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Quasi-Talbot effect of the sub-wavelength Ag grating
Optics Communications 283 (2010) 5231–5235
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Quasi-Talbot effect of the sub-wavelength Ag grating Xia Wan, Qingkang Wang ⁎, Haihua Tao Research Institute of Micro/Nano Science and Technology, Shanghai Jiao Tong University, Shanghai, 200240, China
a r t i c l e
i n f o
Article history: Received 9 April 2010 Received in revised form 18 July 2010 Accepted 20 July 2010 Keywords: Quasi-Talbot effect Surface plasmons Sub-wavelength Ag grating
a b s t r a c t When a TM-polarized beam incidents normally on a thick sub-wavelength Ag grating with small opening ratio, self-images form in the transparent dielectric layer due to contributions not only from waves radiating from the slits, but also from the Surface Plasmons (SPs) created on Ag surface. These two contributions make the self-images connect together along the propagation direction and form continuous stripes with period equal to half of that of the grating. We define this phenomenon as quasi-Talbot effect. For a thin subwavelength Ag grating or a grating with large opening ratio, its intensity distribution conforms to the conventional Talbot effect. Investigation on quasi-Talbot effect of sub-wavelength metal grating can develop new applications in nano-scale devices. © 2010 Elsevier B.V. All rights reserved.
1. Introduction When a monochromatic plane wave illuminates a grating, its selfimaging can be observed at multiple distances in the Fresnel diffraction region, namely Talbot effect of the grating [1]. Talbot effect is one of the most basic optical diffraction phenomena, which has received ongoing attention because of its wide applications in Talbot array illumination [2], temporal Talbot effect [3], Talbot effects of atomic-matter waves [4], and interfereometric using Talbot effect for EUV exposure system [5], etc. At present, discussion on the Talbot effect has been mainly focused on the issue that the period of the grating is larger than the illumination wavelength, which can be explained by the theory of scalar diffraction [6,7]. Talbot effect of a sub-wavelength metal grating placed in air was discussed in [8]. Its self-imaging is confined to be around the grating with the intensity decreasing quickly. However, if there is a transparent dielectric layer near the sub-wavelength Ag grating, a new phenomenon appears due to excitement of the SPs on the mental surface and transformation of SPs into the propagation modes. SPs are electromagnetic waves generated due to the oscillations of the electron plasma in metal surface and they can propagate along the metal–dielectric interface. The intrinsic properties of SPs on nano-structured metal surface make them applicable in nano-photonics [9–13]. In this paper, we will study this new phenomenon, which is called as quasi-Talbot effect in the discussion, generated on a sub-wavelength Ag grating with a transparent dielectric layer. In this case, self-images behind a thick Ag grating with small opening ratio are made up of selfimages from the slits and Ag surface. The connected self-images form continuous stripes perpendicular to the grating plane. The period of the self-images is half of that of the grating. Some factors that could
substantially influence the quasi-Talbot images will be discussed in the following, such as the thickness and opening ratio of the Ag grating. 2. Structure and principle Fig. 1 shows the schematic model for studying the quasi-Talbot effect of a sub-wavelength Ag grating. It consists of two major parts. The part on the top is a quartz slab coated with a sub-wavelength Ag grating. The bottom one is a Si superstrate coated with a transparent dielectric layer. The two parts are fixed together with an air gap h = 50 nm. The period (P) of the Ag grating is 300 nm. The thickness of the transparent dielectric layer (L) is 1.02 μm. The finite-difference time-domain (FDTD) method [14,15] is used to simulate light propagation in the structure. The boundary condition along the x direction is set to be periodical. The perfectly matched layer (PML) is adopted as the absorbing boundary condition along the z direction. The light source is composed of plane waves with Gaussian profile, and its wavelength λ0 is 436 nm. The incident light is TM polarized with its electric vector E along the x and z directions, and magnetic vector H along the y direction. In the calculation, the refractive indices of the quartz and transparent dielectric layer (n) take 1.54, and 1.7, respectively. The relative dielectric constant of Ag is εm = −7.7 + 0.216i. In the case of a TM-polarized beam incidents normally on the subwavelength Ag grating, if the in-plane wave vector of diffractive wave (kin-plane) and that of the SPs (ksp) satisfy [16]: kin−plane = ksp ;
ð1Þ
SPs can be excited and the light is coupled into SPs. Their expressions are defined as: ⁎ Corresponding author. Tel.: +86 21 34206902. E-mail address: [email protected] (Q. Wang). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.07.054
0
kin−plane = k0 n sin θ m2π = P; m = 0; 1; 2; 3:::
ð2Þ
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Fig. 1. Schematic model for generating quasi-Talbot effect of a sub-wavelength Ag grating.
ksp = k0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εd εm = ðεd + εm Þ
ð3Þ
where k0 is the vacuum wave vector, n′ is the refractive index of air, θ is the incident angle, m is an integer, P is the periodicity of Ag
grating, and εd and εm are relative dielectric constants of air and Ag, respectively. If the sub-wavelength Ag grating is placed in air, according to dispersion relationship: (ksp)2 + (kz)2 = (k0)2 and formula (3), kz is an imaginary number. This makes the intensity of SPs attenuate exponentially along the z direction. In this case, SPs are evanescent waves and are confined close to the sub-wavelength Ag grating. However, if there is a transparent dielectric layer near the Ag grating with an air layer between them, (ksp)2 b (nk0)2 can be satisfied. When light penetrates into the transparent dielectric layer from the air layer, based on the tangential boundary condition, we can get k′in-plane = ksp, where k′in-plane is the in-plane wave vector in the transparent dielectric layer. Then, kz is a real number, in terms of (k′in-plane)2 + (kz)2 = (nk0)2. In this case, the SPs can be converted into propagating waves and transmitted to the far field [17]. The air layer between the Ag grating and transparent dielectric layer should be set less than 50 nm, considering the quick exponential intensity attenuation of SPs in the air layer. Due to excitement of SPs on the Ag surface and its conversion into propagation modes, a quasi-Talbot effect shows up. The patterns are influenced by the structure of the Ag grating, such as its thickness and opening ratio. In the following section, we will discuss it in detail.
Fig. 2. Normalized magnetic intensity distributions of light propagating through the sub-wavelength Ag grating with thickness (a) d = 20 nm, (b) d = 40 nm, (c) d = 60 nm, and (d) d = 80 nm, respectively. The period and slit width of the grating are 300 nm and 50 nm, respectively. The red color (the brightness in the press) denotes the maximum intensity and the blue one (the darkness in the press) stands for the minimum intensity. Zt is the Talbot distance.
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3. Results and analysis 3.1. Influence of the grating thickness We first calculate the distributions of the normalized magnetic intensity of the sub-wavelength Ag grating with fixed slit width w = 50 nm and period P = 300 nm. Fig. 2(a)–(d) is the quasi-Talbot effect of the Ag grating varying with its thickness d = 20, 40, 60, and 80 nm, respectively. The patterns in Fig. 2(a) are similar to the Talbot effect. When the thickness increases, the patterns evolve from separated self-images into continuous ones along the z direction. In Fig. 2(b), we can see that when d is 40 nm, the size and intensity of the stripes vary prominently along the z axis. This is the critical thickness for generating Talbot or quasi-Talbot effect. As the thickness increases, the self-images displayed in Fig. 2(c)–(d) conform to the quasi-Talbot effect. The size and intensity of the stripes almost keep constant along the z axis. The quasi-Talbot images are made up of self-images of the radiation modes of SPs created on the Ag surface and waves radiating from the slits. If the slit width of the Ag grating is much less than λ0/2, the optical power localized inside the slits is solely from the surface plasmon waves [18]. When the surface plasmon waves tunneling through the slits arrive at the exit of the slits, they decompose into two parts: SPs on Ag surface and the waves emanating from the slits [9–18]. The SPs are generated on the slit edges, and then propagate away from the slits [9–19]. When the SPs excited at the adjacent slit edges encounter in the center of the Ag surface, they radiate waves along the z direction. This part contributes to the self-images from silver surface. The waves radiating from the slits constitute another contribution to the self-images. There is a position shift of P/2 along the x direction for self-images from these two different contributions. As a result of superposition, continuous patterns with period of P/2 are formed in the transparent dielectric layer. When light travels in the transparent dielectric layer, its wavelength becomes λ0/n. According to the empirical formula for the case that the period of the grating is comparable to the wavelength [8], the Talbot distance Zt of the grating can be expressed approximately as:
Zt =
!1 = 2 !1 = 2 # " λ0 p2 λ2 q2 λ2 = 1− 2 20 − 1− 2 20 ; n P n P n
ð4Þ
where p = 0 and q = 1, represent diffraction orders with integers. Fig. 3(a) shows the Talbot images and quasi-Talbot images of the sub-wavelength Ag grating with thickness d = 20 nm and 60 nm at the position of 2Zt along the x direction. When the thickness of the grating is 20 nm, the light can penetrate through the Ag film. In this case, the intensity of light penetrating through the Ag film is dominant so the self-images are mainly contributed from Ag surface. As the thickness increases up to 60 nm, light almost cannot penetrate through the Ag film. However, the intensity of the SPs tunneling through the slits increases and becomes dominant. As discussed, selfimages are contributed not only from the slits, but also from the Ag surface. In this case, self-images from the Ag surface are mainly from the radiation modes of SPs on the Ag surface, and this could result in the variation of intensity distribution with different grating thickness. In order to further study the influence of grating thickness, curves of the normalized maximum intensity for the self-images from both the slits and the Ag surface at 2Zt as a function of thickness d is plotted in Fig. 3(b). For both curves, the normalized intensity oscillates with the grating thickness in the same step. When d is equal to 60 nm, 200 nm, or 340 nm, the intensities of both self-images get to the peaks. When d is equal to 140 nm or 280 nm, their intensities fall to the valleys.
Fig. 3. (a) Talbot and quasi-Talbot images of the sub-wavelength Ag grating with thickness d = 20 and 60 nm at 2Zt. (b) Curves of the normalized maximum intensity as a function of thickness d for self-images contributed from waves radiating from the slits and SPs created on the Ag surface at 2Z.
The enhancement of SPs in the slit is related to the Fabry–Perot slit resonance [20]. When the Fabry–Perot resonance occurs, it satisfies [21]: k0 Re neff d + argðρÞ = mπ; m = 0; 1; 2; 3:::::;
ð5Þ
where neff is the effective refractive index of the slits, and ρ is the reflection coefficient. Then the space between the peaks can be expressed as [20]: h i h i = λ0 = 2Re neff ; Δd = π = k0 Re neff
ð6Þ
According to the structural parameters of the sub-wavelength Ag grating, Δd is deduced to be 147 nm with Re(neff) of about 1.48 [22]. Despite the inevitable error in calculation, it consists with the period of the intensity curves. This demonstrates that the normalized intensities of the self-images from both the slits and the Ag surface are related to the Fabry–Perot slit resonance.
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3.2. Influence of the opening ratio of grating The opening ratio is another factor that could influence the patterns. In this case, the period P and thickness d of the Ag grating are fixed to be 300 nm and 60 nm, respectively. Fig. 4(a)–(d) shows the calculated normalized magnetic field intensity distributions of the sub-wavelength Ag grating with opening ratios (Г = w/P) taking values of 0.7, 0.5, 0.3, and 0.1, respectively. As shown in Fig. 4(a), when the opening ratio equals to 0.7, the field distribution conforms to Talbot effect. When we decrease the opening ratio to 0.5, continuous periodic patterns reappear along the z direction. However, the normalized intensity and the stripe size change prominently along the z direction. The opening ratio Г = 0.5 is a critical point for quasiTalbot effect. When the opening ratio is further decreased, the quasiTalbot effect can be reproduced. The normalize intensity as well as the stripe size almost keep constant along the z direction as shown in Fig. 4(c) and (d). Fig. 5(a) presents the Talbot and quasi-Talbot images of the subwavelength Ag grating with the opening ratio Г = 0.7 and 0.1 at the position of 2Zt. When the opening ratio of the sub-wavelength Ag grating equals to 0.7, the self-images are mainly contributed from the slits, due to the low excitation efficiency of SPs for large opening ratio [23]. Most of the light transmitting into the slits cannot be coupled into SPs and tunnels through the slits directly. Then, the normalized intensity of the self-images contributed from SPs created on Ag
surface decreases. When the opening ratio of grating is equal to 0.1, the excitation efficiency of SPs is high for this small opening ratio [23]. Furthermore, the SPs tunneling through the slits are dominant [18]. Then the quasi-Talbot images are contributed from both the slits and Ag surface. Fig. 5(b) plots the curves of the normalized maximum intensity of self-images contributed from slits and Ag surface at 2Zt as a function of opening ratio Г. For the curve contributed from slits, it first goes up as the opening ratio increases from 0.1 to 0.5. When the slit width increases, more light can transmit through the slits, which results in the increase of intensity for self-images from the slits. As the slit width further increases, the Fabry–Perot resonance condition cannot be satisfied at a fixed thickness, due to the reduction of neff. As studied, neff is dependent on the slit width (w). When w increases, neff will reduce [22]. In this case, the intensity of the self-images contributed from slits cuts down. For the normalized intensity of self-images contributed from SPs on Ag surface, it first increases with the opening ratio changing from 0.1 to 0.2. Then it begins to drop down as the opening ratio further increases from 0.3 to 0.5. This demonstrates that the SPs excitation efficiency is strongly dependent on the opening ratio. The excitation efficiency of SPs diminishes, as the opening ratio increases. Therefore, improved continuous periodic patterns can be obtained using the grating with small opening ratio. In this way, the feature size can be reduced with the improved optical resolution beyond the limitation of diffraction effect in nanolithography.
Fig. 4. The normalized magnetic intensity distributions of light transmitting through the sub-wavelength Ag grating with the opening ratio (a) Г = 0.7, (b) Г = 0.5, (c) Г = 0.3, and (d) Г = 0.1, respectively. The period and thickness of the grating are 300 nm and 60 nm, respectively. The red (bright in the press) and blue (dark in the press) colors denote the maximum and minimum intensity, respectively. Zt is the Talbot distance.
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fixed small opening ratio, the intensity distribution is similar to Talbot effect when its thickness is less than 20 nm. As the grating thickness increases, continuous periodic patterns with the period of P/2 appear. The self-images are contributed not only from waves radiating from slits, but also from SPs created on Ag surface. For a sub-wavelength Ag grating with fixed appropriate thickness, when the opening ratio of grating is large, the intensity distribution conforms to the Talbot effect. As the opening ratio decreases, continuous periodic patterns reappear. Therefore, the sub-wavelength Ag grating with small opening ratio and suitable thickness could be applicable in nanolithography. Sub-wavelength Ag grating with large opening ratio may also find wide applications in nano-scale devices, just as those of the classical Talbot effect in large scale devices.
Acknowledgments This research was supported by the Shanghai Committee of Science and Technology of China (2008 nanotechnology project No. 0852nm06600).
References
Fig. 5. (a) Talbot and quasi-Talbot images of the sub-wavelength Ag grating with opening ratio Г = 0.7 and 0.1 at 2Zt. (b) Curves of the normalized maximum intensity varying with opening ratio Г for self-images contributed from waves radiating from the slits and SPs created on the Ag surface at 2Zt.
4. Conclusion The influence of the grating thickness and opening ratio on the quasi-Talbot effect is studied. For a sub-wavelength Ag grating with
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Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Quasi-Talbot effect of the sub-wavelength Ag grating Xia Wan, Qingkang Wang ⁎, Haihua Tao Research Institute of Micro/Nano Science and Technology, Shanghai Jiao Tong University, Shanghai, 200240, China
a r t i c l e
i n f o
Article history: Received 9 April 2010 Received in revised form 18 July 2010 Accepted 20 July 2010 Keywords: Quasi-Talbot effect Surface plasmons Sub-wavelength Ag grating
a b s t r a c t When a TM-polarized beam incidents normally on a thick sub-wavelength Ag grating with small opening ratio, self-images form in the transparent dielectric layer due to contributions not only from waves radiating from the slits, but also from the Surface Plasmons (SPs) created on Ag surface. These two contributions make the self-images connect together along the propagation direction and form continuous stripes with period equal to half of that of the grating. We define this phenomenon as quasi-Talbot effect. For a thin subwavelength Ag grating or a grating with large opening ratio, its intensity distribution conforms to the conventional Talbot effect. Investigation on quasi-Talbot effect of sub-wavelength metal grating can develop new applications in nano-scale devices. © 2010 Elsevier B.V. All rights reserved.
1. Introduction When a monochromatic plane wave illuminates a grating, its selfimaging can be observed at multiple distances in the Fresnel diffraction region, namely Talbot effect of the grating [1]. Talbot effect is one of the most basic optical diffraction phenomena, which has received ongoing attention because of its wide applications in Talbot array illumination [2], temporal Talbot effect [3], Talbot effects of atomic-matter waves [4], and interfereometric using Talbot effect for EUV exposure system [5], etc. At present, discussion on the Talbot effect has been mainly focused on the issue that the period of the grating is larger than the illumination wavelength, which can be explained by the theory of scalar diffraction [6,7]. Talbot effect of a sub-wavelength metal grating placed in air was discussed in [8]. Its self-imaging is confined to be around the grating with the intensity decreasing quickly. However, if there is a transparent dielectric layer near the sub-wavelength Ag grating, a new phenomenon appears due to excitement of the SPs on the mental surface and transformation of SPs into the propagation modes. SPs are electromagnetic waves generated due to the oscillations of the electron plasma in metal surface and they can propagate along the metal–dielectric interface. The intrinsic properties of SPs on nano-structured metal surface make them applicable in nano-photonics [9–13]. In this paper, we will study this new phenomenon, which is called as quasi-Talbot effect in the discussion, generated on a sub-wavelength Ag grating with a transparent dielectric layer. In this case, self-images behind a thick Ag grating with small opening ratio are made up of selfimages from the slits and Ag surface. The connected self-images form continuous stripes perpendicular to the grating plane. The period of the self-images is half of that of the grating. Some factors that could
substantially influence the quasi-Talbot images will be discussed in the following, such as the thickness and opening ratio of the Ag grating. 2. Structure and principle Fig. 1 shows the schematic model for studying the quasi-Talbot effect of a sub-wavelength Ag grating. It consists of two major parts. The part on the top is a quartz slab coated with a sub-wavelength Ag grating. The bottom one is a Si superstrate coated with a transparent dielectric layer. The two parts are fixed together with an air gap h = 50 nm. The period (P) of the Ag grating is 300 nm. The thickness of the transparent dielectric layer (L) is 1.02 μm. The finite-difference time-domain (FDTD) method [14,15] is used to simulate light propagation in the structure. The boundary condition along the x direction is set to be periodical. The perfectly matched layer (PML) is adopted as the absorbing boundary condition along the z direction. The light source is composed of plane waves with Gaussian profile, and its wavelength λ0 is 436 nm. The incident light is TM polarized with its electric vector E along the x and z directions, and magnetic vector H along the y direction. In the calculation, the refractive indices of the quartz and transparent dielectric layer (n) take 1.54, and 1.7, respectively. The relative dielectric constant of Ag is εm = −7.7 + 0.216i. In the case of a TM-polarized beam incidents normally on the subwavelength Ag grating, if the in-plane wave vector of diffractive wave (kin-plane) and that of the SPs (ksp) satisfy [16]: kin−plane = ksp ;
ð1Þ
SPs can be excited and the light is coupled into SPs. Their expressions are defined as: ⁎ Corresponding author. Tel.: +86 21 34206902. E-mail address: [email protected] (Q. Wang). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.07.054
0
kin−plane = k0 n sin θ m2π = P; m = 0; 1; 2; 3:::
ð2Þ
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Fig. 1. Schematic model for generating quasi-Talbot effect of a sub-wavelength Ag grating.
ksp = k0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εd εm = ðεd + εm Þ
ð3Þ
where k0 is the vacuum wave vector, n′ is the refractive index of air, θ is the incident angle, m is an integer, P is the periodicity of Ag
grating, and εd and εm are relative dielectric constants of air and Ag, respectively. If the sub-wavelength Ag grating is placed in air, according to dispersion relationship: (ksp)2 + (kz)2 = (k0)2 and formula (3), kz is an imaginary number. This makes the intensity of SPs attenuate exponentially along the z direction. In this case, SPs are evanescent waves and are confined close to the sub-wavelength Ag grating. However, if there is a transparent dielectric layer near the Ag grating with an air layer between them, (ksp)2 b (nk0)2 can be satisfied. When light penetrates into the transparent dielectric layer from the air layer, based on the tangential boundary condition, we can get k′in-plane = ksp, where k′in-plane is the in-plane wave vector in the transparent dielectric layer. Then, kz is a real number, in terms of (k′in-plane)2 + (kz)2 = (nk0)2. In this case, the SPs can be converted into propagating waves and transmitted to the far field [17]. The air layer between the Ag grating and transparent dielectric layer should be set less than 50 nm, considering the quick exponential intensity attenuation of SPs in the air layer. Due to excitement of SPs on the Ag surface and its conversion into propagation modes, a quasi-Talbot effect shows up. The patterns are influenced by the structure of the Ag grating, such as its thickness and opening ratio. In the following section, we will discuss it in detail.
Fig. 2. Normalized magnetic intensity distributions of light propagating through the sub-wavelength Ag grating with thickness (a) d = 20 nm, (b) d = 40 nm, (c) d = 60 nm, and (d) d = 80 nm, respectively. The period and slit width of the grating are 300 nm and 50 nm, respectively. The red color (the brightness in the press) denotes the maximum intensity and the blue one (the darkness in the press) stands for the minimum intensity. Zt is the Talbot distance.
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3. Results and analysis 3.1. Influence of the grating thickness We first calculate the distributions of the normalized magnetic intensity of the sub-wavelength Ag grating with fixed slit width w = 50 nm and period P = 300 nm. Fig. 2(a)–(d) is the quasi-Talbot effect of the Ag grating varying with its thickness d = 20, 40, 60, and 80 nm, respectively. The patterns in Fig. 2(a) are similar to the Talbot effect. When the thickness increases, the patterns evolve from separated self-images into continuous ones along the z direction. In Fig. 2(b), we can see that when d is 40 nm, the size and intensity of the stripes vary prominently along the z axis. This is the critical thickness for generating Talbot or quasi-Talbot effect. As the thickness increases, the self-images displayed in Fig. 2(c)–(d) conform to the quasi-Talbot effect. The size and intensity of the stripes almost keep constant along the z axis. The quasi-Talbot images are made up of self-images of the radiation modes of SPs created on the Ag surface and waves radiating from the slits. If the slit width of the Ag grating is much less than λ0/2, the optical power localized inside the slits is solely from the surface plasmon waves [18]. When the surface plasmon waves tunneling through the slits arrive at the exit of the slits, they decompose into two parts: SPs on Ag surface and the waves emanating from the slits [9–18]. The SPs are generated on the slit edges, and then propagate away from the slits [9–19]. When the SPs excited at the adjacent slit edges encounter in the center of the Ag surface, they radiate waves along the z direction. This part contributes to the self-images from silver surface. The waves radiating from the slits constitute another contribution to the self-images. There is a position shift of P/2 along the x direction for self-images from these two different contributions. As a result of superposition, continuous patterns with period of P/2 are formed in the transparent dielectric layer. When light travels in the transparent dielectric layer, its wavelength becomes λ0/n. According to the empirical formula for the case that the period of the grating is comparable to the wavelength [8], the Talbot distance Zt of the grating can be expressed approximately as:
Zt =
!1 = 2 !1 = 2 # " λ0 p2 λ2 q2 λ2 = 1− 2 20 − 1− 2 20 ; n P n P n
ð4Þ
where p = 0 and q = 1, represent diffraction orders with integers. Fig. 3(a) shows the Talbot images and quasi-Talbot images of the sub-wavelength Ag grating with thickness d = 20 nm and 60 nm at the position of 2Zt along the x direction. When the thickness of the grating is 20 nm, the light can penetrate through the Ag film. In this case, the intensity of light penetrating through the Ag film is dominant so the self-images are mainly contributed from Ag surface. As the thickness increases up to 60 nm, light almost cannot penetrate through the Ag film. However, the intensity of the SPs tunneling through the slits increases and becomes dominant. As discussed, selfimages are contributed not only from the slits, but also from the Ag surface. In this case, self-images from the Ag surface are mainly from the radiation modes of SPs on the Ag surface, and this could result in the variation of intensity distribution with different grating thickness. In order to further study the influence of grating thickness, curves of the normalized maximum intensity for the self-images from both the slits and the Ag surface at 2Zt as a function of thickness d is plotted in Fig. 3(b). For both curves, the normalized intensity oscillates with the grating thickness in the same step. When d is equal to 60 nm, 200 nm, or 340 nm, the intensities of both self-images get to the peaks. When d is equal to 140 nm or 280 nm, their intensities fall to the valleys.
Fig. 3. (a) Talbot and quasi-Talbot images of the sub-wavelength Ag grating with thickness d = 20 and 60 nm at 2Zt. (b) Curves of the normalized maximum intensity as a function of thickness d for self-images contributed from waves radiating from the slits and SPs created on the Ag surface at 2Z.
The enhancement of SPs in the slit is related to the Fabry–Perot slit resonance [20]. When the Fabry–Perot resonance occurs, it satisfies [21]: k0 Re neff d + argðρÞ = mπ; m = 0; 1; 2; 3:::::;
ð5Þ
where neff is the effective refractive index of the slits, and ρ is the reflection coefficient. Then the space between the peaks can be expressed as [20]: h i h i = λ0 = 2Re neff ; Δd = π = k0 Re neff
ð6Þ
According to the structural parameters of the sub-wavelength Ag grating, Δd is deduced to be 147 nm with Re(neff) of about 1.48 [22]. Despite the inevitable error in calculation, it consists with the period of the intensity curves. This demonstrates that the normalized intensities of the self-images from both the slits and the Ag surface are related to the Fabry–Perot slit resonance.
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3.2. Influence of the opening ratio of grating The opening ratio is another factor that could influence the patterns. In this case, the period P and thickness d of the Ag grating are fixed to be 300 nm and 60 nm, respectively. Fig. 4(a)–(d) shows the calculated normalized magnetic field intensity distributions of the sub-wavelength Ag grating with opening ratios (Г = w/P) taking values of 0.7, 0.5, 0.3, and 0.1, respectively. As shown in Fig. 4(a), when the opening ratio equals to 0.7, the field distribution conforms to Talbot effect. When we decrease the opening ratio to 0.5, continuous periodic patterns reappear along the z direction. However, the normalized intensity and the stripe size change prominently along the z direction. The opening ratio Г = 0.5 is a critical point for quasiTalbot effect. When the opening ratio is further decreased, the quasiTalbot effect can be reproduced. The normalize intensity as well as the stripe size almost keep constant along the z direction as shown in Fig. 4(c) and (d). Fig. 5(a) presents the Talbot and quasi-Talbot images of the subwavelength Ag grating with the opening ratio Г = 0.7 and 0.1 at the position of 2Zt. When the opening ratio of the sub-wavelength Ag grating equals to 0.7, the self-images are mainly contributed from the slits, due to the low excitation efficiency of SPs for large opening ratio [23]. Most of the light transmitting into the slits cannot be coupled into SPs and tunnels through the slits directly. Then, the normalized intensity of the self-images contributed from SPs created on Ag
surface decreases. When the opening ratio of grating is equal to 0.1, the excitation efficiency of SPs is high for this small opening ratio [23]. Furthermore, the SPs tunneling through the slits are dominant [18]. Then the quasi-Talbot images are contributed from both the slits and Ag surface. Fig. 5(b) plots the curves of the normalized maximum intensity of self-images contributed from slits and Ag surface at 2Zt as a function of opening ratio Г. For the curve contributed from slits, it first goes up as the opening ratio increases from 0.1 to 0.5. When the slit width increases, more light can transmit through the slits, which results in the increase of intensity for self-images from the slits. As the slit width further increases, the Fabry–Perot resonance condition cannot be satisfied at a fixed thickness, due to the reduction of neff. As studied, neff is dependent on the slit width (w). When w increases, neff will reduce [22]. In this case, the intensity of the self-images contributed from slits cuts down. For the normalized intensity of self-images contributed from SPs on Ag surface, it first increases with the opening ratio changing from 0.1 to 0.2. Then it begins to drop down as the opening ratio further increases from 0.3 to 0.5. This demonstrates that the SPs excitation efficiency is strongly dependent on the opening ratio. The excitation efficiency of SPs diminishes, as the opening ratio increases. Therefore, improved continuous periodic patterns can be obtained using the grating with small opening ratio. In this way, the feature size can be reduced with the improved optical resolution beyond the limitation of diffraction effect in nanolithography.
Fig. 4. The normalized magnetic intensity distributions of light transmitting through the sub-wavelength Ag grating with the opening ratio (a) Г = 0.7, (b) Г = 0.5, (c) Г = 0.3, and (d) Г = 0.1, respectively. The period and thickness of the grating are 300 nm and 60 nm, respectively. The red (bright in the press) and blue (dark in the press) colors denote the maximum and minimum intensity, respectively. Zt is the Talbot distance.
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fixed small opening ratio, the intensity distribution is similar to Talbot effect when its thickness is less than 20 nm. As the grating thickness increases, continuous periodic patterns with the period of P/2 appear. The self-images are contributed not only from waves radiating from slits, but also from SPs created on Ag surface. For a sub-wavelength Ag grating with fixed appropriate thickness, when the opening ratio of grating is large, the intensity distribution conforms to the Talbot effect. As the opening ratio decreases, continuous periodic patterns reappear. Therefore, the sub-wavelength Ag grating with small opening ratio and suitable thickness could be applicable in nanolithography. Sub-wavelength Ag grating with large opening ratio may also find wide applications in nano-scale devices, just as those of the classical Talbot effect in large scale devices.
Acknowledgments This research was supported by the Shanghai Committee of Science and Technology of China (2008 nanotechnology project No. 0852nm06600).
References
Fig. 5. (a) Talbot and quasi-Talbot images of the sub-wavelength Ag grating with opening ratio Г = 0.7 and 0.1 at 2Zt. (b) Curves of the normalized maximum intensity varying with opening ratio Г for self-images contributed from waves radiating from the slits and SPs created on the Ag surface at 2Zt.
4. Conclusion The influence of the grating thickness and opening ratio on the quasi-Talbot effect is studied. For a sub-wavelength Ag grating with
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