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Numerical investigation of the transmission enhancement through subwavelength hole array
Optics Communications 274 (2007) 236–240 www.elsevier.com/locate/optcom
Numerical investigation of the transmission enhancement through subwavelength hole array Yuegang Chen a
a,b
, Yanhua Wang
a,b
, Yan Zhang
a,b,*
, Shutian Liu
b
Department of Physics, Capital Normal University, Xisanhuan Beilu 105, Beijing 100037, PR China b Department of Physics, Harbin Institute of Technology, Harbin 150001, PR China Received 11 August 2006; received in revised form 5 January 2007; accepted 1 February 2007
Abstract The transmission characteristics of a metallic film with subwavelength periodic square hole arrays are investigated by using the threedimensional finite-difference time-domain (3D-FDTD) method. The influences of the hole size, the refractive index of substrate, the refractive index of filled medium, the thickness of film as well as the incident angle on the characters of transmission spectra are studied. It is found that the transmission can be enhanced by filling the holes with higher refractive index medium. This enhancement can be explained by the collaboration of localized waveguide resonance with surface plasmon resonance. Ó 2007 Elsevier B.V. All rights reserved. PACS: 78.66.Bz; 41.20.Jb; 78.20.Bh Keywords: Subwavelength metal structure; Enhanced transmission; Surface plasmon
1. Introduction Recently, experiments which show the extraordinary high transmission of light through the metallic subwavelength hole array are of considerable interests [1,2]. When the size of hole is much smaller than the wavelength of incident light, it can be found that the maximum transmission is about 2–3 times higher than the hole porosity of the structure. This large transmission implies many potential applications. Furthermore, the underlying mechanism for the extraordinary transmission itself is also an intricate and meritorious research. Many theories have been proposed to interpret this phenomenon. It is generally admitted that the enhanced transmission is mainly due to the coupling of surface plasmon *
Corresponding author. Address: Department of Physics, Capital Normal University, Xisanhuan Beilu 105, Beijing 100037, PR China. Tel./fax: +86 1068902178. E-mail addresses: [email protected] (Y. Zhang), stliu@hit. edu.cn (S. Liu). 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.02.001
resonance (SPR) excited on the upper and lower surfaces of the structure through evanescent waves [3–5]. The influence of the hole shape on the transmission are investigated [6–8]. On the other hand, Baibda et al. found that the transmission can be enhanced further by filling the central region of each circular hole with higher refractive index medium [9]. However, the mechanisms for this phenomena has not been clearly explained. In this paper, the transmission of subwavelength square holes filling with higher refractive medium is investigated by using the three-dimensional finite-difference timedomain (3D-FDTD) method. The response of the transmission spectra to the hole size, the refractive index of substrate medium, the refractive index of filled medium, the thickness of film, and the incident angle are shown to elucidate the transmission mechanism. The near field distributions corresponding to the high transmission wavelengths are also given to help us to understand how the wave passes through the suwavelength hole array. This paper is organized as follows: In Section 2, the theoretical model is described. In Section 3, the dependence of
Y. Chen et al. / Optics Communications 274 (2007) 236–240
the transmission spectra on the thickness of film, the refractive index of filled medium as well as the incident angle are investigated. At last, a conclusion is drawn in Section 4.
k (a) side veiw y=d /2 y
θ
region 1 x
0
z=0
2. Theoretical model
2
z=h
The dispersion properties of the metal must be considered here since the absorpbtion and permittivity of the metallic material are frequency dependent. The Drude model [9,10] is used to describe the dependence of the metallic permittivity on the frequency: ! x2p M ðxÞ ¼ 0 1 ; ð1Þ xðx þ icÞ where x is the angle frequency of the incident wave, 0 the permittivity of the vacuum, xp the plasma frequency of the metal, and c represents the damping rate which characters the ohmic absorption loss. The metal gold (Au) is used in this paper, it’s plasma frequency is xp ¼ 1:236 1016 and loss c ¼ 1:4 1014 . The propagation of the light in the metal is described by the Maxwell’s equations, which are coupled with the oscillations of quasi-free electrons (Drude model) [10]: ~ oH ~ ~ ; r E ¼ l0 ot ~ ~ H ~ ¼ 0 oE þ ~ ð2Þ r J; ot o~ J þ c~ J ¼ 0 x2p~ E; ot ~ are the electric and magnetic field vectors, where ~ E and H ~ is the Nabla differential operator. ~ respectively. r J is a current density and equals to the time derivative of the metal polarization, i.e. ~ J ¼ o~ P =ot. l0 is the magnetic permeability of the vacuum. The 3D-FDTD method is employed to simulate the interaction between the metal and incident wave. The structures are illumined normally by a TE-polarized plane wave pulse ðEx; Hy; Hz 6¼ 0Þ centered at k = 600 nm. The spectrum width of this pulse can cover the range desired in the calculation. The Fourier transform of the temporal response is used to obtain the spectrum. Then normalization of the spectrum by the incident wave and the hole porosity of the structure gives the zero-order normalized transmittance spectrum. The schematic of the structure is shown in Fig. 1. The square holes with width d are arranged in the metal film with period p. When a plane wave is incident on the metal film, the SPR enhances energy transmission through the film for [4,11] ~ ~ x jG ~y ¼ ~ k 0 sinðhÞ iG k sp ;
237
ð3Þ
where ~ k 0 is the wave vector of incident light and h is the inci~x and G ~y are the Bragg vectors of the square dent angle. G ~ ~ lattice and jGx j ¼ jGy j ¼ 2p=p. i and j are integers, which express the mode indices. ~ k sp is the surface plasmon vector:
p (b) top veiw z=0
metal Au
region 3
z x d p
d
y
Fig. 1. Schematic of the metal Au structure. (a) xz cross section and (b) top view. The regions 2 in the holes can be filled with different medium.
~0 j j~ k sp j ¼ jk
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1;3 M ; 1;3 þ M
ð4Þ
where 1 and 3 are the dielectric constants of the incident and substrate medium, respectively. M is the dielectric constant of the metal. For the normal incidence, the wavelengths of the excited SPR modes are given approximately by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1;3 M kmax ði; jÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : ð5Þ 2 2 1;3 þ M i þj 3. Simulation results In this Section, the array of square holes in the Au metal film is taken into account. The grating period p = 430 nm is fixed for the whole paper. The thickness of the film is h and the hole width is d. Firstly, the influence of the hole size on the transmission is investigated. The thickness of the Au film is selected as h = 200 nm and the dielectric constants in the regions 1, 2, 3 are 1 ¼ 1:00; 2 ¼ 1:00, and 3 ¼ 2:31, respectively. The sizes of the square holes are dx = dy = 120, 160, 200, 240, and 280 nm, respectively. The normalized zero-order transmission spectra are shown in Fig. 2(a). Two transmission peaks corresponding to the SPR modes (1, 0) and (1, 1) can be obviously seen. It can also be found that increasing the width of holes results a higher transmittance and broader peaks. These results agree with the experiment in Ref. [12] quite well. The red-shift trend with the increasing of the hole width is also the same with the experimental results. These results confirm the validity of the 3D-FDTD code adopted in this paper. The enhanced transmission also depends on the SPR of the substrate interface. The influence of the substrate on the transmission spectrum is shown in Fig. 2(b). The hole width is selected as 200 nm and the thickness of the film is 200 nm, the upper side medium is air. When the dielectric constant of substrate increases, the new transmission peaks
238
Y. Chen et al. / Optics Communications 274 (2007) 236–240
1
(1,1)
(1,0)
0.8
0.6
0.4
0.2
0 400
600 800 Wavelength λ (nm)
Normalized zero−order transmittance
Normalized zero−order transmittance
4 120 160 200 240 280
1000
ε =1.00 3 ε =1.77 3 ε =2.31 3 ε3=3.06
3.5 3 2.5 2 1.5 1 0.5 0 400
600 800 Wavelength λ (nm)
1000
Fig. 2. Transmission spectra for square hole array (a) with different holes size and (b) with different substrate medium.
emerge and shift toward longer wavelength. At the same time, the transmittance decreases because the unmatched SPRs on the upper and lower surfaces of the structure [3]. For the vacuum holes with width of 200 nm, cross sections of the electric field (Ex) are depicted in Fig. 3 for the wavelength k = 741 nm, corresponding to the (1, 0) SPR peak. In the plane of x ¼ d=2, the electric fields are relatively large in the hole and incident surface, as shown in Fig. 3(a). In the plane of y ¼ d=2, the electric fields experiences maxima value in the four corners (see Fig. 3(b)). In the surface of substrate (shown in Fig. 3(c)), electric fields reach maxima at the edge with x ¼ 0 and x ¼ d. The field distributions help us to understand how the light pass through the holes. The strong fields at four edges of the entrance and the exit are due to the SPR on both surface of the Au film. The SPRs enhance the field at two ends of holes to couple the energy into and out of the holes.
2
6
Therefore, the SPR play an important role on transmission enhancement through the subwavelength hole array. When the holes are filled with mediums with permittivity 2 ¼ 1:77; 2 ¼ 2:31; 2 ¼ 2:66, and 2 ¼ 3:06, the transmission spectra behave interestingly, as shown in Fig. 4. It can be found that the peak (1, 1) splits into two branches when the holes are filled with medium with 2 ¼ 1:77. With 2 increasing, the right peak branched from the peak (1, 1) vanishes, while another peak rises in the left edge of the (1, 0) peak. At the same time, the primary (1, 0) peak shifts to the longer wavelength. As a whole, the peak (1, 0) becomes broader and higher. This phenomenon can be explained by the combination of SPR and WR. The hole in film can be considered as a truncated rectangular waveguide with four metal walls. Two sides of the hole are open to the free space and substrate, thus the truncated waveguide forms a low-Q cavity resonator[13]. Therefore, the transmission reaches peaks when the Fabry–Perot resonance condition is satisfied: k 0 Reðneff Þh argðr12 Þ ¼ mp;
where m is an integer, k0 is the wave vector of the incident wave in the vacuum, neff is the effective refractive index of the WR mode [14,15], its value depends on the medium in the hole as well as the metal walls. The first term in Eq. (6) indicates that the resonance occurs only when the cavity length is longer than the half of the WR mode wavelength. The second term is an additional phase shift when the wave is reflected at the interfaces. Filling holes with higher refractive medium, on the one hand, makes the effective size of holes increasing, therefore the cut-off wavelength of the waveguide mode is enlarged. On the other hand, filling holes will also enhance the neff, thus the wavelength which can satisfy Eq. (6) will increase. Below the new cut-off wavelength, a new waveguide mode will generate when Eq. (6) is satisfied and the transmission for the corresponding wavelength will grow. As shown in
6
1.8 5
5
1.4 4
4
1.2 1
3
3
0.8 0.6 0.4
2
2
1
1
0.2 0 (a) x=d/2 plane.
0 (b) y=d/2 plane.
(c) z=h plane.
Fig. 3. Cross sections of Ex amplitude for k = 741 nm with 2 ¼ 1 (a) x ¼ d=2 plane, (b) y ¼ d=2 plane, and (c) z ¼ h plane. The period is 430 nm and the hole width is 200 nm.
Normalized zero—order transmittance
1.8 1.6
ð6Þ
1.6 1.4 1.2
ε2=1.00 ε =1.77 2 ε =2.31 2 ε2=2.66 ε2=3.06
1 0.8 0.6 0.4 0.2 0 400
500
600
700
800
900
1000
Wavelength λ (nm)
Fig. 4. Transmission spectra for square hole array filled with different medium. The period is 430 nm and the hole width is 200 nm.
Y. Chen et al. / Optics Communications 274 (2007) 236–240
Fig. 4, when the dielectric constant increases from 1.77 to 3.06, the WR mode moves from the SPR mode of (1, 1) into the mode of (1, 0). The transmissivity is enhanced distinctly when the hole is filled with medium. The original (1, 0) mode has a redshift. The increase in the transmissivity is attributed to the effective size of the hole becoming large when it is filled with the higher refractive index medium[12]. When the holes are filled with medium with 2 ¼ 3:06, cross sections of the electric fields are plotted in Fig. 5(a– c) for k = 716 nm and Fig. 5(d–f) for k = 884 nm. These two wavelengths correspond to the two peaks in the (1, 0) SPR mode shown in Fig. 4. It can be clearly seen in Fig. 5(a–b), the electric field inside the hole is separated into two regions. The field at the incident surface is stronger than that at the substrate surface. The dark region inside the hole is due to the p’s phase differences between two ends of the hole. So the peak corresponds to the first order WR mode (m ¼ 1) in Eq. (6). The simulation also shows that the corresponding wave passes through the film in the standing wave mode, which also demonstrates the WR participates in the transmission. For the k = 884 nm which corresponds to the initial SPR mode, the field is similar to Fig. 3, which suggests that the right branch is the same with (1, 0) mode of the structure with vacuum holes. Comparing the upper and lower row in Fig. 5, the same character of the near fields for two peaks is that the strong field localizes at the edges along y direction at the entrance and exit. This is due to the SPRs appear at both the upper and substrate surface. The SPRs enhance the field at two ends of the hole and couple the energy into and out of the hole, which is necessary for the transmission enhancement. The effect of film thickness on the transmission has also been investigated. For the film with vacuum holes, the
4
8
z
3 2
0
6
6
4
4
3
2
0 (c) λ=716nm, z=h.
(b) λ=716nm, y=d/2.
(a) λ=716nm, x=d/2.
6
6
2.5 2
5 4
4
1.5 1
3 2
(d) λ=884nm, x=d/2.
2
0.7 0.6
h=200 h=300 h=400 h=500
0.5 0.4 0.3 0.2 0.1
1.8 1.6
h=200 h=300 h=400 h=500
1.4 1.2 1 0.8 0.6 0.4 0.2
2
0.5
1 0 (e) λ=884nm, y=d/2.
where / ¼ Ey ; Ez ; H y ; H z , k x ¼ k 0 sinðhÞ. h is the incident angle of the illuminating light. The parameters for calculation are selected as: The hole width is 200 nm and the thickness of film is 200 nm. The dielectric constant of substrate is 2 ¼ 2:31. The step of the incident angle is 1°. For the vacuum hole array, the dispersion diagram is shown in Fig. 7. One can see that the (1, 0) SPR peak divides into two branches with the increasing of incident angle. Based on Eq. (3), this mode is twofold with i ¼ 1 when h ¼ 0. When h increases the (1, 0) mode shifts to
8
y
2
1
ð7Þ
x
10
5 z
/ðx þ pÞ ¼ expðik x pÞ/ðxÞ;
Normalized zeros−order transmittance
x
transmission spectra are shown in Fig. 6(a) for different film thickness. It can be seen that the transmission peaks decrease swiftly with the metal film thickness increasing. This is because that the energy transmits through the holes in the form of the evanescent wave, which attenuates rapidly with distance increasing, thus the SPR at two surfaces of the metal can not couple well. However, when the holes are filled with the medium with 2 ¼ 3:06, the situation is quite different. As shown in Fig. 6(b), the left transmission peaks do not fall with the thickness increasing. Furthermore, each peak is split into two peaks. For the (1, 0) SPR mode, when the thickness h increases from 200 to 500 nm, the left branch moves to the longer wavelength direction and reaches its maximum in the middle of SPR mode (1, 0), then decreases. While the right branch decreases rapidly with thickness increasing, its behave is quite similar with that of the (1, 0) SPR peak for the structure with vacuum holes. When h = 500 nm, a new WR mode moves into the range of the SPR mode (1, 0). In the SPR mode (1, 1), the peak branches when ðh ¼ 300 and 500 nm, this is also due to the appearance of the WR modes. The influence of the incident angle has also be investigated. For the incline incident, the boundary condition in the x direction is used as
Normalized zero−order transmittance
y
239
(f) λ=884nm, z=h.
Fig. 5. Cross section of Ex amplitude for 2 ¼ 3:06. (a–c) Incident wavelength k = 716 nm and (d–e) incident wavelength k = 841 nm.
0 400
600 800 1000 Wavelength λ (nm)
0 400
600 800 Wavelength λ (nm)
1000
Fig. 6. Transmission spectra of the metal film with different thickness. (a) The vacuum holes array and (b) the holes filled with medium with 2 ¼ 3:06.
240
Y. Chen et al. / Optics Communications 274 (2007) 236–240 0
0.9 0.8
not change with the h increasing, which corresponds the WR mode with m ¼ 1.
Incident angular θ (degree)
5 0.7 10
0.6 SP(—1,0)
0.5
15 0.4 20
0.3 0.2
25 0.1
SP(—1,1)
SP(+1,1) SP(+1,0) 30 500
600
700
800
Wavelength λ (nm)
900
1000
1100
4. Conclusion In conclusion, optical transmission properties of the subwavelength hole array are studied by using the 3DFDTD method. Through filling the hole array with medium with higher refractive index, the effective refractive index of the hole in the metal increases, thus the WR modes can be moved into the SPR range. The collaboration of two mechanisms can enhance the transmission greatly even for the thicker metal film. The corresponding field distributions are also presented for better understanding how the light passes through the holes. Acknowledgements
Fig. 7. Dispersion relation diagram for the vacuum hole array.
0
1.8 1.6
Incident angular θ (degree)
5
The authors thank the anonymous reviewers for valuable comments in improving the manuscript. This work was supported by the National Natural Science Foundation of China (Grant 10604042) and National Basic Research Program of China (Grant 2006CB302901).
1.4
References
10
1.2 1
15
0.8 20
SP(—1,0)
0.4
WR(1)
25 SP(+1,1)
SP(+1,0)
0.6
0.2
SP(—1,1)
30 500
600
700
800
Wavelength λ (nm)
900
1000
1100
Fig. 8. Dispersion relation diagram for the holes filled with medium with 2 ¼ 3:06.
longer wavelength direction, and (+1, 0) mode does to the opposite direction as suggested by the lines. The (1, 1) peaks also divides into ð1; 1Þ branches. The black minimal transmittance regions may be the effect of the SPR. When the holes are filled with medium with 2 ¼ 3:06, the dispersion diagram is shown in Fig. 8. The SPR modes take on similar dispersion relations to that of the vacuum hole array. However, there is a transmission branch that does
[1] T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, P.A. Wolff, Nature (London) 391 (1998) 667. [2] L.B. William, D. Alain, T.W. Ebbesen, Narure 424 (2003) 824. [3] A. Krishnan, T. Thio, T.J. Kim, H.J. lezec, T.W. Ebbesen, P.A. Wolff, J. Pendry, L. Martin-Moreno, F.J. Garcia-Vidal, Opt. Comm. 200 (2001) 1. [4] W.L. Barnes, W.A. Murray, J. Dintinger, E. Devaux, T.W. Ebbesen, Phys. Rev. Lett. 92 (2004) 107401. [5] F.J. Garcı´a-Vidal, H.J. Lezec, T.W. Ebbesen, L. Martı´n-Moreno, Phys. Rev. Lett. 90 (2003) 213901. [6] K.J.K. Koerkamp, S. Enoch, F.B. Segerink, N.F. Hulst, L. Kuipers, Phys. Rev. Lett. 92 (2004) 183901. [7] K.L. Molen, K.J.K. Koerkamp, S. Enoch, F.B. Segerink, N.F. Hulst, L. Kuipers, Phys. Rev. 72 (2005) 045421. [8] H. Cao, A. Nahata, Opt. Express 12 (2004) 3664. [9] F.B. Baida, D. Van Labeke, Phys. Rev. B 67 (2003) 155314. [10] R. Mu¨ller, V. Malyarchuk, C. Lienau, Phys. Rev. B 68 (2003) 205415. [11] H.F. Ghaemi, T. Thio, D.E. Grupp, T.W. Ebbesen, H.J. Lezec, Phy. Rev. B 58 (1998) 6779. [12] K.L. Molen, F.B. Segerink, N.F. Hulst, L. Kuipers, Appl. Phys. Lett. 85 (2004) 4316. [13] Z. Ruan, M. Qiu, Phys. Rev. Lett. 96 (2006) 233901. [14] Q. Cao, P. Lalanne, Phys. Rev. Lett. 88 (2002) 057403. [15] P. Lalanne, J.P. Hugonin, S. Astilean, M. Palamaru, J. Opt. A 2 (2000) 48.
Numerical investigation of the transmission enhancement through subwavelength hole array Yuegang Chen a
a,b
, Yanhua Wang
a,b
, Yan Zhang
a,b,*
, Shutian Liu
b
Department of Physics, Capital Normal University, Xisanhuan Beilu 105, Beijing 100037, PR China b Department of Physics, Harbin Institute of Technology, Harbin 150001, PR China Received 11 August 2006; received in revised form 5 January 2007; accepted 1 February 2007
Abstract The transmission characteristics of a metallic film with subwavelength periodic square hole arrays are investigated by using the threedimensional finite-difference time-domain (3D-FDTD) method. The influences of the hole size, the refractive index of substrate, the refractive index of filled medium, the thickness of film as well as the incident angle on the characters of transmission spectra are studied. It is found that the transmission can be enhanced by filling the holes with higher refractive index medium. This enhancement can be explained by the collaboration of localized waveguide resonance with surface plasmon resonance. Ó 2007 Elsevier B.V. All rights reserved. PACS: 78.66.Bz; 41.20.Jb; 78.20.Bh Keywords: Subwavelength metal structure; Enhanced transmission; Surface plasmon
1. Introduction Recently, experiments which show the extraordinary high transmission of light through the metallic subwavelength hole array are of considerable interests [1,2]. When the size of hole is much smaller than the wavelength of incident light, it can be found that the maximum transmission is about 2–3 times higher than the hole porosity of the structure. This large transmission implies many potential applications. Furthermore, the underlying mechanism for the extraordinary transmission itself is also an intricate and meritorious research. Many theories have been proposed to interpret this phenomenon. It is generally admitted that the enhanced transmission is mainly due to the coupling of surface plasmon *
Corresponding author. Address: Department of Physics, Capital Normal University, Xisanhuan Beilu 105, Beijing 100037, PR China. Tel./fax: +86 1068902178. E-mail addresses: [email protected] (Y. Zhang), stliu@hit. edu.cn (S. Liu). 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.02.001
resonance (SPR) excited on the upper and lower surfaces of the structure through evanescent waves [3–5]. The influence of the hole shape on the transmission are investigated [6–8]. On the other hand, Baibda et al. found that the transmission can be enhanced further by filling the central region of each circular hole with higher refractive index medium [9]. However, the mechanisms for this phenomena has not been clearly explained. In this paper, the transmission of subwavelength square holes filling with higher refractive medium is investigated by using the three-dimensional finite-difference timedomain (3D-FDTD) method. The response of the transmission spectra to the hole size, the refractive index of substrate medium, the refractive index of filled medium, the thickness of film, and the incident angle are shown to elucidate the transmission mechanism. The near field distributions corresponding to the high transmission wavelengths are also given to help us to understand how the wave passes through the suwavelength hole array. This paper is organized as follows: In Section 2, the theoretical model is described. In Section 3, the dependence of
Y. Chen et al. / Optics Communications 274 (2007) 236–240
the transmission spectra on the thickness of film, the refractive index of filled medium as well as the incident angle are investigated. At last, a conclusion is drawn in Section 4.
k (a) side veiw y=d /2 y
θ
region 1 x
0
z=0
2. Theoretical model
2
z=h
The dispersion properties of the metal must be considered here since the absorpbtion and permittivity of the metallic material are frequency dependent. The Drude model [9,10] is used to describe the dependence of the metallic permittivity on the frequency: ! x2p M ðxÞ ¼ 0 1 ; ð1Þ xðx þ icÞ where x is the angle frequency of the incident wave, 0 the permittivity of the vacuum, xp the plasma frequency of the metal, and c represents the damping rate which characters the ohmic absorption loss. The metal gold (Au) is used in this paper, it’s plasma frequency is xp ¼ 1:236 1016 and loss c ¼ 1:4 1014 . The propagation of the light in the metal is described by the Maxwell’s equations, which are coupled with the oscillations of quasi-free electrons (Drude model) [10]: ~ oH ~ ~ ; r E ¼ l0 ot ~ ~ H ~ ¼ 0 oE þ ~ ð2Þ r J; ot o~ J þ c~ J ¼ 0 x2p~ E; ot ~ are the electric and magnetic field vectors, where ~ E and H ~ is the Nabla differential operator. ~ respectively. r J is a current density and equals to the time derivative of the metal polarization, i.e. ~ J ¼ o~ P =ot. l0 is the magnetic permeability of the vacuum. The 3D-FDTD method is employed to simulate the interaction between the metal and incident wave. The structures are illumined normally by a TE-polarized plane wave pulse ðEx; Hy; Hz 6¼ 0Þ centered at k = 600 nm. The spectrum width of this pulse can cover the range desired in the calculation. The Fourier transform of the temporal response is used to obtain the spectrum. Then normalization of the spectrum by the incident wave and the hole porosity of the structure gives the zero-order normalized transmittance spectrum. The schematic of the structure is shown in Fig. 1. The square holes with width d are arranged in the metal film with period p. When a plane wave is incident on the metal film, the SPR enhances energy transmission through the film for [4,11] ~ ~ x jG ~y ¼ ~ k 0 sinðhÞ iG k sp ;
237
ð3Þ
where ~ k 0 is the wave vector of incident light and h is the inci~x and G ~y are the Bragg vectors of the square dent angle. G ~ ~ lattice and jGx j ¼ jGy j ¼ 2p=p. i and j are integers, which express the mode indices. ~ k sp is the surface plasmon vector:
p (b) top veiw z=0
metal Au
region 3
z x d p
d
y
Fig. 1. Schematic of the metal Au structure. (a) xz cross section and (b) top view. The regions 2 in the holes can be filled with different medium.
~0 j j~ k sp j ¼ jk
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1;3 M ; 1;3 þ M
ð4Þ
where 1 and 3 are the dielectric constants of the incident and substrate medium, respectively. M is the dielectric constant of the metal. For the normal incidence, the wavelengths of the excited SPR modes are given approximately by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1;3 M kmax ði; jÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : ð5Þ 2 2 1;3 þ M i þj 3. Simulation results In this Section, the array of square holes in the Au metal film is taken into account. The grating period p = 430 nm is fixed for the whole paper. The thickness of the film is h and the hole width is d. Firstly, the influence of the hole size on the transmission is investigated. The thickness of the Au film is selected as h = 200 nm and the dielectric constants in the regions 1, 2, 3 are 1 ¼ 1:00; 2 ¼ 1:00, and 3 ¼ 2:31, respectively. The sizes of the square holes are dx = dy = 120, 160, 200, 240, and 280 nm, respectively. The normalized zero-order transmission spectra are shown in Fig. 2(a). Two transmission peaks corresponding to the SPR modes (1, 0) and (1, 1) can be obviously seen. It can also be found that increasing the width of holes results a higher transmittance and broader peaks. These results agree with the experiment in Ref. [12] quite well. The red-shift trend with the increasing of the hole width is also the same with the experimental results. These results confirm the validity of the 3D-FDTD code adopted in this paper. The enhanced transmission also depends on the SPR of the substrate interface. The influence of the substrate on the transmission spectrum is shown in Fig. 2(b). The hole width is selected as 200 nm and the thickness of the film is 200 nm, the upper side medium is air. When the dielectric constant of substrate increases, the new transmission peaks
238
Y. Chen et al. / Optics Communications 274 (2007) 236–240
1
(1,1)
(1,0)
0.8
0.6
0.4
0.2
0 400
600 800 Wavelength λ (nm)
Normalized zero−order transmittance
Normalized zero−order transmittance
4 120 160 200 240 280
1000
ε =1.00 3 ε =1.77 3 ε =2.31 3 ε3=3.06
3.5 3 2.5 2 1.5 1 0.5 0 400
600 800 Wavelength λ (nm)
1000
Fig. 2. Transmission spectra for square hole array (a) with different holes size and (b) with different substrate medium.
emerge and shift toward longer wavelength. At the same time, the transmittance decreases because the unmatched SPRs on the upper and lower surfaces of the structure [3]. For the vacuum holes with width of 200 nm, cross sections of the electric field (Ex) are depicted in Fig. 3 for the wavelength k = 741 nm, corresponding to the (1, 0) SPR peak. In the plane of x ¼ d=2, the electric fields are relatively large in the hole and incident surface, as shown in Fig. 3(a). In the plane of y ¼ d=2, the electric fields experiences maxima value in the four corners (see Fig. 3(b)). In the surface of substrate (shown in Fig. 3(c)), electric fields reach maxima at the edge with x ¼ 0 and x ¼ d. The field distributions help us to understand how the light pass through the holes. The strong fields at four edges of the entrance and the exit are due to the SPR on both surface of the Au film. The SPRs enhance the field at two ends of holes to couple the energy into and out of the holes.
2
6
Therefore, the SPR play an important role on transmission enhancement through the subwavelength hole array. When the holes are filled with mediums with permittivity 2 ¼ 1:77; 2 ¼ 2:31; 2 ¼ 2:66, and 2 ¼ 3:06, the transmission spectra behave interestingly, as shown in Fig. 4. It can be found that the peak (1, 1) splits into two branches when the holes are filled with medium with 2 ¼ 1:77. With 2 increasing, the right peak branched from the peak (1, 1) vanishes, while another peak rises in the left edge of the (1, 0) peak. At the same time, the primary (1, 0) peak shifts to the longer wavelength. As a whole, the peak (1, 0) becomes broader and higher. This phenomenon can be explained by the combination of SPR and WR. The hole in film can be considered as a truncated rectangular waveguide with four metal walls. Two sides of the hole are open to the free space and substrate, thus the truncated waveguide forms a low-Q cavity resonator[13]. Therefore, the transmission reaches peaks when the Fabry–Perot resonance condition is satisfied: k 0 Reðneff Þh argðr12 Þ ¼ mp;
where m is an integer, k0 is the wave vector of the incident wave in the vacuum, neff is the effective refractive index of the WR mode [14,15], its value depends on the medium in the hole as well as the metal walls. The first term in Eq. (6) indicates that the resonance occurs only when the cavity length is longer than the half of the WR mode wavelength. The second term is an additional phase shift when the wave is reflected at the interfaces. Filling holes with higher refractive medium, on the one hand, makes the effective size of holes increasing, therefore the cut-off wavelength of the waveguide mode is enlarged. On the other hand, filling holes will also enhance the neff, thus the wavelength which can satisfy Eq. (6) will increase. Below the new cut-off wavelength, a new waveguide mode will generate when Eq. (6) is satisfied and the transmission for the corresponding wavelength will grow. As shown in
6
1.8 5
5
1.4 4
4
1.2 1
3
3
0.8 0.6 0.4
2
2
1
1
0.2 0 (a) x=d/2 plane.
0 (b) y=d/2 plane.
(c) z=h plane.
Fig. 3. Cross sections of Ex amplitude for k = 741 nm with 2 ¼ 1 (a) x ¼ d=2 plane, (b) y ¼ d=2 plane, and (c) z ¼ h plane. The period is 430 nm and the hole width is 200 nm.
Normalized zero—order transmittance
1.8 1.6
ð6Þ
1.6 1.4 1.2
ε2=1.00 ε =1.77 2 ε =2.31 2 ε2=2.66 ε2=3.06
1 0.8 0.6 0.4 0.2 0 400
500
600
700
800
900
1000
Wavelength λ (nm)
Fig. 4. Transmission spectra for square hole array filled with different medium. The period is 430 nm and the hole width is 200 nm.
Y. Chen et al. / Optics Communications 274 (2007) 236–240
Fig. 4, when the dielectric constant increases from 1.77 to 3.06, the WR mode moves from the SPR mode of (1, 1) into the mode of (1, 0). The transmissivity is enhanced distinctly when the hole is filled with medium. The original (1, 0) mode has a redshift. The increase in the transmissivity is attributed to the effective size of the hole becoming large when it is filled with the higher refractive index medium[12]. When the holes are filled with medium with 2 ¼ 3:06, cross sections of the electric fields are plotted in Fig. 5(a– c) for k = 716 nm and Fig. 5(d–f) for k = 884 nm. These two wavelengths correspond to the two peaks in the (1, 0) SPR mode shown in Fig. 4. It can be clearly seen in Fig. 5(a–b), the electric field inside the hole is separated into two regions. The field at the incident surface is stronger than that at the substrate surface. The dark region inside the hole is due to the p’s phase differences between two ends of the hole. So the peak corresponds to the first order WR mode (m ¼ 1) in Eq. (6). The simulation also shows that the corresponding wave passes through the film in the standing wave mode, which also demonstrates the WR participates in the transmission. For the k = 884 nm which corresponds to the initial SPR mode, the field is similar to Fig. 3, which suggests that the right branch is the same with (1, 0) mode of the structure with vacuum holes. Comparing the upper and lower row in Fig. 5, the same character of the near fields for two peaks is that the strong field localizes at the edges along y direction at the entrance and exit. This is due to the SPRs appear at both the upper and substrate surface. The SPRs enhance the field at two ends of the hole and couple the energy into and out of the hole, which is necessary for the transmission enhancement. The effect of film thickness on the transmission has also been investigated. For the film with vacuum holes, the
4
8
z
3 2
0
6
6
4
4
3
2
0 (c) λ=716nm, z=h.
(b) λ=716nm, y=d/2.
(a) λ=716nm, x=d/2.
6
6
2.5 2
5 4
4
1.5 1
3 2
(d) λ=884nm, x=d/2.
2
0.7 0.6
h=200 h=300 h=400 h=500
0.5 0.4 0.3 0.2 0.1
1.8 1.6
h=200 h=300 h=400 h=500
1.4 1.2 1 0.8 0.6 0.4 0.2
2
0.5
1 0 (e) λ=884nm, y=d/2.
where / ¼ Ey ; Ez ; H y ; H z , k x ¼ k 0 sinðhÞ. h is the incident angle of the illuminating light. The parameters for calculation are selected as: The hole width is 200 nm and the thickness of film is 200 nm. The dielectric constant of substrate is 2 ¼ 2:31. The step of the incident angle is 1°. For the vacuum hole array, the dispersion diagram is shown in Fig. 7. One can see that the (1, 0) SPR peak divides into two branches with the increasing of incident angle. Based on Eq. (3), this mode is twofold with i ¼ 1 when h ¼ 0. When h increases the (1, 0) mode shifts to
8
y
2
1
ð7Þ
x
10
5 z
/ðx þ pÞ ¼ expðik x pÞ/ðxÞ;
Normalized zeros−order transmittance
x
transmission spectra are shown in Fig. 6(a) for different film thickness. It can be seen that the transmission peaks decrease swiftly with the metal film thickness increasing. This is because that the energy transmits through the holes in the form of the evanescent wave, which attenuates rapidly with distance increasing, thus the SPR at two surfaces of the metal can not couple well. However, when the holes are filled with the medium with 2 ¼ 3:06, the situation is quite different. As shown in Fig. 6(b), the left transmission peaks do not fall with the thickness increasing. Furthermore, each peak is split into two peaks. For the (1, 0) SPR mode, when the thickness h increases from 200 to 500 nm, the left branch moves to the longer wavelength direction and reaches its maximum in the middle of SPR mode (1, 0), then decreases. While the right branch decreases rapidly with thickness increasing, its behave is quite similar with that of the (1, 0) SPR peak for the structure with vacuum holes. When h = 500 nm, a new WR mode moves into the range of the SPR mode (1, 0). In the SPR mode (1, 1), the peak branches when ðh ¼ 300 and 500 nm, this is also due to the appearance of the WR modes. The influence of the incident angle has also be investigated. For the incline incident, the boundary condition in the x direction is used as
Normalized zero−order transmittance
y
239
(f) λ=884nm, z=h.
Fig. 5. Cross section of Ex amplitude for 2 ¼ 3:06. (a–c) Incident wavelength k = 716 nm and (d–e) incident wavelength k = 841 nm.
0 400
600 800 1000 Wavelength λ (nm)
0 400
600 800 Wavelength λ (nm)
1000
Fig. 6. Transmission spectra of the metal film with different thickness. (a) The vacuum holes array and (b) the holes filled with medium with 2 ¼ 3:06.
240
Y. Chen et al. / Optics Communications 274 (2007) 236–240 0
0.9 0.8
not change with the h increasing, which corresponds the WR mode with m ¼ 1.
Incident angular θ (degree)
5 0.7 10
0.6 SP(—1,0)
0.5
15 0.4 20
0.3 0.2
25 0.1
SP(—1,1)
SP(+1,1) SP(+1,0) 30 500
600
700
800
Wavelength λ (nm)
900
1000
1100
4. Conclusion In conclusion, optical transmission properties of the subwavelength hole array are studied by using the 3DFDTD method. Through filling the hole array with medium with higher refractive index, the effective refractive index of the hole in the metal increases, thus the WR modes can be moved into the SPR range. The collaboration of two mechanisms can enhance the transmission greatly even for the thicker metal film. The corresponding field distributions are also presented for better understanding how the light passes through the holes. Acknowledgements
Fig. 7. Dispersion relation diagram for the vacuum hole array.
0
1.8 1.6
Incident angular θ (degree)
5
The authors thank the anonymous reviewers for valuable comments in improving the manuscript. This work was supported by the National Natural Science Foundation of China (Grant 10604042) and National Basic Research Program of China (Grant 2006CB302901).
1.4
References
10
1.2 1
15
0.8 20
SP(—1,0)
0.4
WR(1)
25 SP(+1,1)
SP(+1,0)
0.6
0.2
SP(—1,1)
30 500
600
700
800
Wavelength λ (nm)
900
1000
1100
Fig. 8. Dispersion relation diagram for the holes filled with medium with 2 ¼ 3:06.
longer wavelength direction, and (+1, 0) mode does to the opposite direction as suggested by the lines. The (1, 1) peaks also divides into ð1; 1Þ branches. The black minimal transmittance regions may be the effect of the SPR. When the holes are filled with medium with 2 ¼ 3:06, the dispersion diagram is shown in Fig. 8. The SPR modes take on similar dispersion relations to that of the vacuum hole array. However, there is a transmission branch that does
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