1. Introduction

Recently there has been a renewed interest in one-dimensional (1-D) transmission gratings due to extraordinary optical transmission and potential device applications [1–12]. These structures have been investigated for many years and anomalies in their optical response have been identified and described [13–15]. In these past works, the individual optical modes, both propagating and surface-confined, have been identified and efforts to describe their effects on reflectance, transmission and redistribution of energy into diffracted components have been attempted using phenomenological approaches [13–15]. More recently, it has been suggested that surface plasmons and cavity modes play significant role in enhanced transmission, but there has been considerable disagreement in their relative roles and the physics behind this phenomenon remains in dispute [1–11]. In this paper, we reexamine classical 1-D transmission gratings and three new structures to explain the physics underlying the enhanced transmission and the use of these structures for possible device applications. In the authors’ past works on electromagnetic resonance enhanced devices, it had been concluded that horizontally oriented surface plasmons (HSP) are undesirable for certain optoelectronic device applications, such as silicon-based metal-semiconductor-metal photodetectors (MSM-PD), for several reasons [10, 11]. It has been observed in these works that an excitation of a HSP, whether on the air/contact interface or Si/contact interface usually, but not always, leads to a reflectance maxima, transmittance minima, increased absorption within the lossy metal contacts, and undesirable electromagnetic field profiles and charge-carrier generation profiles. Because of these undesirable characteristics of HSPs in these structures, techniques were developed to eliminate HSPs and use other optical modes to perform the one useful thing that HSPs do perform in these device structures, namely to localize the electromagnetic field near the contact/Si interface. In these works, the authors were always careful to qualify their observations to structures with dissimilar top and substrate materials, i.e., air and silicon respectively. However, the important issue of whether HSPs in any lamellar grating profile, with dissimilar or identical top and substrate materials, can ever produce significantly enhanced transmission was never directly and thoroughly addressed in these works [10, 11]. This issue is important because of the commonly held assumption in the surface plasmon research community that HSPs, or coupled HSPs can lead to enhanced transmission and because optoelectronic device engineers have occasionally been using this assumption as a starting point in time consuming and expensive development of surface plasmon enhanced optoelectronic devices [1,2]

There are several works that address the issue of which electromagnetic modes are responsible for peaks in transmittance [3–15]. Examples of these modes include HSPs, Wood-Rayleigh (WR) anomalies (i.e., the onset of a diffracted mode) [13], diffracted modes and cavity modes (CM), sometimes called waveguide modes, vertical surface resonances or vertical surface plasmons (Fig. 1). Two works in particular specifically address this issue and come to different conclusions [3,4]. On one side of the issue is Porto et al., who proposed two separate mechanisms for enhanced transmission through slit arrays: 1) Coupled HSPs on top and bottom interface 2) Cavity modes located in the slits. Taking a somewhat opposing view is Cao and Lallane who agree with Porto et al., in that CMs produce enhanced transmission but conversely state that excitation of HSPs invariably leads to a minima in transmittance. Both of their works use analytical models that make a few simplifying assumptions but otherwise appear sound, but their two sets of competing conclusions cannot both be simultaneously and entirely correct.

 

Fig. 1. Three electromagnetic resonance modes produced by a normal incident TM polarized plane wave for a Si lamellar grating structure with 0.96μm thick, 1.14μm wide Au contacts with a period of 1.75μm. Top: A 0.21eV contact/Si HSP, Middle: A 0.28eV CM, Bottom: A 0.69eV air/contact HSP.

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The first part of this paper addresses this issue of whether HSPs enhance transmission, as promulgated by Porto et al., or inhibit transmission, as promulgated by Cao and Lallane. It is concluded in this section that HSPs can both strongly inhibit and weakly enhance transmission. However, the predominant effect is a strong inhibition of transmission and interference with the transmission-enhancing properties of CMs, WRs and diffracted modes. The second part builds upon these conclusions and discusses methods to eliminate HSP modes, which may be undesirable for certain optoelectronic devices. Finally, device applications including chemical and biological sensors and InGaAs MSM-PDs are discussed.

2. Re-examination of a structure with hypothetical HSP-enabled enhanced transmission

To address the question of whether HSPs enhance or inhibit transmission, the structure that was analyzed by Porto et al., and lead them to conclude HSPs enhance transmission is reexamined [3]. Their work specifically addressed the issue of HSP-enabled enhanced transmission and reached a conclusion that is different than Cao and Lallane as well as all of the authors’ past observations [10, 11]. Therefore, no work on this issue would be complete without an analysis and reexamination of their results and conclusions. The structure that was studied is a typical lamellar grating structure as shown in Fig. 1 (but with air as the substrate) with Au contacts [16], air for the top, bottom and groove materials, a contact window opening of width c_II=0.5μm, a pitch of d=3.5μm, and several contact thicknesses in the range of h=0.6→4μm. The electromagnetic (EM) fields are modeled using the surface impedance boundary condition (SIBC) technique [8]. The EM fields in the top and bottom layer are expressed in pseudo-Fourier expansion according to Floquet’s theorem [8, 10, 11]. The EM fields inside the slit cavity are expressed as a linear combination of orthonormal modes [8, 10, 11]. The reader is referred to ref. [8,10,11] for a detailed description of the EM calculations. In their work, Porto et. al., looked at a structure with h=0.6μm and first observed a narrow transmission peak at 0.354eV for a normal incident TM polarized input beam. Second, they looked at the electromagnetic field distribution of this mode and observed a distribution that is largely consistent with an HSP mode. Third, they studied the dispersion of this mode and observed that it has a dispersion that is similar to the dispersion of an HSP. Because of these three characteristics that are all consistent with the behavior of an HSP, they concluded that the mode is a pure HSP (actually coupled HSP modes) and therefore concluded that HSPs can produce enhanced transmission. They then looked at a structure with a deeper groove, i.e., h=3μm and observed that the transmission peak is broad, has a high electromagnetic field within the grooves, and has a very small dispersion. These three characteristics are consistent with a CM. They then made the conclusion that the mode is a pure CM and therefore that CMs can also produce enhanced transmission. They therefore concluded that both a CM mode and coupled HSPs can enhance transmission. However, in their work, they do not clearly explain how the CM mode with an energy of 0.17eV for h=3μm evolves into a HSP mode with an energy of 0.354eV for the same structure except for a smaller contact thickness of h=0.6μm. This mode must be a CM/HSP hybrid mode with CM and HSP characteristics for heights between these two contact heights. But then the question arises as to the contributions to the enhanced transmission from the CM and HSP components of the hybrid mode and whether the 0.354eV mode with small-enhanced transmission is a pure HSP mode or if its small CM component is responsible for the enhanced transmission. This issue is analyzed in this work in the animations in Figs. 2 and 3.

Figure 2 is an animation that shows how the primarily CM mode at 0.302eV (excited by a normal incident TM polarized plane wave) in the lamellar grating structure with h=1.25μm, evolves into a primarily HSP mode at 0.354eV when the contact thickness is reduced to h=0.6μm along with the resulting affect on the enhanced transmission. Figure 2 shows that the transmittance is a high value of over 80% when the hybrid mode is primarily a CM mode for higher contact heights but as the mode approaches the energy of the narrow HSP mode slightly beyond 0.354eV, the hybrid mode assumes more of a HSP component (as seen from the electromagnetic field profile), the transmittance decreases, and the wavelength range of the peak decreases. Porto et al., erroneously assumed that the CM/HSP hybrid mode at 0.354eV is a pure HSP mode and that the small transmission peak is therefore caused by a coupled HSP modes (i.e., HSP modes on the top and bottom interfaces that are coupled via a CM or fringing fields within the groove). What is now clear from Fig. 2 is that the enhanced transmission at 0.354eV for the grating structure with h=0.6μm was simply the result of the small CM component of the CM/HSP hybrid mode at that energy. As the contact thickness is reduced even further, the hybrid mode acquires even more HSP characteristics, the CM characteristics are further decreased, and the transmittance rapidly approaches zero.

 

Fig. 2. Left: The reflectance and transmittance of the lamellar grating profile with a pitch of 3.5μm and Au contacts that are 3μm wide and decreasing thickness starting at 1.25μm and ending at 0.1μm. Right: The magnetic field intensity profile for the peak in transmission (identified by the green cross in the right-hand graph). [Media 1]

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A second issue that needs to be addressed is the dispersion of the hybrid CM/HSP mode. Porto et al. concluded that because the dispersion was similar to the dispersion of a HSP mode, this was further evidence that the enhanced transmission was produced by HSP modes. However, this is not the correct interpretation of this observed behavior. By observing zero order transmittance as a function of the energy and parallel wave vector component (k_x), as shown in the animation in Fig. 3, typical energy band behavior can be studied including the observation that a CM band will not cross a HSP band and as the two such bands approach each other, they will interact by hybridizing and altering their dispersions in typical ways. Concerning the dispersion, the CM band can pushed up or down in the E-k_x diagram by a neighboring HSP band resulting in a dispersion of the CM/HSP hybrid band that may resemble the dispersion of a pure HSP band. However, the enhanced transmission associated with the CM/HSP hybrid band in this situation is entirely due to the CM-aspects of the mode.

 

Fig. 3. The zero-order transmittance as a function of the energy and in-plane component of the wavevector (k_x) of the incident beam. The behavior of the CMs, HSPs and hybrid CM/HSP modes and their effects on enhanced transmission are clearly illustrated. Media 2]

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Therefore, the results of this analysis primarily support Cao and Lallane and the authors’ past conclusions that HSPs generally inhibit transmission and that any significant transmission peak is produced by CMs. However, it will be shown in the next section that some HSPs in certain structures can slightly enhance transmission, but in these situations the enhancement is small. Also, HSPs are also undesirable for certain optoelectronic device applications for several reasons including the narrowness of this electromagnetic resonance mode, their increased absorption of the incident beam within the lossy metal contacts, their large angle of incidence and wavelength dependencies, and other aspects of their electromagnetic field profiles. Therefore, methods to eliminate and control HSPs will now be described.

3. Minimizing HSPs in periodic grating structures

There are many optoelectronic devices that use a periodic arrangement of metallic contacts similar to a lamellar grating. [1,2, 10, 11, 17, 18, 19] In these structures, different optical modes, including WRs, CMs, diffracted modes and Fabry-Perot cavity modes, can be used to perform different functions. In lamellar grating-like structures, HSP modes usually have energies and in-plane momentum that are very close to desirable WRs and the onset of diffraction modes; these two modes occur in pairs, i.e., HSP/WR pairs. Because of this fact, a WR is often confused with a HSP but the two modes have important differences:

These differences can be used to selectively eliminate or minimize the role of HSPs while leaving diffracted modes, WRs, and CMs unchanged. In particular, Difference #2 listed above can be used to selectively “turn off” certain HSP modes while leaving WR modes relatively unaffected. Figure 4 shows four structures that are examined in this paper and include the classical lamellar grating (denoted by S1), a grating structure with small notches composed of a dielectric material at the bottom of the contacts (denoted as S2), a grating structure with small notches composed of a dielectric material at the top of the contacts (denoted as S3), a grating structure with small notches composed of a dielectric material at both the top and bottom of the contacts (denoted as S4). The small notches at the surfaces strongly inhibit HSPs along the interface while leaving the WRs relatively unaffected, thereby allowing the contribution to, or inhibition of, transmission by HSPs to be analyzed. More specifically, comparing the optical characteristics of S1 and S2 allows the effects of the contact/substrate HSPs on the transmission to be determined. Likewise, comparing the optical characteristics of S1 and S3 allows the effects of the air/contact HSPs on the transmission to be determined.

The structure that will be studied is the one analyzed in [6] where the authors numerically modeled a lamellar grating (S1) and tried to extrapolate the results to describe the experimental optical characteristics of a structure similar to S2. The S1 structure they studied had h_II=0.96μm thick, 1.14μm wide Au contacts with a period of d=1.75μm (therefore c_II=0.6μm). The structure S2 studied in this work has the same dimensions but with SiO₂ in the c_III=0.74μm wide, h_III=0.2μm thick notches at the bottoms of the contacts. The SiO₂ is assumed to have a frequency independent dielectric constant ε=2. The EM fields in the top and bottom layer are expressed in pseudo-Fourier expansion according to Floquet’s theorem [8, 10, 11]. The EM fields inside the slit cavity and the notches are expressed as a linear combination of orthonormal modes [8, 10, 11]. By introducing the notches in the contacts, additional CM modes associated with the notches can be produced that have interesting and useful optical effects, an example of this is given in the next section in which a 1.33μm to 1.5μm InGaAs MSM detector is described that can channel 93% of the normal incident TM polarized beam into the semiconductor. However these additional CMs are purposely avoided in the structures S2 and S3 in order to clearly identify the optical effects of HSPs on enhanced transmission.

 

Fig. 4. The four grating structures that will be analyzed in this work. S1 is the classical lamellar grating profile that has HSPs on the air/contact and substrate/contact interfaces. S2 will largely eliminate the HSPs on the substrate/contact interface, S3 will largely eliminate the HSPs on the air/contact interface, and S4 will eliminate both sets of HSPs.

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Figure 5 shows the transmittance for a normal incident, TM polarized input plane wave, incident on the classical grating structure (S1) and the structures S2 and S3. The results are in good agreement with those of [6]; the transmittance of S1 agrees with their theoretical results and the transmittance of S2 agrees with their experimentally measured results. The main difference between S1 and S2 is the removal of bottom metal interface between the contacts and Si and the HSP modes associated with this interface are eliminated. Figure 5(a) shows that the HSP-produced transmittance minima at energies 0.21eV and 0.42eV in the S2 structure are eliminated. The top air/contact HSP at 0.69eV is unaffected by the small notches in S2. One very interesting result is that the transmittance peak at 0.62eV that is present in S1 is absent in S2. This strongly suggests that the contact/Si HSP mode at 0.62eV produced a weakly enhanced transmission. This contradicts the hypothesis of Cao’s and Lallane’s that states that any excitation of an HSP leads to a strong inhibition of transmission. Likewise, the difference between S1 and S3 is the removal of the top metal interface between the contact and air and the HSP modes associated with this interface are eliminated. Figure 5(b) shows that the HSP-produced transmittance maximum at 0.62eV is eliminated. The bottom Si/contact HSPs are unaffected by the notches in S3. It is therefore seen that neither Porto et. al., or Cao and Lallane are entirely correct because HSPs can enhance and inhibit transmission. This conclusion produces many questions including how HSPs are affecting the transmission in these two opposite ways.

A short note is in order at this point concerning the phenomenological approaches described in refs. [14,15]. These results may initially be viewed as agreeing with refs. [14,15] that concluded that Wood anomalies can produce either a maximum or a minimum. In these references a phenomenological approach was developed that has worked well for reflection gratings and dielectric gratings but does not accurately predict enhanced transmission and surface plasmon effects in more complicated transmission grating profiles as studied in this work as well as not studying the various modes individually and at a fundamental physical level. In the phenomenological approach, the resonant modes, including HSPs, CMs, Fabry-Perot resonant modes are described by a Lorentzian behavior 1/λ-λ^p, with λ^p being a pole corresponding to the resonance phenomenon [14,15]. Also, the optical behavior of the propagating orders, i.e., transmitted, reflected and diffracted orders, is proportional to λ-λ^z/λ-λ^p with the pole λ^p and a zero λ^z so that the amplitude, both the pole and the zero being complex [14,15]. The problem with this approach is that it treats a HSP/WR pair as a single resonance in the optical behavior and states that no general rule about maximum and minimum transmission can be obtained. This is true because the enhanced transmission will be affected by different amounts by the overlapping HSP and WR modes. For simple optical gratings applications, the effects on enhanced transmission of the HSP and WR components separately of the HSP/WR pair may not be necessary because what is usually of interest is the resulting far-field profile. In more recent results including this work, the individual effects on the optical behavior of the HSP and WR of a HSP/WR pair are analyzed. This aspect is of increasing interest because, as has already been described and will be described further in the Device Applications sections, HSPs and WRs have different properties that make selective use of one while eliminating the other desirable.

 

Fig. 5. Left: The zero-order transmittance of the classical grating structure (S1) and the modified structure (S2). It is seen that the HSPs associated with the substrate/contact interface are largely absent in structure S2. Right: The zero-order transmittance of the classical grating structure (S1) and the modified structure (S3). It is seen that the HSPs associated with the air/contact interface are absent in structure S3.

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Fig. 6. Left: The Poynting vector showing that the 0.21eV Si/contact HSP causes the energy flow to form a vortex that is in a direction that inhibits energy flow through the groove. Right: The Poynting vector showing that the 0.62eV Si/contact HSP causes the energy flow to form a vortex that is in a direction that enhances energy flow through the groove.

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Further work can be done to develop device design guidelines to produce HSPs with the proper direction of the vortices of energy to inhibit or enhance transmission depending on what is desired but this is out of the scope of this current work. Below, the main theoretical results of this work are summarized:

4. Device applications

There are many optoelectronic devices that can be developed or improved using the optical effects described in this paper. These applications include the practical development of high bandwidth and high responsivity metal-semiconductor-metal photodetectors (MSM-PD), grating couplers in GaAs quantum well infrared photodetectors (QWIPs), and chemical and biological sensors to name a few. Two examples of these applications will now be described.

Surface plasmons have been used in the past to fabricate biological and chemical sensors. Usually the SP sensor is a flat metal film in a Kretschmann configuration [19], where energy of the SP resonance is measured as the top of the metal film is exposed to the fluid that is to be analyzed for the presence of the chemical. If the chemical absorbs on the surface of the metal film, it will change the local index of refraction and change the energy of the SP resonance. Another configuration for this type of device can use the structure S1 with a high aspect ratio of 10 for the grooves (i.e., 500nm deep and 50nm wide) in aluminum. With this high aspect ratio, the CMs produce narrow and dramatic resonant dips in the reflectance. Figure 7 shows the reflectance for a TM polarized normal incidence beam and shows the dramatic reflection dip caused by the CM mode. For chemical and biological sensors, the index of refraction can be expected to be in the range of 1 to 1.25. The inset in Fig. 7 shows the how the wavelength of the resonance changes as the index of refraction in the grooves is increased from 1 to 1.25. It is seen that there is almost a 200 nm shift in the sharp reflectance minimum thereby allowing for high-resolution detectors.

 

Fig. 7. Reflectance of the grating structure for normal incidence and an index of refraction of 1.14. The CM mode at 1.05μm produces a sharp minimum in the reflectance. The inset shows the wavelength of the reflectance minimum as the index of refraction of the material in the groove changes from 1 to 1.25

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The structure S1 can also be used for the development of high bandwidth and high responsivity MSM-PDs as outlined in [10, 11]. The structures S2, S3 and S4 can also be used for this application and have several advantages. As mentioned earlier S2, S4 will avoid the undesirable HSPs at the metal/semiconductor interface that have been shown to produce large absorption in the metal wires, peaks in the reflectance and high field intensities in the semiconductor directly under the contact (where the static field produced by an applied bias is small) [10, 11]. The techniques to eliminate HSPs and controlling the distribution of light into various propagating transmitted modes should be used in different ways depending if the semiconductor is a direct bandgap material with a large absorption constant such as In_0.53Ga_0.43As for the important telecommunication wavelengths of 1.33μm to 1.5μm or if it is an indirect bandgap semiconductor with a small absorption constant such as Si in devices for a variety of applications in the visible and near IR spectral range (imaging, 850nm wireless or very short range (VSR) fiber-based communication systems, single photon counting devices…). For Si, a high zero-order transmitted wave is undesirable because this component would propagate 10s of microns into the Si and would greatly decrease the bandwidth of the device. Hence for Si, the aim is to have the excited CM transfer all the energy into higher diffracted orders that travel as parallel to the semiconductor/metal interface as possible, as described in [10, 11]. Also, since structures S2, S4 eliminate the dips associated with HSPs, the CM mode around a particular energy location is broadened. A broad and enhanced CM mode is very useful from a design point of view for an MSM-PD as the light shining from a fiber has wavelengths in and around the energy locations for which the device is designed.

As for the direct bandgap semiconductor InGaAs and MSM-PD made of this material, the responsivity is dependent on the total amount of light transmitted into the substrate because the high absorbance causes all of the light to be absorbed within a few microns of the substrate and is fairly independent of how this total energy is distributed among the zero-order and higher order propagating modes. Because of this high absorbance, the bandwidth is less dependent on the distribution of energy among the zero-order and higher order propagating modes in the substrate and more dependent on contact spacing, as compared to the Si MSM-PD. Keeping this in mind, let us study optical response of an InGaAs MSM-PD with a S4 contact design. The top and bottom sub-grooves of S4 serve different purposes and are independent of each other. The excitation of resonance in the top sub-groove helps in collecting light from the air region and channeling it through the main cavity (i.e., the groove) into the substrate. This is a new type of resonance associated with top sub-groove, which would otherwise not exist for a classical grating (S1), and for this particular structure occurs at an energy location slightly less than first WR mode for the air/metal interface (these two modes are independent of each other). Also, the top sub-groove will eliminate the top air/contact HSP. On the other hand the bottom sub-groove eliminates the bottom contact/substrate HSPs that may be surrounding the energy value of interest depending on the structural dimensions. Figure 8 shows the reflection and total intensity of light transmitted into the substrate region for the InGaAs MSM-PD with S1 and S4 contact designs. This structure is composed of Au contacts, with c_II=0.4μm c_III=0.53μm, h_II=0.76, d=1.33μm, height of both sub-grooves is 0.2μm and InGaAs as the substrate and the material in bottom sub-groove and air in the top sub-groove. It can be clearly seen that a broad peak in the total transmission was achieved with a new resonance associated with top sub-groove and also the top HSP located at 0.93eV in S1 was eliminated. In the energy location studied the bottom HSP does not exist for the given structural dimensions, nevertheless a bottom sub-grove is helpful in avoiding it if it exists for a different structural geometry.

Figure 9 shows the magnetic field intensity and Poynting vector for the InGaAs MSM-PD 0.84eV (1.47μm) for the dimensions mentioned above. There is strong localization of light right under the window opening and very little field under the contacts and in the substrate region, which is ideal for the enhancing the performance of the MSM-PD. The Poynting vector shows the excitation of top sub-groove resonance and how it concentrates the energy of the incident beam within the sub-grooves and the subsequent channeling of the energy through the main groove and into the substrate. An efficiency of 93% is obtained for the percentage of the total energy of the incident beam that is transmitted into the substrate. This is a large percentage considering the fact that approximately 70% of the surface of the InGaAs is covered with thick metallic contacts.

 

Fig. 8. Right: The reflectance and total energy transmitted for the S1 structure showing a HSP/WR resonant pair at 0.93eV. Left: The HSP at 0.93eV is eliminated and subgroove CMs are produced that channel light through the main groove producing a high value of 93% for the total energy transmitted. The transmission peak at 0.84ev is large and broad making this device desirable for use as a MSM-PD.

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Fig. 9. Top: The magnetic field profile for a TM polarized, 0.84eV normal incidence plane wave. The geometry of the subgrooves create CM modes that concentrates the energy of the incident beam within the subgrooves. Bottom: The Poynting vector shows that energy is being channeled from the subgroove CMs to and through the main groove.

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5. Conclusion

An analysis of the role of optical modes in the transmission of energy through 1D periodic grating structures was performed. A reexamination of a situation where HSPs were hypothesized to enhance transmission was performed and concluded that the strong enhanced transmission was caused by the CM component of a HSP/CM hybrid optical mode. Properties of HSP/CM modes were described. It was then shown that HSPs can both strongly inhibit and weakly enhance transmission depending on the direction of energy flow that is established within the structure. Methods to minimize HSPs were described and control light was described. A new type of resonance effect is described for enhanced transmission by the use of a simple structure, which consists of a notch on the top metal surface. Applications of these structures and optical modes were briefly described including chemical and biological sensors and MSM-PDs.