1. Introduction

When a periodic structure is illuminated by a monochromatic light, an image of the structure will appear at some positions behind it. This was discovered by Talbot in 1836 [1] and thus, called the Talbot effect or self-imaging effect. The recurring length is called Talbot distance Z _t, which can be expressed as 2d ²/λ under the paraxial approximation condition, where d is the period and λ is the wavelength of the light. The Talbot effect for matter waves has also been studied extensively [2, 3]. The Talbot effect is an attribute of all kinds of waves. It is natural that it must occur in surface plasmon polaritons (SPPs), which are electromagnetic waves produced by the strong interaction between the electromagnetic field and free electron oscillations at a metal-dielectric interface [4]. Some researchers have theoretically studied the plasmonic Talbot effect [5–7]. The theoretical study by Dennis et al. [5] was based on the system consisting of a row of holes drilled in a metal film. When the film is illuminated from the back side, the SPPs are excited from the holes propagating and deploying a Talbot carpet. They numerically computed the plasmon Talbot carpets for different lattice spacings. Niconoff et al. [6] used the representation for optical modes in homogeneous media to obtain a more general mode solution and to describe the self-imaging phenomenon for the SPPs. On the basis of an impedance boundary condition approach, Maradudin and Leskova [7] theoretically proved that when the SPPs are scattered by periodic metal or dielectric dots, they will display self-images of those dots.

As the plasmonic Talbot effect has wide potential applications, it is important that the effect be properly studied in experiments. In this paper, we study the plasmonic Talbot effect using the SPP launching gratings (SPPLGs), which consist of periodic grooves drilled on a gold film. When the laser beam is illuminated on the SPPLG, the SPPs will be excited at the grooves and propagate to form a Talbot carpet. The 3λ _SPP and 6λ _SPP are selected as the periods of the SPPLGs. The paraxial approximation, which is although widely used in analyzing the Talbot effect, is no longer applicable in this case. The SPP intensity distributions are calculated numerically and the results agree well with the experiments. In order to obtain strong amplitude-modulated revivals, multi-layer and multi-level-phase SPPLGs are designed. Intensive multiple focal spots with subwavelength transverse full widths at half maximum (FWHMs) are implemented by using a three-layer four-level-phase SPPLG.

2. Experimental demonstration of the plasmonic Talbot effect

The schematic of a SPPLG is shown in Fig. 1 (a) which is composed of periodical grooves located on a gold film with a length a, a width w, and a period d. The “opening” ratio α is defined as a: d. When a y-polarized laser beam normally illuminates the SPPLG, the SPPs are launched at the grooves deploying a Talbot carpet. The width of every groove w = λ _SPP/2 in order to obtain a high coupling efficiency [8]. In our experiment, the SPPLGs are fabricated using a focused ion-beam milling system on a 50 nm thin gold film thermally evaporated onto a glass substrate. The exciting wavelength is 830 nm, which corresponds to the SPP wavelength of 814 nm. Figure 1 (b) shows a part of the scanning electron micrograph of the SPPLG with d = 6λ _SPP and α = 1/2. The Talbot carpets are detected by a leakage radiation microscopy (LRM) system [9, 10], which is described in detail in Ref [11].

 

Fig. 1 (a) Schematic of the SPPLG. (b) Scanning electron micrograph of the SPPLG with period d = 6λ _SPP and α = 1/2 (only 2.5 periods are shown).

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Two SPPLGs are fabricated with the periods of 3λ _SPP and 6λ _SPP and the numbers of periods 20 and 16, respectively. In addition, the opening ratio was α = 1/2. The experimental results are shown in Figs. 2 (a) and 2 (b), respectively. Only the Talbot carpets at the right sides of the SPPLGs are shown because the systems are symmetrical with respect to the SPPLGs. The light ribbons on the left of the images around the positions of the SPPLGs come from the direct contribution of the laser beam to the LRM image through the gratings. The similarity the main parts of the wave vector with the SPP waves accounts for this and it cannot be filtered out by the spatial filter of LRM [10]. The obvious transversal and longitudinal periodical intensity distributions are observed in Figs. 2 (a) and 2 (b).

 

Fig. 2 Experimental Talbot carpets for the SPPLGs with d = 3λ _SPP (a) and 6λ _SPP (b), respectively. (c) Experimental and theoretical transversal intensity distributions at Z _t /2 for the SPPLG with d = 6λ _SPP. (d) Theoretical Talbot carpet for the SPPLG with d = 6λ _SPP; white dashed lines from left to right indicate the positions of Z _t /2, d ²/λ, Z _t and 2d ²/λ, respectively.

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Since the periods of the SPPLGs are compatible with the wavelength, the paraxial approximation cannot be applied to determine the Talbot distance. The repeating length Z _n of the ±n order beams of the diffracted SPPs is

For the SPPLG with d = 3λ _SPP and α = 1/2, only the 0 and ±1 order beams of the diffracted SPPs are left in the far-field. The longitudinal period is decided by the repeating length of the ±1 order beams Z ₁ = 17.5λ _SPP = 14.2 μm. This result agrees with the experimental result of 14.2 μm that is obtained by the Fourier transformation of the longitudinal intensity distribution in Fig. 2 (a). In the case where d = 6λ _SPP and α = 1/2, the 0, ±1, ±3, and ±5 order beams are observed. Usually, their repeating lengths are no longer coincident. We can obtain the Talbot distance Z _t approximately. Since the phase of the high order beam changes rapidly, Z _t is determined by the exact repeating length of the ±5 order beams, which is close to the repeating lengths of the ±1 and ±3 order beams. Consequently, we obtain Z _t = 30Z ₅ = 54.6 μm, which is close to 9Z₃ (54.6 μm) and Z₁ (58.2 μm). In the experiment, we have found the recurring images at the positions mZ _t (m = 1, 2, 3… …) from the SPPLG. At the same time, the half-period revivals also show up properly at positions (m – 0.5) Z _t with half a period lateral shift. Figure 2 (c) shows the experimental transversal intensity profile of Fig. 2 (b) at Z _t/2. The transverse FWHMs of the “pillars” are around 1.9 μm, which is smaller than the half period 2.4 μm. There are wavy structures at the top of these pillars, which might be indications of a true diffractive effect. The noise in the experiment is mainly from the interference created between the leakage radiation and the scattering noisy light.

To be able to compare with the experimental results, the Talbot carpets are numerically calculated and the propagation loss is considered. The SPPLG can be considered as a series of points of SPP secondary sources. Moreover, the SPP intensity distribution in the two-dimensional plane can be obtained by applying the Huygens-Fresnel principle [11]. Figure 2 (d) is the calculated Talbot carpet corresponding to the case in Fig. 2 (b), which matches well with the experimental result. The 4 dashed lines in Fig. 2 (d) indicate the positions of Z _t/2, d ²/λ, Z _t and 2d ²/λ from left to right, respectively. These clearly show that the paraxial approximation is invalid in this case. The theoretical intensity profile is drawn at Z _t/2 in Fig. 2 (c) and the theoretical FWHM 1.85μm is obtained.

The traveling SPPs are damped with a propagation length l _SPP(ω) = 1/[2Imk _SPP(ω)] depending upon the metal/dielectric configuration, where k _SPP(ω) is the complex wave number of the SPP of frequency ω [4]. The permittivity of gold at the working wavelength of 830 nm is ε = −29.3 + i2.05 [12]. In order to detect the SPPs by using the LRM system, the thickness of the gold film in the experiment is set at 50 nm and the resulting propagation length l _SPP = 33 μm. The short propagation length compared to the Talbot distance leads to the rapid decay of the intensities of the successive self-images with the increase in the distance from the source, which is the case in Figs. 2 (b) and 2 (d). However, for a practical application, the thickness of 50 nm of the gold film is not necessary. The propagation length can be 52 μm for a 200-nm-thick gold film at the vacuum wavelength 830 nm. Furthermore, increasing the wavelength can dramatically improve the propagation length.

3. Enhancement of intensities of the amplitude-modulated revivals

In order to enhance the intensity of the plasmonic Talbot carpet, the easiest way is to add more SPPLGs behind the original one to form a multi-layer SPPLG [shown schematically in Fig. 3 (a) ]. The distance between two adjacent layers is Δ. In the case where d = 6λ _SPP and α = 1/2, the numerical simulation shows that the best constructive coherence of the SPPs exited from different layers happens when Δ is slightly smaller than any multiple of λ _SPP without considering the transmission loss. In our experiment, a three-layer SPPLG with d = 6λ _SPP, α = 1/2 and Δ = λ _SPP is fabricated. An enhancement factor of ~5 is obtained at Z _t/2.

 

Fig. 3 (a) Schematic of the multi-layer SPPLG with the distance between two adjacent layers Δ. (b) Schematic of a two-level-phase SPPLG with an optical path difference δ (only 2.5 periods are shown).

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Multilevel phase gratings are usually used in the array illuminators to obtain a high efficiency based on the fractional Talbot effect [13, 14]. In this study, we use multilevel phase SPPLGs to increase the intensities of the amplitude-modulated revivals, which are similar to the optical phase grating for the array illumination. Figure 3 (b) schematically shows an example which is a two-level-phase SPPLG with an optical path difference of δ and a = b = d/2. When the paraxial approximation is valid, δ is set to be λ/8 to ensure the conversion from a phase grating to a 100% amplitude-modulated image at Z _t/4 [15]. In our case, although the paraxial approximation is not satisfied, the numerical simulation indicates that δ = λ _SPP/8 is also applicable for the two-level-phase SPPLG. We fabricate a two-level-phase SPPLG as shown in Fig. 3 (b) with d = 6λ _SPP and δ = λ _SPP/8, and the experimental amplitude-modulated image at Z _t/4 shows an enhancement factor of ~2 with respect to the single-phase case.

To combine these two methods, we fabricate a three-layer two-level-phase SPPLG with d = 6λ _SPP, Δ = λ _SPP, and δ = λ _SPP/8 [partly shown in Fig. 4 (a) ]. The experimental Talbot carpet is shown in Fig. 4 (b) with a better contrast. A nearly 100% amplitude-modulated revival at Z _t/4 with a FWHM ~2 μm is obtained, which is similar to the self-imaging FWHM of the single-phase SPPLG in Fig. 2 (b). The average intensity of the amplitude-modulated image at Z _t/4 is about one-order higher than that of the self-imaging of a single-layer single-phase SPPLG at Z _t/2 for the same exciting intensity. As mentioned above, the vertical light ribbon around the position of the SPPLG comes from the direct contribution of the laser beam to the LRM image. The Talbot carpet is no longer symmetrical with respect to the SPPLG due to the asymmetric structure of the phase grating.

 

Fig. 4 (a) Scanning electron micrograph of the three-layer two-level-phase SPPLG (only three periods are shown). (b) SPP image.

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4. Implementation of effective plasmonic focusing

Talbot focusing or array illumination is one important application of the Talbot effect [16]. Although a single-phase SPPLG with a small size of a in Fig. 1 (a) can result to plasmonic focusing at the self-imaging positions, the coupling efficiency is fairly low when the opening ratio is small. In order to obtain effective plasmonic focusing, a three-layer four-level-phase SPPLG is designed, of which one period is schematically shown in Fig. 5 (a) . In this particular case, d = 6λ _SPP and each period is averagely divided into six segments with the equal lengths of l ₀ = l ₁ = l ₂ = l ₃ = l ₄ = l ₅ = d/6. The light path difference between groove 0 and groove p (p = 1-5) is s _p. Thus, we set s ₁ = λ _SPP/12, s ₂ = λ _SPP/3, s ₃ = 3λ _SPP/2, s ₄ = λ _SPP/3, and s ₅ = λ _SPP/12, which correspond to the phase differences of π/6, 2π/3, 3π/2, 2π/3, and π/6, respectively [17]. The distance between two adjacent layers Δ = λ _SPP.

 

Fig. 5 (a) Schematic of one period of the three-layer four-level-phase SPPLG. (b) SPP image. (c) Experimental and theoretical transverse intensity profiles of one focal spot atl –Z _t/6.

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The experimental Talbot carpet is shown in Fig. 5 (b), in which four plasmonic focus arrays located at −7Z _t/6, -Z _t/6, 5Z _t/6, and 11Z _t/6, respectively, can be seen clearly. The strongest focusing is obtained at -Z _t/6. The experimental and theoretical transverse intensity profiles across one focal spot at -Z _t/6 are shown in Fig. 5(c), which show the experimental and theoretical transverse FWHMs of 713 nm and 661 nm, respectively. The positions of the focus arrays and the separation between the two adjacent focal spots can be adjusted by changing the parameters of the SPPLG. This method provides a new way to obtain the plasmonic focus array.

5. Conclusion

The study experimentally demonstrated the SPP analogue to the classical Talbot effect. The SPPs are currently of intense interest in nano-optical integrated devices which necessitates that the whole structure should be quite small. In this study, 3λ _SPP and 6λ _SPP are selected as the periods of the SPPLGs. The paraxial approximation, which is widely used in the analysis of the Talbot effect, has been found to be inapplicable. Talbot distance was given by comparing the repeating lengths of all order beams of the diffracted SPPs. Two methods, adding more gratings and using phase gratings, were used to increase the intensities of the amplitude-modulated revivals. For the application of the Talbot effect, strong SPP focus array with a FWHM of 713 nm was implemented by using a three-layer four-level-phase SPPLG when the wavelength of the incident light is 830 nm. More SPP-based applications can be expected in nano-optics [18] and optical trapping [19].