Using finite-difference-time-domain simulation, we have studied the near-field effect of Germanium (Ge) subwavelength arrays designed in-plane with a normal incidence. Spectra of vertical electric field component normal to the surface show pronounced resonance peaks in an infrared range, which can be applied in a quantum well infrared photodetector. Unlike the near-field optics in metallic systems that are commonly related to surface plasmons, the intense vertical field along the surface of the Ge film can be interpreted as a combination of diffraction and waveguide theory. The existence of the enhanced field is confirmed by measuring the Fourier transform infrared spectra of fabricated samples. The positions of the resonant peaks obtained in experiment are in good agreement with our simulations.
© 2013 Optical Society of America
Surface plasmon (SP) subwavelength optics have attracted considerable interests since Ebbesen et al. reported the extraordinary transmission . SPs are electromagnetic (EM) waves localized at a metallic surface through interaction with free electrons . There are a wide range of SP-related applications such as near-field microscopy, light emitters, solar cells, semiconductor lasers, and photodetectors [3–5]. A major feature of SPs is their excellent near-field effect or the way in which SPs concentrate and channel light while simultaneously generating an evanescent field through subwavelength structures . The near-field effect can lead to an electric field enhancement, which has been exploited by Wei Wu et al. to enhance the detectivity of a quantum well infrared photodetector (QWIP) . QWIP is only sensitive to the electric field component normal to the quantum well surface (TM mode) , which can be effectively supported by SPs. However, the imaginary part of the dielectric function is large for metallic materials, causing inherent absorption loss or SP loss . SP loss can lead to lower field enhancement and a broadening of resonant peaks , thus reducing the performance of the device. On the other hand, metallic subwavelength structures always cause radiation loss into the SP modes when they are applied in light generating devices such as organic light-emitting diodes .
Besides SP theory, there are other instructive interpretations of subwavelength optics such as composite diffracted evanescent waves (CDEWs) , Bloch-wave modes of dynamic diffraction  and vertical resonant cavities that act as efficient waveguides . In addition, the SP-related structure is not the only way to cause the useful near-field effect. For example, an optical microfiber/nanofiber can provide a strong near-field interaction, which has been exploited in nanophotonic devices such as polymer sensors for humidity detection . A photonic crystal surface-emitting laser utilizes the constructive interference of in-plane propagating waves to induce the surface emission property . A photonic crystal slab can also be applied in the QWIP to form an optical resonator providing an enhanced vertical electric field .
In this paper, we achieve an alternative generation of near-field effect in the infrared range using a non-metallic structure—Germanium (Ge) subwavelength arrays—for overcoming SP loss. Through the three-dimensional finite-difference-time-domain (3D-FDTD) method, an intense, vertical electric field (indicated as Ez field in the following paragraphs) was observed on the surface of a free-standing film, which is comparable to that of an Au film with similar parameters. After adding an InP substrate below the free-standing structure, we investigated the potential applications of the enhanced Ez field for designing a real device such as a QWIP.
This paper is organized as follows. Section 2 begins with the simulation of free-standing Ge subwavelength arrays using the 3D-FDTD method. Section 3 presents the obtained EM field response and the field distributions in the cross-section. Section 4 presents the parameter investigation and the basic understanding of the phenomena. Section 5 compares our simulations and the fabricated samples when an InP substrate is added for a practical application. Section 6 states the conclusion.
2. Simulation method and structures
We use the FullWAVE simulation tool (Rsoft product) to perform the 3D-FDTD analysis. For simplicity and generalization, free-standing Ge subwavelength array consisting of periodic Ge stripes is studied at a polarized normal incidence. We study the optical properties of the structure in an infrared range from 1.6 to 8 μm. In this range, Ge has a high real part of refractive index (n>4, much larger than usual materials) and a zero imaginary part (no absorption), which is beneficial for light confinement. After obtaining the characteristics and the mechanism of the free-standing subwavelength arrays, an InP substrate with a much larger thickness than that of the Ge film was added below the structure for studying the application potential. Details of the simulated structures are shown in Fig. 1. Periodic stripes are arranged along the x-axis with an infinite length along the y-axis. Therefore, in the simulation model, we build one period of the structure in the x domain, much longer length in the y domain, and about 4 μm in the z domain (large enough compared with the thickness), then we use the periodic boundary condition in the x and y domain to achieve the periodic symmetry of the structure, and the perfectly matched layers (PML) condition in the z domain for the removal of stray light. The normal incidence is x-polarized so that the electric field is perpendicular to the stripe direction. The field response can be recorded by a monitor set at the edge of the opening just beneath the outgoing surface, as indicated in Fig. 1. In our simulation, we apply frequency-dependent dielectric constants obtained from literature .
3. EM field response and field distribution
We investigate the EM fields of periodic Ge stripes. As a comparison, we further investigate the EM fields of a free-standing periodic Au stripes with the lattice constant a = 2.9 μm, the width d = 1.5 μm, and the thickness t = 0.24 μm. The monitors record enhanced Ez field responses with much larger intensity than the normal incidence in both structures. The normalized intensities, obtained by dividing the monitor values of the Ez field by the normal incidence, are shown in Fig. 2.
The Ez field generated at the interface of Ge and air is comparable to that of an Au subwavelength array with similar parameters. We can divide the curves into two parts, as indicated by the vertical dashed lines in Fig. 2. In part 1, both of the structures have a major peak (3350 cm−1 for Ge and 3400 cm−1 for Au) in the frequency range of 3000 – 4000 cm−1 whose intensity can be up to ten times the normal incidence. The lowered resonant peak at 3400 cm−1 in the periodic Au stripes is caused by the SP loss in the infrared range. In part 2, a group of small peaks occur at a higher frequency range beyond 4000 cm−1 in the case of Ge stripes (e.g. ~5150 cm−1 and ~6200 cm−1 peak), which can never be observed in Au structure. Furthermore, the ~5150 cm−1 peak in the Ge stripes can be as obvious as the Part 1 peak and its performance will be further improved if the parameters are adjusted properly. The characteristics of this Part 2 peak will be studied and exploited later in this paper.
To look further into the differences of these two kinds of peaks, the cross-section field distributions of one period of the array are recorded, providing the intuitive visions of the spatial patterns. The Ez field distribution of the Part 1 peak in periodic Ge stripes is presented in Fig. 3(a). The field pattern shows an evanescent feature with most of the intensity concentrated at the corners. The corner-preferred Ez field and the similar evanescent pattern can also be observed at the resonant peak in periodic Au stripes, which are caused by SPs , as Fig. 3(c) indicates. The major difference between the Ge stripes and the Au stripes is the Hy field distribution. As indicated in Fig. 3(b) and Fig. 3(d), the Hy field mainly concentrates inside the Ge stripe because of the zero absorption. In Au stripes, the Hy field can only stay on the surface.
For the Part 2 peak, the field distributions along the cross-section of periodic Ge stripes at the resonant frequency of 5150 cm−1 are presented in Fig. 4(a) and Fig. 4(b). In Fig. 4(a), the Ez at 5150 cm−1 shows similar evanescent features as the Part 1 peak, but the intensity maxima distribute along the surface of the stripe instead of concentrating only at the corners. This property is more suitable for a QWIP device because it helps to increase the effective area of the Ez field, an essential factor of detectivity . The surface-preferred distribution is reminiscent of the similar EM properties of a waveguide mode propagating inside an optical microfiber/nanofiber. A more obvious example can be seen in Fig. 4(b). There is a strong Hy field (indicated by 19.0657 on the color scale) forming a waveguide-mode-like pattern. This pattern has three rounds concentrating only inside the Ge stripe indicating a propagation length of 3π (in simulation, one round represents a phase shift of π). At the other Part 2 peak (6200 cm−1 in Fig. 2), a five-round Hy pattern is observed, as shown in Fig. 4(c). This type of Hy pattern is totally different from that of the Part 1 peak that has “tails” left outside the Ge stripe (Fig. 3(b)). In addition, there is a π /2 phase shift between the Hy and Ez fields due to the half-round dislocation observed in the patterns of Fig. 4(a) and Fig. 4(b).
4. Parameter investigation and mechanism interpretation
4.1. Parameter investigation
The position and the intensity of the peaks vary as the change of structure parameters. Three main parameters are investigated in periodic Ge stripes: the lattice constant a, the stripe width d, and the thickness t. During the investigation, a standard structure with a = 2.9 μm, d = 1.5 μm, and t = 0.24 μm was chosen and then one of the parameters was changed while the other two remained the same.
The performance of the peaks is strongly affected by the thickness t. If the thickness is small, there is little Ez field recorded. When the thickness is as large as 0.15 μm, the peaks start to be observable, as shown in Fig. 5(a). The Part 1 peak, indicated by the asterisk, stays nearly at the same position; however, the Part 2 peaks quickly grow and shift to lower frequencies as the thickness increases. When the thickness reaches 0.4 μm, one Part 2 peak moves to a frequency even lower than the Part 1 peak frequency, and the peak intensity becomes much stronger (see the 2800 cm−1 peak of the green line in Fig. 5(a)). Throughout the investigation of the thickness, the Part 2 peak performs better with larger thickness. This is another difference from the Au film, in which the generated Ez field by SPs is stronger with thicknesses less than 0.2 μm [18–20]. It should be noted that only the Part 2 peak shows the waveguide-mode-like Hy pattern, so we can easily tell whether a peak belongs to the Part 1 or the Part 2 peak by checking its Hy pattern.
When the width d changes, the Part 1 peak still does not move, as can be seen in Fig. 5(b). On the other hand, this peak increases with the increase of width, at first, and then it decreases as the continuous increase of the width d. Finally, the peak diminishes to a small amount until the stripe is wide enough to make a narrow air slit (d = 2.5 μm, a-d = 0.4 μm). On the contrary, the Part 2 peaks rise and shift to lower frequencies monotonically with new peaks coming into the frequency range one by one, indicating that more modes are appearing. The position of the Part 1 peak only shifts to a lower frequency when the lattice constant a increases, as indicated in Fig. 5(c). As the Part 1 peak moves to the lower frequency, it seems that the Part 2 peaks (see the red line and blue line in Fig. 5(c)) also feel this change and follow with smaller shifts.
4.2. The origins of the two kinds of peaks
In the case of periodic Au stripes, the position of the Part 1 peak is caused by SP excitations . According to the momentum-matching conditions, the position of the Part 1 peak in Au stripes can be given in a first approximation by:11, 21, 22]. The position of the Part 1 peaks in Ge stripes can be estimated according to the grating equations:23], which attributes the phenomena to conversions between continuous radiation states and a discrete localized in-plane state . As a result, they show similar evanescent features of the Ez pattern.
On the other hand, the origin of the Part 2 peaks in periodic Ge stripes was studied in detail. As indicated in Fig. 4(a) and Fig. 4(b), the field patterns of the Part 2 peak show the features of a waveguide mode. In order to verify whether it is a waveguide mode, a simple Ge waveguide model was introduced in the FDTD simulation. Figure 6(a) shows the schematic of this model. One Ge stripe is extracted and turned by 90 degrees as a short planar waveguide with two normal sources, one irradiated each end of the stripe. The length of the waveguide coincides with the stripe width d and thickness t. We also investigate the case of only one source irradiating at the bottom end. The Hy field response is recorded using a monitor at the lateral interface. Even with only one end of the stripe irradiated, many guided modes are generated inside the Ge stripe.
The details are studied through the field distribution along the cross-section. Only the Hy field patterns are shown in Fig. 6 because of their strong dependence on the Ge material thus providing a clear view of the guided mode. The three-round pattern is observed at the frequency of 5150 cm−1 (Fig. 6(b) and Fig. 6(c)), coinciding with the result of the periodic Ge stripes. The maximum of Hy at 5150 cm−1 (indicated on the color scale) is enlarged in the case of both ends irradiated. There is also a four-round pattern at a frequency of 5750 cm−1 (Fig. 6(d)), and a five-round pattern at 6200 cm−1 (not shown here). A similar enhancement of Hy for both ends irradiated is also observed at 6200 cm−1 as in the case of 5150 cm−1, indicating constructive interference. However, the magnitude of Hy at the frequency of 5750 cm−1 decreases (the appearance of the pattern also changes) for both ends irradiated, indicating destructive interference (Fig. 6(e)). Note that the field response of the periodic Ge stripes in Fig. 2 only shows two Part 2 peaks, 5150 cm−1 and 6200 cm−1, which have a three-round pattern and a five-round pattern, respectively. There is no peak for an even-round pattern, which is destructive interference. Therefore, the comparison of Hy patterns at different peaks indicates that the guided modes generated from both ends of a Ge stripe will experience constructive or destructive interference, but only the constructive interference contributes to the Part 2 peak.
4.3. Explanation based on diffraction and waveguide theory
A more intuitive way to find out the coupling mechanism can be done by simply simulating a single slit in a wide Ge film with a normal incidence and PML boundary conditions. The result is shown in Fig. 7(a). There is propagation of guided mode observed in the lateral direction inside the Ge film. According to all the above proofs, we can explain the generation of the Part 2 peak in this way, as shown in Fig. 7(b): when the incident light comes across the surface of a subwavelength array, it will experience strong diffraction, since λ > a-d (the air opening), and will create diffracted modes in different directions , as the red arrows in Fig. 7(b) indicate. Since Ge has a relatively high refractive index (~4.1), the large difference in refractive index between Ge and the nearby layer (here is air) makes it a good waveguide for confining light. Every Ge stripe behaves as a short planar waveguide in the x-direction providing efficient channels for most of the diffracted modes coupled into the in-plane evanescent modes. As a result, the normal incident wavevector is turned by 90 degrees and becomes the in-plane wavevector (the blue arrows in Fig. 7(b)), thus the diffracted light is guided inside the Ge stripe. At certain frequencies such as 5150 cm−1, the guided modes from the two ends of the stripe can have constructive interference and then form standing waves. Therefore, an enhanced Ez field is generated.
The position of the resonant peaks can be verified according to waveguide theory  whose main function can be expressed as:Equation (3) is a transcendental equation that can be solved by graphic method in MATLAB. For example, in the current model, we obtain for the position of 5150 cm−1 (equal to 0.515 μm −1 in the calculation). Then the wavevector of the guided mode in Ge can be estimated by:
5. Experimental verification of the application potential
In applications of semiconductor devices, a substrate is always necessary. For example, a QWIP has InGaAs/AlGaAs multi-quantum-well structure on a substrate of InP. However, when such a semiconductor layer is added beneath the Ge subwavelength arrays, the difference of the refractive index between Ge and the nearby layer (InP) becomes much smaller and the confinement of the Ge structure declines. As a result, the intensity of the Ez field at the Part 2 peak decreases drastically and many peaks disappear. Again, this situation can be explained by waveguide theory. The mode number in a waveguide can be estimated as:
A substrate with a high refractive index (nsub ~3.2) will cause many modes to leak from the Ge layer and thus the waveguide becomes easily cutoff. One way to alleviate this deterioration is to increase the thickness t as both Eq. (6) and the results in Section 4.1 indicate. For example, when the thickness was increased from 0.24 μm to 0.8 μm, a large Ez field was observed. Further increase of the thickness will certainly enlarge the Ez field, but such a large thickness will make the fabrication of narrow slits (a-d = 0.4 μm) difficult to achieve with the current techniques. As a result, we limit the thickness to less than 1 μm.
We adjust the parameter to generate an optimal resonance of Ez field coinciding with the 5-μm absorption peak of an intended device. One of our designs contains the parameters: the lattice constant a = 3.2 μm, the width d = 2.8 μm, and the thickness t = 0.8 μm. The simulation results are shown in Fig. 8(c). In the simulation, two monitors are placed at different positions along the Ge-InP interface to record the field response. As the upper part of Fig. 8(c) shows, two peaks can always be recorded, one of which (at 2000 cm−1) is our target of 5 μm. Since existing equipments cannot directly measure the Ez field, we use the zero-order transmission spectrum to demonstrate the positions of the peaks. As indicated in Fig. 8(c), a Fano-type profile of the simulated transmission occurs at frequencies where there are strong Ez peaks.
We fabricate arrays of 1D periodic Ge stripes on raw InP. First, a smooth Ge film was deposited by evaporation on a plain InP plate. Periodic stripes were then fabricated through the film by sputtering using a focused-ion-beam (FIB) system (40 keV Ga ions, resolution 5 nm). The quality of the fabricated structures was tested using a scanning electron microscope (SEM). A SEM image of the sample is shown in Fig. 8(a). The uniformity and the surface conditions are excellent, except for some inevitable slight slopes in the milled slits. It should be noted that the length of the stripe was designed to be 500 μm to approach the idealized condition of infinity in the y-axis, but it is difficult to keep the uniformity of the narrow slits on such a large scale. The sample was fabricated by composing 7 repeats of 70-μm-long stripes with a small gap of 2 μm between each repeat. Over 150 repeats were fabricated to ensure the periodicity in the simulation, so that the whole area for one sample is 500 × 500 μm. We fabricate three same structures at different positions of the Ge film to avoid the fluctuation or random error caused by surface defects and other reasons. Measurement results show similar performance of these samples indicating good fabrication uniformity.
The zero-order transmission spectra of the fabricated samples were recorded using a NICOLET 8700 FTIR spectrometer. It should be noted that the light path of the configuration has a focused angle of 5° which differs from the normal incidence condition in our simulation. However, the obtained spectra still show the main characteristics. Figure 8(b) shows the FTIR zero-order transmitted intensities of the fabricated periodic Ge stripes, the smooth Ge (no stripes) and the source. Two distinct absorption peaks (1970 and 2850 cm−1) are observed in the periodic Ge stripes, which are labeled by asterisks to indicate that they share similar positions as the simulation results (2000 and 2800 cm−1) in Fig. 8(c). The deviation can be caused by processing errors, surface roughness and the 5° angle in the measurement.
The instinct absorption peaks (~1600 cm−1 and ~3700 cm−1: water absorption; ~2350 cm−1: CO2 absorption) are also observed. The absorption of the designed peaks caused by the subwavelength array can be nearly half of the CO2 level indicating a strong localization of the EM field. A more obvious indication can be found in the transmission spectra in Fig. 8(d), which is obtained by dividing the source intensity. Though some fake peaks (~2300 cm−1 and ~3750 cm−1) are caused by the instinct absorption, two peaks with over 20% absorption are observed. The positions of these peaks also coincide with the simulation (the simulated transmission spectrum is copied to overlay with the real spectrum for an easy comparison).
Resonant peaks showing excellent near-field effects in the Ez and Hy field can be obtained through periodic Ge subwavelength arrays in the FDTD simulations. These peaks are located at different frequency positions from the Part 1 peak calculated by the grating equation. The peak intensities are comparable to that of the Au case, but there are no SPs participating in Ge material in the infrared range. Field distributions show that the evanescent Ez field mainly concentrates along the surface of the Ge structure, which is different from the Ez field for Au that focuses only at the corners of the opening. Analyzing the waveguide-mode-like Hy pattern, the phenomena can be interpreted by coupled diffraction and waveguide theory. The property of the structure is considered to be an alternative for making a QWIP.
According to literature [25, 26], gratings have already been used to improve the detectivity of QWIPs since 1990s. For example, a lamellar grating  can make the change of the wavevector through only diffraction, while the method of crossed grating and a waveguide  makes an advancement by combining diffraction and waveguide. The distinction of our work is that we not only exploit diffraction and waveguide, but also use a high refractive index material (nGe>nsub>nair) to improve the light confinement. In addition, the simultaneously generated constructive interference will also help to enhance the near field, thus amplify the vertical field drastically. Through parameter investigations, it is found that increasing the thickness t and the width d will help generate a stronger Ez field for the device. If possible, it is better to use a free-standing configuration to improve the performance , as we also indicate in our study.
Although the mechanism of the near-field effect in Ge subwavelength arrays can be easily understood by our interpretation, the in-plane periodic subwavelength structure is reminiscent of a photonic crystal. Further study should be arranged by introducing photonic bandstructure to interpret this phenomenon. It will help to better exploit the near-field effect in novel applications such as infrared sensing, near-field microscopy and other semiconductor devices.
We thank Shohei Hayashi for FIB fabrication of our samples, and Yoshitaka Kurosaka and Kazuyoshi Hirose for discussions. We are also grateful for some advice given by T. W. Ebbesen regarding FTIR measurements.
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