, we spin coat a microscope cover slip (refractive index: $n_{\textrm {g}}=1.52$) with the positive tone resist poly(methyl methacrylate) (PMMA) and define the layout of the director and the reflector in the resist layer by electron beam lithography. After the development, either electron-beam evaporation (HfO$_2$) or sputter evaporation (ITO, TiO$_2$) is used to deposit the respective dielectric material. A lift-off process completes the first step. In the second step, the colloidal quantum dots are deposited in the gap between the director and the reflector. To that end, the sample is spin coated again with PMMA and a 300 nm $\times$ 300 nm area between the two elements is defined by electron beam lithography. Following the development, the quantum dots are chemically linked to the unveiled substrate and the PMMA resist is removed. A scanning electron micrograph of a HfO$_2$ antenna with quantum dots fabricated along these lines is depicted in Fig. 1(a).
Fig. 1. a) Scanning electron micrograph of a HfO$_2$ antenna deposited on a microscope cover slip. Quantum dots are deposited in the feed-gap region (see red box). b) Scheme of the Fourier plane imaging microscopy setup.
The emission properties of the antennas are characterized by Fourier plane imaging microscopy. A scheme of the setup employed for this task is depicted in Fig. 1(b). The quantum dots of a single antenna are excited by focusing either a blue or a green pump laser (wavelength $\lambda =$ 453 nm and $\lambda =$532 nm, respectively) through the glass substrate with a high resolution oil immersion objective ($100 \times$ magnification, numerical aperture NA=$1.45$) on the gap region. Both the lasers have a spot size of less than 400 nm. The near-infrared quantum dot fluorescence is centered around a wavelength of 780 nm. It is collected with the same immersion objective. The back focal plane of the objective is imaged with a 100 mm lens onto an electron multiplying charge-coupled device camera (EMCCD camera). The pump light is blocked by the combination of a dichroic beam splitter and a longpass filter in front of the camera. A polarizer behind the dichroic beam splitter is used to analyze the polarization of the emitted light.
The angular distribution of the emitted light and the spatial intensity distribution in the back-focal plane of the objective are related by the sine condition, i.e., all light emitted by the antenna in a particular direction $(\theta , \varphi )$ is imaged onto the same position $(x,y)$ of the camera chip with
Here, $\kappa$ is the scaling factor of our imaging system, $\theta$ is the polar angle measured with respect to the optical axis (normal to the substrate interface), and $\varphi$ is the azimuthal angle. The coordinate system is chosen such that $(\theta =90^\circ , \varphi =0^\circ )$ corresponds to the antenna axis. The numerical aperture $\textrm {NA}$ of the microscope objective specifies the maximum collection angle $\theta _{\textrm {NA}}=\arcsin \left (\textrm {NA}/n_g\right )$ and hence also the corresponding maximum distance $\rho _{\textrm {max}}$ of a signal on the camera chip from the optical axis. We can use this property to determine the scaling factor of our imaging system as $\kappa =\rho _{\textrm {max}}/\sin (\theta _{\textrm {NA}})$.3. Emitter-antenna coupling
Before we address the experimental results, it is instructive to numerically study the coupling of the quantum dots to a dielectric optical antenna. In particular, we want to investigate the effect of the dipole orientation and position of a quantum dot on the angular far-field intensity distribution of the antenna, similar to the work in [30]. The calculations employ the software package CST Studio Suite for the near and far field calculations to derive the angular resolved emission into the glass substrate. The length, width, and height of the director/reflector of the antenna structure made of HfO$_2$ are chosen to be 2200 /180 nm, 600 /785 nm, and 140 /140 nm, respectively. The two elements are separated by a gap of 260 nm. By fitting the near-field distribution within the director, we find that the effective mode index is $n_{\mathrm {eff}}=1.51+0.04i$. Hence, the director acts as a leaky waveguide for these parameters. A Hertzian dipole with an emission wavelength of 780 nm serves as an emitter and is placed in the feed gap between the director and the reflector.
In the first set of simulations, the dipole is centrally placed 40 nm in front of the director on the substrate. This position has been found in a parameter sweep to optimize the directivity of the antenna. The random orientation of the quantum dots occurring in the experiments is taken into account by performing three simulation runs, in which the dipole is successively directed along the $x$-, $y$- and $z$-axis, respectively. The calculated angular far-field intensity distributions for these three dipole orientations are shown in Fig. 2 (a). The largest far-field intensity signal is found for the dipole oriented along the $y$-axis. In this case, we observe a strong main lobe at ($\theta _{\mathrm max} = 66^\circ$, $\varphi _{\mathrm max} = 0^\circ$), which results from leakage radiation emitted along the director into the substrate [45]. A considerably weaker signal is produced by the dipole oriented along the $z$-axis. Finally, the dipole oriented along the $x$-axis does not give rise to an appreciable far-field signal. Therefore, we will concentrate in the following on the $y$-oriented dipole, as it provides the most relevant data for characterization of the emission patterns of our antenna.
Fig. 2. Calculated angular far-field intensity distributions of a dipole coupled to a HfO$_2$ antenna. All distributions are shown with the same color scale and are normalized to the maximum signal found in the calculations. a) Variation of the dipole polarization (left: $x$-polarization, middle: $y$-polarization, right: $z$-polarization). b) Variation of the $x$-position of the dipole. c) Variation of the $y$-position of the dipole.
In the experiments, the quantum dots are deposited in an area of approximately 300 nm $\times$ 300 nm between the director and the reflector (see red box in Fig. 1(a)). Hence, it is interesting to investigate the effect of displacing the dipole from its optimal position on the coupling efficiency and the emission pattern. We first discuss a displacement along the $x$-axis keeping the $y$-position constant. Figure 2(b) shows the angular far-field intensity distributions for three dipoles placed 120 nm, 160 nm and 200 nm in front of the director, respectively. As the separation between the dipole and the director increases, the intensity in the main lob decreases and back reflection becomes more important. The reduced directivity can be attributed to the decreasing coupling efficiency of the dipole to the director with increasing separation. As a consequence, the emission pattern starts resembling more to that of an uncoupled dipole.
Next, we consider the effect of displacing the dipole in $y$-direction. Figure 2(c) displays the angular far-field intensity distributions for three dipoles with an offset of $ {-100}$ nm, $ {+100}$ nm and $ {200}$ nm, respectively, along the $y$-axis from the symmetry plane of the antenna. The distance to the director in $x$-direction is in each case 40 nm. We observe for all three configurations a pronounced main lobe. However, the azimuthal position of the maximum emission changes. For a $y$-offset of $ {-100}$ nm, the main lobe is redirected by $\Delta \varphi =+5^{\circ }$ and similarly, for a $y$-offset of $ {+100}$ nm and $ {+200}$ nm, the main lobe is redirected by $\Delta \varphi =-5^{\circ }$ and $\Delta \varphi =-10^{\circ }$, respectively.
In the experiments, the quantum dots in the feed gap radiate independently of each other such that the total signal is given by the incoherent addition of the individual intensities. Based on the calculation presented above, we anticipate the strongest experimental signals for TE-polarization (analyzer axis is set perpendicular to the antenna axis). Moreover, we expect that the spatial distribution of the quantum dots will result in a slight angular broadening of the emission lobes with the strongest contribution from quantum dots close to the optimal coupling position.
4. Variation of the antenna parameters
In the following, we will study the dependence of the emission properties of the antenna on the width and length of the director as well as on the material the antenna is composed of. For this purpose, we will compare the measured and calculated angular far-field intensity distributions.
4.1 Variation of the director’s width
We will start our discussion on the influence of different antenna parameters on the emission properties with the width of the director. All antennas analyzed in this section are made of HfO$_2$ and have a constant reflector size with a length of 180 nm, width of 785 nm, and height of 140 nm. The emitter is placed 40 nm from the director and 220 nm from the reflector. The length and height of the director are chosen to be 2200 nm and 140 nm, respectively. The director’s width is increased from $w={600}$ nm to $w= {1200}$ nm and finally to $w= {1800}$ nm.
Figure 3(a) displays the calculated angular far-field intensity distributions for the three different values of the director’s width $w$ for TE polarization. The antenna with the $w= {600}$ nm wide director exhibits a pronounced main lobe at ($\theta _{\mathrm max} = 66^\circ$, $\varphi _{\mathrm max} = 0^\circ$) and has a directivity of $D_{\mathrm {max}}=29.2$ in magnitude. By increasing the director’s width to $w= {1200}$ nm, the polar angle of main lobe increases to $\theta _{\mathrm {max}} = 69^\circ$. The corresponding directivity is $D_{\mathrm {max}}=28$. Finally, for the antenna with $w= {1800}$ nm, we observe several emission lobes with a main peak at ($\theta _{\mathrm max} = 49^\circ$, $\varphi _{\mathrm max} = 0^\circ$).
Fig. 3. a) Calculated and b) measured angular far-field intensity distributions of HfO$_2$ antennas for different director widths recorded using TE analyzer setting. The white circles at $\theta _{\textrm {NA}}=72^{\circ }$ mark the experimentally accessible angular range. Experimental and theoretical intensities are independently normalized to their respective maximum.
Measured angular far-field intensity distributions for the corresponding antenna widths $w= {600}$ nm, $w= {1200}$ nm, and $w= {1800}$ nm are presented in Fig. 3(b). The experimental data confirms the main trends predicted by theory. The emission pattern of the antenna with the width $w= {600}$ nm features a main lobe at around ($\theta _{\mathrm max} = 65^\circ$, $\varphi _{\mathrm max} = 2^\circ$). Additionally, we observe a ring-like feature at $\theta =41.5^\circ$ that can be attributed to the emission of uncoupled quantum dots [45]. For $w= {1200}$ nm, the polar angle of the main lobe increases to approximately 69°. The measured angular far-field intensity distributions of the antenna with the largest width ($w= {1800}$ nm) shows, like in the calculations, a multi-peaked emission pattern with a maximum at approximately 49°.
To interpret the observed features, it is instructive to perform an eigenmode study. In this study, based on the Finite Element Method, the finite length of the director is neglected. Figure 4 displays the effective mode index of the guided modes of a HfO$_2$ waveguide with a height of 140 nm as function of its width. For light with a wavelength of 780 nm, i.e., for the quantum dot fluorescence, the first mode is only guided if the width exceeds 850 nm. Hence, the first antenna with director width $w= {600}$ nm is a leaky wave antenna and the main lobe results from leaky wave emission along the director [45]. In contrast, the second antenna width $w= {1200}$ nm supports a guided mode and the emission predominately originates from the end facet of the director. The stronger spatial localization of emission goes hand in hand with a reduction of the directivity. Finally, for the antenna width $w= {1800}$ nm the second guided mode becomes relevant leading to multi mode emission with three emission peaks in the direction of the director.
Fig. 4. Calculated effective mode index of the guided modes of a HfO$_2$ ridge waveguide as a function of its width for a height of 140 nm. The insets show cross sections of the intensity distributions of the first three guided modes.
4.2 Variation of the director’s length
In this section, we analyze the influence of the director length on the emission properties of the antenna. For this purpose, we consider a director that supports a single guided mode (height: 180 nm, width: 600 nm) and vary its length from 1400 nm to 3000 nm. The dimensions of the reflector and the position of the emitter are the same as in the previous section. The corresponding calculated and measured angular far-field intensity distributions together with snapshots of the calculated absolute value of the electric near-field distributions are depicted in Fig. 5.
Fig. 5. a) Calculated snapshots of the absolute value of the electric field (linear scale) for three HfO$_2$ antennas with different director lengths. A dipole in the feed gap between director and reflector acts as the source. b) Calculated and c) measured angular far-field intensity distributions of HfO$_2$ antenna recorded as function of the director length using TE analyzer setting. The white circles at $\theta _{\textrm {NA}}=72^{\circ }$ in the patterns mark the experimentally accessible angular range. Experimental and theoretical intensities in the images are independently normalized to their respective maximum.
The calculated far-field intensity distributions show two main trends as the director length increases. First, the polar angle of the main lobe increases from 63° over 69° to 72°, accompanied by an increasing directivity ranging from 20.9 over 25.4 to 28.8, respectively. And second, a pattern of side lobes in the $\varphi =0^\circ$-direction is formed, where the number of side lobes increases with the length of the director. In the experimental data, we do not observe a clear increase of the polar angle of the main lobe with increasing director length. This seeming discrepancy between theory and experiment can be easily traced back to the finite collection angle of the objective lens, i.e., the main lobe of the longest antennas is not fully captured in the experiment. As in the calculations, we observe in the experimental data in addition to the main lobe, a series of side lobes in the $\varphi =0^\circ$-direction.
The observed increase of the polar angle of the main lobe is at first sight surprising, as one might expect that for a single mode director the emission profile is only determined by the diffraction at the end facet and hence should not depend on its length. However, the dipole in the feed gap does not only excite the guided mode but also leaky higher order modes in the director, which quickly emit into the substrate (see near-field distributions in Fig. 5(a)). For the shortest director, both emission channels play a significant role and the main lobe is influenced by the interference of the fields emitted along the director and at its end facet. With increasing antenna length, leakage radiation becomes less relevant and the antenna emission is dominated by diffraction at the end facet. In this regime, the polar angle of the main lobe stays practically constant with a value of $\theta _{\mathrm max} = 80^\circ$ as determined by additional numerical calculations for antenna length up to $ {10000}$ nm (not shown).
The pattern of side lobes can also be attributed to the excitation of higher order modes. As mentioned above, these modes couple to radiating modes in the substrate and hence lead to emission along the waveguide. In this process, the director acts as a finite sized aperture, where the interference of the leakage radiation emitted from different parts of the aperture results in the pattern of side lobes.
4.3 Variation of the antenna material
The discussion in connection with the director’s width has shown that the effective mode index of the director has a large influence on the emission properties of the antenna. In this section, we want to further elaborate this point and compare three antennas made from different materials (HfO$_2$, ITO and TiO$_2$) but with the same real part of the mode index. As reference, we choose the leaky HfO$_2$ antenna with effective index $n_{\mathrm {eff}}^{\mathrm {HfO_2}}=1.51+0.04i$ already discussed in section 4.1. To determine the geometry parameters for the directors of the other two antennas, we performed numerical calculations for different geometry parameters and extracted for each configuration the effective mode index by fitting the respective electric field distribution within the director. From these calculations, we find that suitable dimensions for the director of the ITO-antenna are $\mathrm {width}\times \mathrm {height}\times \mathrm {length}= {900} \; \textrm{nm} \times {110}\; \textrm{nm} \times {2200}$ nm resulting in an effective index $n_{\mathrm {eff}}^{\mathrm {ITO}}=1.51+0.06i$. For the TiO$_2$-antenna, we choose a director with the dimensions $\mathrm {width}\times \mathrm {height}\times \mathrm {length}= {800}\;\textrm{nm} \times {40}\;\textrm{nm} \times {2200}$ nm leading to an effective index $n_{\mathrm {eff}}^{\mathrm {TiO_2}}=1.50+0.05i$.
Figure 6(a) depicts the calculated angular far-field intensity distributions of the three antennas for TE analyzer setting. We observe in all three cases a pronounced main lobe at ($\theta _{\mathrm max} = 66^\circ$, $\varphi _{\mathrm max} = 0^\circ$). This underscores, that the direction of maximum emission of a leaky wave antenna is mainly determined by the real part of the effective index [46]. The corresponding imaginary part affects the directivity of the antenna. This difference in the directivity corresponds well to the ratio of the power decay factor, $\exp {(-2\mathrm {Im}(n_{\mathrm {eff}})2\pi \mathrm L/\lambda )}$ of the mode at the end facet after propagating the director length L, which is approximately 24% for HfO$_2$, 17% for TiO$_2$ and 12% for ITO. The HfO$_2$-antenna with the smallest imaginary part of the mode index features the largest directivity $D_{\mathrm {max}}^{\mathrm {HfO_2}}=29.2$, followed by the TiO$_2$-antenna with $D_{\mathrm {max}}^{\mathrm {TiO_2}}=19.4$, and the ITO-antenna with $D_{\mathrm {max}}^{\mathrm {ITO}}=13.6$. Experimental angular far-field intensity distributions for the three antennas made from HfO$_2$, ITO, or TiO$_2$, respectively, are shown in Fig. 6(b). The dimensions of the director correspond in each case to those used in the respective numerical calculation (see above). In line with the numerical results, all three measured data sets feature a pronounced signal in forward direction at approximately $\theta =66^\circ$. Additionally, we observe in particular for the ITO-antenna and the TiO$_2$-antenna a strong ring-like feature that can be attributed to uncoupled quantum dots. The latter aspect indicates that the quality of the dielectric antennas fabricated by our sputtering process (ITO and TiO$_2$) is inferior compared to the antennas deposited by electron beam evaporation (HfO$_2$).
Fig. 6. (a) Calculated and (b) measured angular far-field intensity distributions of a HfO$_2$ antenna (left), an ITO-antenna (middle) and a TiO$_2$ antenna (right) recorded using TE analyzer setting. The dimensions of the director of the three antennas have been chosen such that the real part of the respective mode index is the same. The white circles at $\theta _{\textrm {NA}}=72^{\circ }$ indicate the experimentally accessible angular range. Experimental and theoretical intensities in the images are independently normalized to their respective maximum.
5. Conclusion
In conclusion, we present a parameter study for highly directional dielectric travelling wave antennas. The antennas are composed of two dielectric building blocks, i.e., the reflector and the director, and use quantum dots as an internal light source. Our experiments and numerical calculations show that the emission properties of the antennas are mainly determined by the effective mode index of the director. The most directive antenna investigated has a directivity of $D_{\textrm {max}}$=29.2 in magnitude. Its director has a width, length and height of 600 nm, 2200 nm and 140 nm, respectively and is composed of Hafnium dioxide. Most efficient coupling is accomplished with a y-polarized dipole. We anticipate that the directivity of the antenna can be further improved using optimization algorithms.
Funding
Deutsche Forschungsgemeinschaft (231447078 - TRR142, Project C05, ML4Q, EXC 2004/1 - 390534769, Cluster of Excellence Matter and Light for Quantum Computing); Bundesministerium für Bildung und Forschung (13N14150); Horizon 2020 Framework Programme (724306).
Acknowledgement
The authors gratefully acknowledge the computing time support provided by the Paderborn Center for Parallel Computing (PC2).
Disclosures
The authors declare no conflicts of interest.
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