1. Introduction

Generally, light displays strong diffraction with aperture in subwavelength scale. Recently, H.J. Lezec et al. reported experimentally that light beaming from a subwavelength aperture with periodic corrugation on a metallic film [1]. They also pointed that this surprising diffraction phenomenon is caused by the diffraction of the excited surface plasmon (SP) wave on the corrugated metal surface. Later, this explanation is further developed by Yu et al. with rigorous coupling wave analysis from the reciprocal optical problem with respect to the abnormal grating spectral reflectance [2]. The calculated beaming direction agrees with the measured value in experiment. L. Martin-Moreno et al. presented a physical model of calculating the electromagnetic (EM) field behind the slit-grooves screen [3]. The model utilizes boundary conditions of perfect conductor to connect the excited EM field in grooves and slit with the radiated EM field in the free space. By using this method, they investigated the slit-grooves’ unique focusing property with elongated focal length [4]. Similar diffraction phenomena can also be observed numerically and experimentally in photonic crystal waveguides with corrugated output surface [5, 6]. As an example of application, corrugated metallic surface is utilized for the control of spontaneous emission [7]. Particularly, an investigation by J. B. Pendry et al shows the surface EM wave on nano-structrued metal surface displays similar property with SP [8]. This provides a convenient way for designers to obtain specific SP dispersion relation just by tuning the geometrical parameters.

In this paper, we investigate the excitation and diffraction of surface EM wave by a subwavelength slit and periodically positioned metallic grooves at the output surface. The investigation is based on the theoretical model in Ref. [3], in which some basic principles are presented. Here we provide a detailed and systematic analysis for the angular spectra of the slit-grooves structure for different geometrical parameters and wavelength. More importantly, we would try to explain the relation between the excitation and the resulted abnormal diffraction angular spectrum in the far field.

2. Theoretical formalism

The schematic of the metallic structure investigated in this paper is shown in Fig. 1. A thin metallic film is perforated with a subwavelength slit, and on the output side symmetrically patterned with finite grooves at both sides of the slit. The groove and slit width is a, the period is d, groove depth h and a total groove number 2N. The normal incident light from left is TM polarized with magnetic field H _y for the used coordinate depicted in the figure.

Although the theoretical formalism for the diffraction of this structure can be read in Ref. [3], we think a brief review and analysis of the theory is necessary and would help to understand the following numerical investigation and discussion. The key point is the EM field on the metallic surface. By assuming perfect conductor and fundamental mode approximation, the continuous EM boundary condition leading to the following set of linear equations

E _α is the parallel component of the electric field E _x at the αth groove openings and α=0 for slit. ε _α=cot(kh) represents the surface admittance at groove openings and ε ₀=-i. $G_{\alpha \beta }=\frac{\mathit{ik}}{2a}\iint {\varphi }_{\alpha }^{*}\left(x\right){\varphi }_{\beta }\left(x\text{'}\right)H_0^{\left(1\right)}\left(k\mid x-x\text{'}\mid \right)dxdx\text{'}$ can be viewed as the effective mutual admittance between groove α and β. k=2π/λ with λ the wavelength of light for the slit. $H_0^{\left(1\right)}$ is the zero order first class hankel function. ϕ _α stands for the tangent electric field at the metallic surface which equals E _α at indentation openings and zero otherwise.

 

Fig. 1. Schematic of slit-grooves structure for surface EM wave excitation and diffraction

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Then the field in the free space can be calculated as a summation of excited fields by all the grooves and the central slit

If a≪λ, G _αβ can be approximated as $G_{\alpha \beta }=\frac{ika}{2}{H}_0^{\left(1\right)}\left(k\mid \alpha -\beta \mid d\right)$ [9] and the angular distribution of magnetic field in the far field can be further simplified as

To clearly see the difference of light diffraction of a subwavelength slit with and without corrugations on the exit plane, the angular spectrum is defined as $I_N\left(\theta \right)=\frac{{\mid H_{yN}\left(\theta \right)\mid }^2}{{\mid H_{y0}\left(\theta \right)\mid }^2}$, which denotes the angular spectral gain of radiation from the viewpoint of antenna engineers.

The diffraction field of slit-grooves structures can be regarded as the coherently superimposing of light emitted from the 2N grooves and central slit with certain complex amplitude E _α describing the surface EM field along the metallic surface. From Eq.(1), the two factors G _αβ and ε _α play an important role in determining the excitation of surface EM field. G _αβ is mainly related to the ratio of period d and λ if a is very small. The surface admittance ε _α , on the other hand, represents the characteristic of the EM field inside grooves with certain shape and is only determined by h for rectangular groove profile.

3. Slit-groove diffraction’s dependence with geometrical parameters

In this section, we will show how wavelength, groove space, depth, width and number influence the behavior of slit-groove diffraction in the far field. Since the excitation equation is scale invariable, it is convenient to express the geometrical parameters in the unit of wavelength.

3.1 Groove space

Although the rigorous analysis of surface EM wave diffraction with subwavelength scaled structures usually requires complicated vector EM field theory, we discuss it here in a simple manner just from the Raleigh expansion. Considering a one-dimensional metal grating with square grooves, the periodic modulated field along the metal structure can be expressed as a superposing of a set of harmonic waves, $H_y(x,z)=\sum _{n=-\infty }^{\infty }{A}_{\alpha }\exp \left[-i\left({\beta }_{n}x+\sqrt{k_0^2-{\beta }_n^2}z\right)\right]$ and ${\beta }_n={\beta }_0+\frac{2n\pi }{d}$. As |βn|>k ₀, the harmonic wave displays evanescence property in the direction normal to the grating surface and the corresponding EM field is confined to the grating top surface. Otherwise, it represents a plane wave with a direction angle determined by sin θ=β _n /k ₀.

The ratio of d and λ determines if any plane wave can be radiated from the grating surface or the surface EM field can be excited by an incident plane wave. As λ/d>2, no scattering can be observed since no β _n is localized in the region [-k ₀,k ₀]. If 2>λ/d>1, only one β _n meeting the momentum conservation equation. But there even exists more than one β _n for λ/d<1, implying surface EM wave scattering or being coupled with plane wave at multiple directions.

 

Fig. 2. Diffraction angular spectra for variant wavelength ranging from 0.4d to 2.4d for a slit-grooves structure configured with a=0.1λ, h=λ/7, and N=10.

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To give a definite understanding of the unconventional slit-groove diffraction behavior, Fig. 2 presents a group of angular spectra for different illumination light wavelength. We can see that strong radiation can be observed at directions around the diffraction orders defined by k ₀sinθ _n =k ₀-2nπ/d. Note that the light beaming comes from both sides of grooves and they appear alternatively in the spectra. For some specific wavelengths, the directions of diffraction orders from both sides maybe coincide to each other, such as λ=d at 0 degree and λ=2d/3 at ±19.47 degree. These coinciding often result in an increased beaming intensity around the coinciding angles due to the constructive combination of both diffraction orders. The case of coinciding at 0 degree is very interesting and potentially applicable because of its single light beaming from a subwavelength aperture at the normal direction with small divergence and high gain of up to 30 (14.8dB) as depicted in Fig. 2.

3.2 Groove depth

In addition to the period of grooves, which plays a key role in determining the diffraction pattern, such as beaming angle, groove depth is also an important geometrical parameter. From the excitation equation, we can see that different excitation can be obtained by tuning h. Thus the correspondent angular spectra would give different presentation. In the following analysis, we will see that the complex amplitude of electric field at groove openings, both its magnitude and phase distribution, together with the complex amplitude at the central slit opening, determine the definite diffraction angular spectrum.

 

Fig. 3. Evolution of angular spectra for variant groove depth ranging from 0 to 0.5λ. Here (a) λ=1.6d and (b) λ=0.8d. The other parameters are a=0.1λ and N=10.

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To see this clearly, Fig. 3 provides a comprehensive angular spectra evolution picture for λ=1.6d and λ=0.8d with h ranging from 0 to 0.5λ. They possess 2 and 4 beaming directions respectively. But the most obvious aspect they both displayed is three types of diffraction, corresponding to the three regions depicted in Fig. 3. I. Obvious light beaming at certain direction with small divergence; II. Light concentration into regions confined by two adjacent beaming directions; III. Reversed beaming spectra (in contrast with the ‘positive’ beaming, here ‘reversed’ means that obvious dip, instead of a peak for normal diffraction, is observed in the spectrum, closely around the expected beaming direction). Usually, there is not a clear distinction between the above three types of spectrum. The diffraction behavior evolves slowly from one type to another. For h≪λ, it can be expected as shown in Fig. 3 that the diffraction spectrum exhibits nearly uniform radiation in all directions, just like that for a slit aperture with smoothed metal screen. As h increases, obvious but not strong peak appears in the spectra, closely at the beaming direction θ _-1=arcsin[1-λ/d]. For further deeper groove, the diffraction peak gets stronger and, at the same time, the beaming angle shifts slightly toward the zero degree direction. At about 0.2λ, the two parts seem to get even overlapped and united peaks appear in the regions between the two adjacent directions of diffraction orders, two +1 orders in Fig. 3(a) and +1 and +2 order in Fig. 3(b). Note that the two overlapped orders come from grooves at different sides of the central slit. But the combination does not prevent the damping tendency of the peak. In the following set of h, the united peak decays quickly and two diffraction dips around θ _-1=arcsin[1-λ/d] characterizes the angular spectra. Also the dips display decaying property for further deeper groove depth. Again, as h approaches λ/2, the dips disappear completely and the diffraction behavior returns to the case of a single slit.

To explain the above phenomena, it is necessary to analyze the excitation state of surface EM wave in detail. For the sake of simplicity, the detailed analysis is referred to Fig. 3(a). Fig. 4(a) and Fig. 4(b) present the complex amplitude of E _α for variant groove depth. Clearly, variant groove depth results in different excitation state. We are interested in two features of the excitation state. The first is the intensity of excited surface EM wave. Here we introduce a factor of surface EM wave excitation efficiency defined as $\eta =\frac{\sum _{\alpha \ne 0}{\mid E_{\alpha }\mid }^2}{\sum _{\alpha }{\mid E_{\alpha }\mid }^2}$, which indicates the portion of energy being excited to surface EM mode. As shown in Fig. 5(a), the excitation efficiency origins from zero at h=0, reaches maximum at some depth around 0.2λ and returns to zero when h=0.5λ. Another feature is related with the phase of the surface field. From Fig. 4(b), we can see that the phases of E _α (α≠0) displays an approximately linear distribution on both sides of the central slit. But the phase of E ₀ usually diverts from this linear distribution. Fig. 5(b) gives the calculated phase deviation of E ₀ versus the groove depth. If the phase of E ₀ follows the linear distribution principle, the slit displays positive contribution to the diffraction and two intensified diffraction peaks are expected to be seen in the angular spectra. Otherwise, this positive effect is weak and may be reversed if the deviation exceeds π.

For groove depth in region I, the excitation is weak but can be spread into far grooves in a slowly decaying manner. The phase deviation of E ₀ is small and the contribution from the central slit is positive. So the light can be diffracted from a large region of grooves and obvious beaming with small divergence can be observed. While the depth gets deeper and crosses about 0.15λ, it enters region II. Here the excitation of surface EM mode is stronger but is localized at the few grooves around the central slit. For far grooves, the excitation decays quickly. This implies narrowed radiation source and light is diffracted into wider angles. At the same time, the phase deviation of E ₀ increases greatly and this brings the shift of light to large diffraction orders. Since the orders from both sides appear alternatively in the angular coordinate, the radiation is mainly observed in the directions confined by two adjacent beaming directions. This can be well seen from both Fig. 3(a) and Fig. 3(b). For groove depth in region III (h>0.25λ), the excitation is weak and the phase deviation approaches and exceeds π, so the two obvious dips appear in the angles around the diffraction orders.

Another feature of the diffraction behavior we are interested in is its diffraction efficiency. Although it is hard to obtain a rigorous and explicit expression for calculation, the factor η can be regarded as a reasonable way to approximately evaluate the efficiency. From Fig. 5(a), the attainable diffraction efficiency is usually below 30% for λ=1.6d. In fact, this implies that only about 10% light is beaming at each diffraction angle for the positive diffraction type. But a great amount of light may be concentrated in the regions confined by two adjacent beaming angles for groove depth around 0.2λ. This feature may be useful for some application like beam shaping through surface corrugations.

 

Fig. 4. Excited electric field E _α (amplitude (a) and argument (b)) at groove openings for variant groove depth. The other parameters of calculation is the same as that of Fig. 3(a).

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Fig. 5. Groove depth dependence of (a) surface EM mode excitation efficiency and (b) the phase deviation of slit from the linear distribution of grooves’ phases for the structure defined in Fig. 3(a).

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3.3 Groove width

As known for square-groove structured gratings, the groove width usually does not play a key role in determining the grating’s optical property. We can easily see that this characteristic also holds true for the case of slit-groove diffraction. For α≠0, By utilizing the approximation form of G _αβ , the groove mode excitation equation (1) takes the form $E_{\alpha }=\frac{\sum _{\beta \ne \alpha }{H}_0^{\left(1\right)}\left(k\mid \alpha -\beta \mid d\right)E_{\beta }}{\frac{2}{ika}\cot \left(kh\right)-H_0^{\left(1\right)}\left(0\right)}$, so changing groove width would not deliver the critical change of the linear relations of all E _α . The obvious influence can be predicted from the term $\frac{2}{ika}\cot \left(kh\right)$ in the denominator that increasing groove width would deliver elongated range of groove depth for high surface mode resonance excitation. As illustrated in Fig. 6(a) and Fig. 6(b), grooves with small width display a much narrowed resonance range of h, while great beaming phenomena can be observed in an elongated range of h if a is increased (Fig. 6(c) and Fig. 6(d)). Noting that a is too large in Fig. 6d, the approximation of G _αβ is not valid and the original definition is used for this case. Another feature associated with increased grooved width is considerably decrease of maximum beaming intensity, especially for large a.

 

Fig. 6. Contour plots of functions of angular spectra with groove depth for variant groove width, (a) a=0.01λ, (b) a=0.06λ, (c) a=0.12λ, (d) a=0.3λ. Here λ=1.6d and N=10.

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3.4 Groove number

So much for the investigation of the diffraction pattern evolution with different groove depth, we will discuss the angular spectra’s dependence on the number of grooves positioned at both sides of central slit. Conventionally, increasing the number of grating periods implies increased diffraction area thus much narrowed and stronger diffraction peak can be observed in the angular spectrum. For the slit-groove diffraction, however, this is not always the case. As stated in the previous section, the diffraction behavior is mainly governed by the excitation state of surface wave along grooves. The change of diffraction spectra for different groove number also reflects the property of surface wave excitation.

To see this clearly, Fig. 7 presents the evolution of angular spectra with increasing N for three representatives of groove depth in correspondence with the three types of diffraction in Fig. 3(a). The great change of angular spectra profiles can be seen and well understood for small groove numbers. But for groove numbers already large enough (N>5), increasing N would deliver different results for the three groove depths. For the depth in positive diffractive region as plotted in Fig. 7(a) and Fig. 7(b), the beaming peak value continues to grow by adding grooves. This phenomenon results from that light can be spread into grooves far away from the central slit. So very strong beaming can be obtained in this way, reaching about 14 for N=200 in this case. Further numerical investigation (Fig. 8(a)) shows the dependence of maximum beaming intensity with groove number. The correspondent beaming divergence (full width at half maximum, FWHM) also displays a steady decrease with increased N (Fig. 8(b)). In addition, the angular peak shifts slightly toward large angles with increasing N due to the intensified grating-like diffraction.

For the other two representatives in Fig. 7(c) and Fig. 7(d), however, the spectra keep nearly unchanged as N exceeds 5. This also indicates that the excited light is mainly localized in the few grooves close to the slit. Generally, no great change of angular spectrum can be expected by just increasing the groove number while other parameters are fixed. The angular spectra display similar characteristic, that is to say, beaming at certain direction in a ‘positive’ or ‘negative’ manner is mainly controlled by the groove depth.

 

Fig. 7. Angular spectra of light beaming for variant groove numbers. (a) and (b) h=0.1λ, (c) h=0.2λ, (d) h=0.3λ. The other parameters are λ=1.6d, a=0.1d.

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Fig. 8. Beaming intensity (a) and divergence (FWHM) (b) for groove number ranging from 0 to 200 with a step of 20. The other parameters here are λ=1.6d, a=0.1d and h=0.125λ.

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4. Conclusion

In summary, the dependence of diffraction behavior in the far field with slit-groove structure parameters is investigated numerically. It is found that the illustrating wavelength, groove space and groove depth mainly govern the light diffraction in the far field. The first two parameters approximately determine the beaming direction or the peak position observed in the angle spectra. The groove depth, if the other parameters are fixed, displays a dominating way to influence surface electromagnetic wave excitation and its diffraction behavior. Furthermore, we found that the angular spectra exhibits positive and negative diffraction features for variant groove depth, which can be explained from the complex electric field distribution associated with certain surface mode excitation state. These results, we believe, would help design appropriate slit-groove structures for potential applications such as multiple wavelength dividing and surface beam shaper for subwavelength light source.