By out-of-particle surface plasmon (SP) excitation in the near infrared range, the influences of key parameters on the basic optical properties of the Au crescent are qualitatively studied from the mode dispersion. Based on the coherent control of SP wave, a crescent pair sensor with the intensified extracted signal and the controllability of sensing is proposed. In addition, the crescent half replaced by Ag functioning as a position detector is also proposed. The particular phase of the detecting light as a detection parameter is used to improve the accuracy of the position detection.
© 2011 OSA
As an important spectroscopic tool, surface plasmon resonance (SPR) [1–3] has been of great interest to many fields, such as biology, medicine, biomedical engineering, environmental and industrial monitoring, as well as defence and security [4–7]. The local field enhancement, which is associated with the plasmon resonance in metallic nanostructures [8,9], especially attracts intense research interest. Lee’s group proposed the nanophotonic crescent, which combined the priorities of nanotips and nanorings . This standalone structure realized the local field enhancement by the intra-particle plasmonic coupling, and was used in the ultrasensitive biomolecular detection. Effective two dimensional (2D) approximation was employed to theoretically analyze the hybridization between the tip mode and the cavity mode , and the local field enhancement maximization . Recently, they analyzed optical properties of the crescent-shaped nanohole theoretically and experimentally by tuning the geometrical parameters . However, most of their analyses are based on the intra-particle excitation, and the analysis of mode hybridization is a little complex.
The near-field excitation is a non-resonant excitation which always induces the interferences in different areas of the metal surface and forms the complex field distribution . However, there is little influence on the nanocrescent because the field is highly localized near the nanotips. The out-of-particle excitation avoids the intricate intra-particle mode coupling, though some inner nodes are excited because of the plasmon being coupled from the outer surface to the inner surface due to the small thickness, which is comparable with the skin depth of about 25 nm in our interest wavelength range. Therefore, we can focus on the analysis of optical properties more applicably. The coherent control of SP wave based on the constructive and destructive interferences is beneficial to the optical signal extraction and control, and can be used in optical sensing, addressing and optical storage. Furthermore, utilizing the interference to detect the position of the light-emitting object is of referential value to the design of the position detector.
In this article, the near-field excitation from the outer surface of the crescent in the near infrared range is used to form the tip mode, also localize and enhance the field. Key parameters, such as the inner radius, the spacing between the nanotips and the refractive indices of the cavity and the background, are analyzed in order to qualitatively evaluate the basic optical properties from the mode dispersion. The coherent control of SP wave based on the crescent is also investigated for the first time. A pair of crescent cylinders, one used as a sensing and controlling unit and the other as a signal extracting unit, is proposed as a sensor and studied under the coherent control. In addition, the crescent partially replaced by Ag which functions as a position detector is also proposed and typical positions of the signal light are detected by the device. The phase of the detecting light that makes the partial interference intensity reach the maximum is used as a detected parameter in order to improve the accuracy of the position detector efficiently.
2. Simulation model
The 2D approximation is used in the simulation since it can be easily extended to crescent cylinders of finite height or crescent spheres. As shown in Fig. 1(a) , the outer radius r 1 of the crescent is 100 nm. The inner radius and the spacing between the nanotips are represented by r 2 and h, respectively. The following Drude model is applied to calculate the dielectric parameter:
in which = 1, plasmon frequency = 1.37 × 104 THz and γ = 40.7 THz for gold . The relatively decentered hole and the background are full of air and the corresponding refractive index n 0 = 1. A transverse magnetic (TM) polarization Gaussian light with the width 10 nm is incident from the left side with the distance 25 nm away from the outer surface along the horizontal symmetric axis. By the near-field excitation which is mainly at the wide section of the crescent, different electric components of the incident light excite periodically oscillating charges, forming the SP wave which propagates around to the claws without being reflected. The field is highly localized and enhanced near the tips. Specific optical properties are analyzed in the following sections. The simulation is performed using the finite-difference time-domain method (Rsoft Design). The calculated area is 1.1 × 1.1 μm2 with the grid size of 0.5 nm. The perfect matching layer with the width 0.3 μm is used in our simulation.
3. Analysis of basic optical properties
When r 2 = 80 nm and h = 20 nm, the obtained resonant peak is at a wavelength of 1.754 μm with the quality factor about 12. Figures 1(b), (d) and (f) show the field distributions of components Ex, Ez and Hy, respectively. Figures 1(c) and (e) are the correspondingly magnified field distributions of Ex and Ez near the tips. From the field distributions of the electric component, it is clear that multiple nodes are formed in the outer surface and the inner surface because the near-field repeatedly converges and diverges horizontally and vertically. The field distribution of Ex has a large difference with that of Ez for the distributions of Ex near the top tip and the bottom tip are independent, while that of Ez depend strongly on the interaction of the two tips. The Ex component is an equivalent dipole with the oscillation along the x direction, which creates positive and negative charges in the outer and inner surfaces, also excites the corresponding dipole-like near field. Where the charges are nearer to the tips, the crescent becomes thinner, hence the nodes are nearly symmetric with respect to the z direction, which is displayed by the two opposite surfaces shown in Fig. 1(c). As for the component Ez, it is an equivalent dipole with oscillation along the z direction. Positive and negative charges are distributed symmetrically about the x direction thus excite the corresponding dipole-like near field. The nodes are symmetric with respect to the x direction, and the lighting rod effect appears distinctly at the tips (see Fig. 1(e)). It is also clear that the maximum of Ez is larger than that of Ex due to the field distribution difference between Ez and Ex as described above. The total electric field distribution is the hybridization effect of the two types of dipole-like modes along the x direction and the z direction, respectively. As shown in Fig. 1(f), the intensive part of the magnetic component field is distributed in the cavity. From the inner surface to the outer surface, the magnetic amplitude is decreased due to the influence of the skin depth. The maximum of the magnetic component is much less than that of the electric component. Figure 1(g) shows the enhancement of the Poynting vector which is defined as , whereis the Poynting vector amplitude in free space. The maximum of the enhancement reaches 24.6 dB. In the calculation, the electron tunneling effect, which may reduce the field enhancement , has not been taken into account where the tips are at a desirable thinness. Most of the energy is localized near the tips. Excellent energy accumulation may make the crescent utilized in the solar cells or nanoantennas.
Some key parameters have been changed to analyze the influences on optical properties. In the case of r 2 = 80 nm, the increase of h from 20 nm to 90 nm makes the coupling of the dipole-like mode along the z direction weaker. Meanwhile, the thickness of the crescent is increased and thus the sharpness of the tips are decreased, which results in the weakened coupling of the dipole-like mode along the x direction. The effective refractive index is decreased with the diffusion of the field, hence the resonant peak blue shifts and the amplitude is decreased, which is shown in Fig. 2(a) .
As shown in Fig. 2(b), h = 20 nm, the resonant peak red shifts as r 2 is increased from 40 nm to 80 nm. Since the refractive index of Au in this wavelength range is much less than air, the increase of inner radius results in a thinner crescent and thus the increase of effective refractive index, which leads to the decrease of resonant frequency. The tips become sharper and the coupling of the dipole-like mode along the x direction is strengthened. As a result, the mode localization near the tips is stronger. There is a tradeoff between the enhancement of localization and the absorption of the Au crescent, because the imaginary part of the dielectric function of metal becomes important to the mode with high localization. Consequently, when r 2 is 80 nm, the absorption loss, which is beyond the enhancement, causes the decrease of the resonant amplitude.
Two types of influences of the refractive index n sensor change on the resonant spectrum are studied. One is that the light is incident from the left side, and the out-of-particle excitation leads to the shift of the resonant peak by changing n sensor of the cavity, which is shown in the left lower inset of Fig. 3(a) . The other is that light is incident from the right side, and the change of the background n sensor causes the shift of the resonant peak by the intra-particle excitation, which is shown in the left lower inset of Fig. 3(b). Increasing n sensor of the cavity or the background causes the effective index to be increased and the field to be more concentrated. As a result, the red shift of the resonant peak happens. It is clear that the shift of the peak in Fig. 3(a) is larger than that in Fig. 3(b). Right upper insets in Fig. 3(a) and Fig. 3(b) show the concrete shifts of the resonant peaks when n sensor is increased from 1.0 to 1.5. The shifts are nearly linearly increased, thus the simple linear fit is utilized to calculate the slopes, which are 0.950 and 0.285, respectively. That is to say the sensitivities of the crescent as a sensor working in two ways reach 950 nm/RIU and 285 nm/RIU, where RIU is the refractive index unit. In Fig. 3(a), with the increase of n sensor, some fluctuations appear in the resonant spectra, because the increase of impedance causes the Fabry-Perot effect of the SP. While in Fig. 3(b), since the intra-particle coupling modes in the inside cavity formed by the airhole hybridize with the modes supported at the Au-medium interface of the outside cavity formed by the detected medium with larger n sensor, some wiggles occur in the resonant spectra.
Except the superiority of sensitivity in the out-of-particle excitation, some comparisons between the two sensing mechanism are studied. As the refractive index of the background is detected, in order to avoid the direct influence of the incident light on the detected medium, the light is incident from the right side towards the split formed by tips, which limits the size of source and requires a strict alignment. But when the detection of the refractive index in the cavity is implemented, the limitation of the source size and the requirement of the alignment become somewhat relaxed. Some measure is taken to detect the magnetic component or the electric component of the signal light . M. Burresi et al. fabricated a near-field split-ring probe to detect the magnetic field at optical frequencies. The magnetic field of light was converted to an electric field in the probe, then coupled to the fiber of the probe and detected in one channel of the microscope. In the other channel, the electric field related optical signal was detected. Thus in our proposed sensing and position detection below, the magnetic or electric near field can be detected by their detecting method. If the electric component of the resonant signal is detected, the scanning area of the probe is limited, since the electric field is mainly localized near the tips. And if we extract the magnetic component of the signal, the high detecting resolution is necessary for the probe because of little enhancement of the magnetic field. As regards broadening the scanning area and intensifying the resonant signal (mainly the magnetic field), we will discuss in the following content.
4. Interference and coherent applications
The interference in small size is relatively easy to be realized, especially to the structure supporting the SP wave, which overcomes the diffraction limitation. To demonstrate this, two TM sources of the same amplitude are incident from the left side of the crescent, with the distance 2 nm away from the outer surface on both sides of the horizontal symmetric axis, which is shown in the right lower inset of Fig. 4 . The width of each source is 10 nm and the spacing between the physical centers is 20 nm.
The total phase difference of the interference of two identical frequency beams satisfies
where is the initial phase difference between two light sources, and are the phase difference introduced by the difference of the light path length in the air and the difference of the SP wave path length , respectively. , and , where and are the wavelength of the light in the air and the SP wavelength, respectively. In the case of the same amplitude interference, if = 2n π (n is an integer), there is a constructive interference, whereas if = (2n + 1) π, the destructive interference is obtained. Thus the interference intensity can be expressed as
where is the resonant intensity of the single source.
When the two sources have the same initial phase, = 0. Although there is a distance of 20 nm between the two sources, two SP waves are excited at nearly the same point. Hence small and would result in near zero and . According to Eq. (2), the obtained can be nearly zero. The constructive interference is obtained, which is shown as the black solid line in Fig. 4. Compared with the resonant peak of the single source which is shown as the black dashed line, the intensity of the constructive interference peak is just four times larger at the same peak wavelength. Once the phases of two sources are 0, π respectively, equals π, thus is nearly the same value as π, leading to the destructive interference, shown as the red solid line in Fig. 4. Though some peaks occur due to the differences of the excited points and the calculated grid size, their intensities are so low that they can be neglected. The extinction ratio reaches at the most 118 dB, where and are the constructive and the destructive interference intensities, respectively. Consequently, tuning the phase of incident light can realize the coherent control of SP wave on the crescent.
Based on the control of the interference, the applications in chemical sensing are discussed. Another mirror symmetry crescent is added to form a pair for coupling field to broaden the detectable area for the probe, which is shown in the inset of Fig. 5 . There is a distance of 50 nm between the top tips and the bottom tips, and 20 nm between the left tips and the right tips. Such a crescent pair is divided into the sensing and controlling unit on the left and the signal extracting unit on the right. The detected nanofluid flows in a tube throughout the left cavity, and the right cavity is used as the scanning area of the probe. The preliminary calculation results of crescent pair coupling indicate that the detected resonant intensity of the crescent pair is higher than that of the single crescent due to stronger electric field concentration between the tips. Here the coherent control is utilized to intensify the extracted signal in the constructive interference, and to recognize the controllability of sensing. If some unexpected disturbance appears during the data collection, we can conveniently tune to π for the destructive interference without frequently turning on/off the light and other detecting instruments. All these are at the cost of the decrease of sensitivity due to the mode expansion between the two cavities of crescents. The sensitivity is decreased from 950 nm/RIU of single crescent to 412 nm/RIU (see Fig. 5). In terms of sensors, the figure of merit (FOM) introduced by Sherry et al.  is about 8.4 for the single crescent and 2.8 for the crescent pair. The calculated sensitivities and FOMs are comparable with the recent reported local SPR sensor data [19–22]. The detected medium lying in the cavity is a challenge for the fabrication. But as reported before, the size of gold nanoholes for detecting the nanofluid may be as small as 200 nm , providing the opportunities for the crescent pair sensor to be actualized in the future.
In addition, some optical properties of the crescent are used to reflect the position of the light-emitting object. Here, the light-emitting object is represented by the signal source, which is characterized with θ and d. As shown in Fig. 6(a) , θ is the azimuth angle which is anti-clockwise and is increased from 0° to 360°, d is the near-field distance away from the outer surface of the crescent. To achieve the position detection, a part of the Au crescent is replaced by Ag. The asymmetry of the crescent is not introduced by the structure but the material, for the structural asymmetry always destroys the field distribution and brings complexities to the analysis.
The moving single signal source with the fixed phase is investigated in which some typical positions of the signal source are detected, which is shown in Table 1 . The detected parameters are the resonant peak wavelength and the magnetic amplitude. From Table 1, it is shown that the resonant peak wavelengths are different, but the detected magnetic amplitudes are generally small, which raises a high requirement for the magnetic field probe.
To increase the resonant intensity, the partly constructive interference is employed. A source with the changeable phase, i.e. the detecting source, is fixed at the left side along the horizontal symmetric axis with d = 2 nm and θ = 180° (see Fig. 6(b)). The resonant wavelength must be included in the spectrum range of the two sources. Their phases are identical at first, for example 0, thus = 0. When the signal source is of the different θ, the varieties of and lead to the change of , whereas is nearly unchanged except that the signal source moves along the radial direction. The calculation results are shown in Table 2 . It is clear that when d = 2 nm, θ = 90°, 270° and d = 7 nm, θ = 270°, the resonant wavelengths are identical, and there are almost no differences of amplitude. This case is not ideal for the position detection, so we assign another detected parameter for achieving more information in order to improve the detecting accuracy. The parameter is the phase of the detecting source which maximizes the resonant intensity in the partial interference. The various and need the matched to form the partly constructive interference. Accordingly, in the course of position detection, the phase scanning will be done on the detecting source. If the intensity reaches the maximum, this intensity, as well as the corresponding resonant wavelength and the phase of the detecting source will be recorded. The obtained three detected parameters when θ = 90° and 270° are shown in Table 3 , which demonstrates that different positions can be identified with the crescent. As a result, we can ascertain the position of the light-emitting object by the three detected parameters. Compared with the direct detection of the single signal source, the coherent detection increases the resonant amplitude. Meanwhile, introducing another detected parameter improves the accuracy of the position detection. However, because the SP waves are only near-field excited, the detected range is limited to the near-field for the operating wavelength unless the exciting method is changed or replaced by the particle array excitation, e. g. the far-field and crescent sphere array excitation.
The simulations and analyses above are all based on 2D calculations. We give a discussion about how the 2D results are expected to apply in a more realistic 3D model. The crescent cylinders of finite height have the similar optical properties to 2D crescents in the case that the incident direction of the source is along the normal line of side wall of the crescent. Because the TM light has the magnetic component along the direction of the height of the cylinder, the charge distribution in this direction is not influenced. Consequently, the 2D analyses of the chemical sensor and the position detection are suitable for crescent cylinders. In a 3D nanocrescent sphere, the local field distribution with the high local field enhancement extending around the sharp edge is similar to that of crescents in 2D. In the analysis of optical properties, the 3D crescent sphere follows trends similar to 2D crescents . So the analysis of position detection in 2D can be applied in a 3D crescent sphere. As long as the crescent sphere is made of multiple noble metals to break the symmetry, the 3D position detection can be achieved. However, for the crescent sphere array, only the single signal source position detection is applicable.
In the near infrared range, by the out-of-particle SP excitation, we analyze the field distributions of different components of the incident TM light and qualitatively investigate the influences of key parameters on the optical properties of the crescent from the mode dispersion. Moreover, the coherent control of the SP waves is also studied, and the extinction ratio as high as 118 dB is obtained. In that case, the crescent pair is proposed as a sensor with the intensified extracted signal and the controllability of sensing. Based on the interference of SP waves, the position detection of light-emitting object is demonstrated by our simulated results. Including the phase of the detecting light source as a detected parameter can distinctively improve the accuracy of position detector. If the change of phase is as small as arc minute, even arc second, the accuracy may be higher. By modifying the geometric size, the resonant wavelength can be expanded into the biological window or the terahertz range, i.e. the size effect . Enlarging the size of the crescent which results in the red shift of the resonant wavelength would bring more convenience to the chemical sensing and the position detection at the longer wavelength range. The coherent control of SP wave based on the crescent also benefits the light signal modulation, optical storage and the high order harmonic wave generation, so can be applied in future photoelectric integrated circuits.
This work is supported by the Chinese National Key Basic Research Special Fund/CNKBRSF (Grant No. 2011CB922000), the National Natural Science Foundation of China (Grant Nos. 61025025 and 60838003), and the National High Technology Research and Development Program of China (Grant Nos. 2007AA03Z410 and 2007AA03Z408).
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