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Point imaging through one-dimensional graded metal–dielectric photonic crystals
Optik 126 (2015) 400–402
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Point imaging through one-dimensional graded metal–dielectric photonic crystals Gaige Zheng a,b,∗ , Aigen Xie b , Linhua Xu a,b , Min Lai b , Yigen Wu b , Yuzhu Liu c a b c
School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, Jiangsu, China Optics and Photonic Technology Laboratory, Nanjing University of Information Science & Technology, Nanjing 210044, China Paul Scherrer Institute, Villigen CH5232, Switzerland
a r t i c l e
i n f o
Article history: Received 31 December 2013 Accepted 20 September 2014 Keywords: Photonic crystals Metal–dielectric multilayers Point imaging
a b s t r a c t The subwavelength imaging effects in two kinds of one-dimensional (1D) graded metal–dielectric (MD) photonic crystals (PCs) are studied theoretically. When considering the losses, the structure can still act as a superlens. Compared with the structure consists of units with equal thickness, the result shows that the intensity of transmitted waves is larger. Inside the slab, the field is highly concentrated, and the structure behaves as a waveguide, in a scheme that helps the formation of a super-resolved image of an object. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Before the concept of perfect lens was proposed [1], diffraction limited resolution in optical imaging systems had been considered as an unconquerable nature law for a long time. In 2000, Pendry predicted that a flat slab of an isotropic material with both negative permittivity (ε) and permeability () would make a perfect lens capable of focusing both the far and the near field components of a point object, thus achieving super-resolution. However, there are no known natural materials that exhibit negative permittivity and permeability simultaneously. In order to circumvent this shortcoming, Pendry considered the case where electric and magnetic effects may be decoupled such that for TM-polarization only the requirement that ε = −1 needs to be satisfied. This is a welcome relaxation of the requirements especially at the optical frequencies where material may have a negative value for ε, but show no magnetic activity. But the transmittance through a single metal layer is rather low, and gets rapidly worse in the visible range. In order to reduce the losses incurred in the single metal lens, a structure was proposed consisting of alternating layers of metal and dielectric materials having thickness much smaller than the incident wavelength [2–10].
∗ Corresponding author at: School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, Jiangsu, China. Tel.: +86 13851781730. E-mail address: [email protected] (G. Zheng). http://dx.doi.org/10.1016/j.ijleo.2014.09.013 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
In order to change the direction of light propagation, it is necessary to include defects into the structure [11]. By contrast, graded photonic crystals (GPhC) offer a way of curving light without requiring the incorporation of defects [12,13]. Graded photonic crystals (GPCs) are obtained by appropriate gradual modifications of PCs parameters such as the filling factor, the optical index or the lattice period. They rely on gradual modifications of the PhC structure parameters that gradually change their dispersive properties and, thus, result in a smooth reorientation of the light propagation direction as light flows [14]. Following previous theoretical studies which clearly predicted the extreme capability and versatility of GPhCs [12,13], we present an analysis of point imaging in a periodic layered GPhC. In contrast with the previous work, we consider the modification of the PCs parameters. The results show that in a 1D GPhCs, the superlensing effect still exists and the GPhCs enhance versatility in the molding of the flow of light.
2. Structure and methods The structure of one-dimensional (1D) GPhCs is shown in Fig. 1, it is composed of alternating layers of metal and dielectric, periodic with respect to the z-axis, and invariant along the x and y directions. Suppose that the periodic multilayer S contains metal and dielectric materials a and b, i.e., S = (ab)n , where n is the repeated number of the unit “ab”. The thicknesses of two kinds of layers are da and db respectively, the period of the structure is d (d = da + db ). In order to take into account realistic material parameters, we consider the losses in the metal. At the operating wavelength = 400 nm, the
G. Zheng et al. / Optik 126 (2015) 400–402
401
Fig. 1. Geometry of the one-dimensional graded metal–dielectric photonic crystals which is composed of alternating layers of metal and dielectric, periodic with respect to the z-axis, and invariant along the x and y directions. The period is d = da + db .
dielectric constants for silver is εsilver = −4 + 0.2i [15], the dielectric layer is assumed to be HfO2 , and εHfO2 = 4, the permeability is constant (silver = HfO2 = 1). In the considered model, the incident wave is limited only along one direction, thus the incident electric field is given by:
∞
Hiy =
dkx exp [i(kx x + kiz z)] (kx )
(1)
−∞
where (kx ) =
g √ exp 2
−
g 2 (kx − kix ) 4
2
(2)
The incident beam centered about ki = xˆ kix + zˆ kiz = xˆ k0 sin i + zˆ k0 cos i , i is the incident angle, g represents the width of the waist, in the following simulation work, we choose g = 0.6. The incident point source located at 100 nm away in air from the left slab surface. Here we only consider the H-polarization case.
Fig. 3. The field intensity distribution for subwavelength imaging of 1D-MD GPCs (structure 2). The difference from structure 1 is the thickness of the unit.
The calculated transverse intensity distributions for imaging through different structures are presented in Fig. 4, the solid curve is for the point sources, the dash curve is for structure 1, the dot curve is for structure 2, and the dash-dot one (Image 3) is for the structure that was composed of four periods of layers with equal thicknesses (35 nm). One can see that the intensity of Image 1 is lager than the intensities of Image 2 and Image 3. By smooth modifications of the periodicity, GPCs can provide a good level of transmission, in order to increase the intensity of the image, we should choose 1D GPCs rather than PCs contain periods of layers with equal thicknesses. Fig. 5 is the steady-state magnetic field Hy distribution inside the slab for structure 1, one can see that the field is highly concentrated in the structure. The structure displays strong anisotropic
3. Results and discussions First, we consider the case that the thicknesses of the unit cell are increased gradually (structure 1). We choose d1 = 20 nm, d2 = 30 nm, d3 = 40 nm, d4 = 50 nm. The calculated field pattern for the structure with four periods of layers is displayed in Fig. 2, a well-shaped bright image spot is formed (at a distance w from the right surface of the slab). This image spot is peaked at (0.89 m, 0.85 m), very close to the right surface of the slab, the image distance w = 0.31 m. When d1 = 50 nm, d2 = 40 nm, d3 = 30 nm, d4 = 20 nm (structure 2), the field distribution of the structure is shown in Fig. 3, the image spot is peaked at (1.08 m, 0.85 m) and the image distance w = 0.48 m. Fig. 4. The calculated transverse intensity distributions for imaging through different structures. The solid curve is for the point sources, the dash curve is for structure 1, the dot curve is for structure 2, and the dash-dot one (Image 3) is for the structure that was composed of four periods of layers with equal thickness.
Fig. 2. The field intensity distribution for subwavelength imaging of 1D-MD GPCs (structure 1). We use arbitrary units for the power density in all the maps, the left and right arrows indicate source and image, respectively.
Fig. 5. The intensity distribution of the magnetic field Hy inside the slab.
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G. Zheng et al. / Optik 126 (2015) 400–402
properties that make it possible for it behaves as a waveguide, with little or no diffraction taking place, in a scheme that helps the formation of a super-resolved image of an object. 4. Conclusion In summary, the subwavelength effect is found in the 1D-MD GPCs, the existing wave guide mode can be used to amplify the decaying evanescent fields from the source, leading to partial recovery of the evanescent field components of an object. 1D-MD GPCs can be fabricated using classical film deposition technology, they also provide an opportunity to expand the exploration of wave propagation phenomena in metals, both in the linear and nonlinear regimes. Acknowledgements This work is partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 13KJB140006). The authors would especially like to thank Dr. Zhang Wei from Nanjing University of Science & Technology for his useful discussions. References [1] J.B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett. 85 (2000) 3966–3969.
[2] S.A. Ramakrishna, J.B. Pendry, M.C.K. Wiltshire, W.J. Stewart, Imaging the nearfield, J. Mod. Opt. 50 (2003) 1419–1430. [3] P.A. Belov, H. Yang, Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal–dielectric structure operating in the canalization regime, Phys. Rev. B 73 (2006) 113110. [4] H. Shin, S.H. Fan, All-angle negative refraction and evanescent wave amplification using one-dimensional metallo-dielectric photonic crystals, Appl. Phys. Lett. 89 (2006) 151102. [5] M. Scalora, G. D’Aguanno, N. Mattiucci, M.J. Bloemer, D. Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M.G. Cappeddu, M. Fowler, J.W. Haus, Negative refraction and sub-wavelength focusing in the visible range using transparent metallo-dielectric stacks, Opt. Express 15 (2007) 508–523. [6] J.L. Zhang, H.T. Jiang, B. Gralak, S. Enoch, G. Tayeb, M. Lequime, Towards-1 effective index with one-dimensional metal–dielectric metamaterial: a quantitative analysis of the role of absorption losses, Opt. Express 15 (2007) 7720–7729. [7] K.J. Webb, M. Yang, Subwavelength imaging with a multilayer silver film structure, Opt. Lett. 31 (2006) 2130–2132. [8] B. Wood, J.B. Pendry, D.P. Tsai, Directed subwavelength imaging using a layered metal–dielectric system, Phys. Rev. B 74 (2006) 115116. [9] X. Li, S.L. He, Y. Jin, Subwavelength focusing with a multilayered Fabry–Perot structure at optical frequencies, Phys. Rev. B 75 (2007) 045103. [10] M.Y. Li, P.F. Gu, J.L. Zhang, Y.Y. Li, X. Liu, Subwavelength imaging in a onedimensional metal–dielectric structure, Chin. Phys. Lett. 24 (2007) 3539–3542. [11] W. Prather, S. Shi, J. Murakowski, G.J. Schneider, A. Sharkawy, C. Chen, B.L. Miao, R. Martin, Self-collimation in photonic crystal structures: a new paradigm for applications and device development, J. Phys. D 40 (2007) 2635. [12] E. Centeno, D. Cassagne, Graded photonic crystals, Opt. Lett. 30 (2005) 2278. [13] E. Centeno, D. Cassagne, J.P. Albert, Mirage and superbending effect in twodimensional graded photonic crystals, Phys. Rev. B 73 (2006) 235119. [14] E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, J.M. Lourtioz1, Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect, Appl. Phys. Lett. 92 (2008) 133501. [15] P.B. Johnson, R.W. Christy, Optical constants of the noble metals, Phys. Rev. B 6 (1972) 4370–4379.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Point imaging through one-dimensional graded metal–dielectric photonic crystals Gaige Zheng a,b,∗ , Aigen Xie b , Linhua Xu a,b , Min Lai b , Yigen Wu b , Yuzhu Liu c a b c
School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, Jiangsu, China Optics and Photonic Technology Laboratory, Nanjing University of Information Science & Technology, Nanjing 210044, China Paul Scherrer Institute, Villigen CH5232, Switzerland
a r t i c l e
i n f o
Article history: Received 31 December 2013 Accepted 20 September 2014 Keywords: Photonic crystals Metal–dielectric multilayers Point imaging
a b s t r a c t The subwavelength imaging effects in two kinds of one-dimensional (1D) graded metal–dielectric (MD) photonic crystals (PCs) are studied theoretically. When considering the losses, the structure can still act as a superlens. Compared with the structure consists of units with equal thickness, the result shows that the intensity of transmitted waves is larger. Inside the slab, the field is highly concentrated, and the structure behaves as a waveguide, in a scheme that helps the formation of a super-resolved image of an object. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Before the concept of perfect lens was proposed [1], diffraction limited resolution in optical imaging systems had been considered as an unconquerable nature law for a long time. In 2000, Pendry predicted that a flat slab of an isotropic material with both negative permittivity (ε) and permeability () would make a perfect lens capable of focusing both the far and the near field components of a point object, thus achieving super-resolution. However, there are no known natural materials that exhibit negative permittivity and permeability simultaneously. In order to circumvent this shortcoming, Pendry considered the case where electric and magnetic effects may be decoupled such that for TM-polarization only the requirement that ε = −1 needs to be satisfied. This is a welcome relaxation of the requirements especially at the optical frequencies where material may have a negative value for ε, but show no magnetic activity. But the transmittance through a single metal layer is rather low, and gets rapidly worse in the visible range. In order to reduce the losses incurred in the single metal lens, a structure was proposed consisting of alternating layers of metal and dielectric materials having thickness much smaller than the incident wavelength [2–10].
∗ Corresponding author at: School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, Jiangsu, China. Tel.: +86 13851781730. E-mail address: [email protected] (G. Zheng). http://dx.doi.org/10.1016/j.ijleo.2014.09.013 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
In order to change the direction of light propagation, it is necessary to include defects into the structure [11]. By contrast, graded photonic crystals (GPhC) offer a way of curving light without requiring the incorporation of defects [12,13]. Graded photonic crystals (GPCs) are obtained by appropriate gradual modifications of PCs parameters such as the filling factor, the optical index or the lattice period. They rely on gradual modifications of the PhC structure parameters that gradually change their dispersive properties and, thus, result in a smooth reorientation of the light propagation direction as light flows [14]. Following previous theoretical studies which clearly predicted the extreme capability and versatility of GPhCs [12,13], we present an analysis of point imaging in a periodic layered GPhC. In contrast with the previous work, we consider the modification of the PCs parameters. The results show that in a 1D GPhCs, the superlensing effect still exists and the GPhCs enhance versatility in the molding of the flow of light.
2. Structure and methods The structure of one-dimensional (1D) GPhCs is shown in Fig. 1, it is composed of alternating layers of metal and dielectric, periodic with respect to the z-axis, and invariant along the x and y directions. Suppose that the periodic multilayer S contains metal and dielectric materials a and b, i.e., S = (ab)n , where n is the repeated number of the unit “ab”. The thicknesses of two kinds of layers are da and db respectively, the period of the structure is d (d = da + db ). In order to take into account realistic material parameters, we consider the losses in the metal. At the operating wavelength = 400 nm, the
G. Zheng et al. / Optik 126 (2015) 400–402
401
Fig. 1. Geometry of the one-dimensional graded metal–dielectric photonic crystals which is composed of alternating layers of metal and dielectric, periodic with respect to the z-axis, and invariant along the x and y directions. The period is d = da + db .
dielectric constants for silver is εsilver = −4 + 0.2i [15], the dielectric layer is assumed to be HfO2 , and εHfO2 = 4, the permeability is constant (silver = HfO2 = 1). In the considered model, the incident wave is limited only along one direction, thus the incident electric field is given by:
∞
Hiy =
dkx exp [i(kx x + kiz z)] (kx )
(1)
−∞
where (kx ) =
g √ exp 2
−
g 2 (kx − kix ) 4
2
(2)
The incident beam centered about ki = xˆ kix + zˆ kiz = xˆ k0 sin i + zˆ k0 cos i , i is the incident angle, g represents the width of the waist, in the following simulation work, we choose g = 0.6. The incident point source located at 100 nm away in air from the left slab surface. Here we only consider the H-polarization case.
Fig. 3. The field intensity distribution for subwavelength imaging of 1D-MD GPCs (structure 2). The difference from structure 1 is the thickness of the unit.
The calculated transverse intensity distributions for imaging through different structures are presented in Fig. 4, the solid curve is for the point sources, the dash curve is for structure 1, the dot curve is for structure 2, and the dash-dot one (Image 3) is for the structure that was composed of four periods of layers with equal thicknesses (35 nm). One can see that the intensity of Image 1 is lager than the intensities of Image 2 and Image 3. By smooth modifications of the periodicity, GPCs can provide a good level of transmission, in order to increase the intensity of the image, we should choose 1D GPCs rather than PCs contain periods of layers with equal thicknesses. Fig. 5 is the steady-state magnetic field Hy distribution inside the slab for structure 1, one can see that the field is highly concentrated in the structure. The structure displays strong anisotropic
3. Results and discussions First, we consider the case that the thicknesses of the unit cell are increased gradually (structure 1). We choose d1 = 20 nm, d2 = 30 nm, d3 = 40 nm, d4 = 50 nm. The calculated field pattern for the structure with four periods of layers is displayed in Fig. 2, a well-shaped bright image spot is formed (at a distance w from the right surface of the slab). This image spot is peaked at (0.89 m, 0.85 m), very close to the right surface of the slab, the image distance w = 0.31 m. When d1 = 50 nm, d2 = 40 nm, d3 = 30 nm, d4 = 20 nm (structure 2), the field distribution of the structure is shown in Fig. 3, the image spot is peaked at (1.08 m, 0.85 m) and the image distance w = 0.48 m. Fig. 4. The calculated transverse intensity distributions for imaging through different structures. The solid curve is for the point sources, the dash curve is for structure 1, the dot curve is for structure 2, and the dash-dot one (Image 3) is for the structure that was composed of four periods of layers with equal thickness.
Fig. 2. The field intensity distribution for subwavelength imaging of 1D-MD GPCs (structure 1). We use arbitrary units for the power density in all the maps, the left and right arrows indicate source and image, respectively.
Fig. 5. The intensity distribution of the magnetic field Hy inside the slab.
402
G. Zheng et al. / Optik 126 (2015) 400–402
properties that make it possible for it behaves as a waveguide, with little or no diffraction taking place, in a scheme that helps the formation of a super-resolved image of an object. 4. Conclusion In summary, the subwavelength effect is found in the 1D-MD GPCs, the existing wave guide mode can be used to amplify the decaying evanescent fields from the source, leading to partial recovery of the evanescent field components of an object. 1D-MD GPCs can be fabricated using classical film deposition technology, they also provide an opportunity to expand the exploration of wave propagation phenomena in metals, both in the linear and nonlinear regimes. Acknowledgements This work is partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 13KJB140006). The authors would especially like to thank Dr. Zhang Wei from Nanjing University of Science & Technology for his useful discussions. References [1] J.B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett. 85 (2000) 3966–3969.
[2] S.A. Ramakrishna, J.B. Pendry, M.C.K. Wiltshire, W.J. Stewart, Imaging the nearfield, J. Mod. Opt. 50 (2003) 1419–1430. [3] P.A. Belov, H. Yang, Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal–dielectric structure operating in the canalization regime, Phys. Rev. B 73 (2006) 113110. [4] H. Shin, S.H. Fan, All-angle negative refraction and evanescent wave amplification using one-dimensional metallo-dielectric photonic crystals, Appl. Phys. Lett. 89 (2006) 151102. [5] M. Scalora, G. D’Aguanno, N. Mattiucci, M.J. Bloemer, D. Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M.G. Cappeddu, M. Fowler, J.W. Haus, Negative refraction and sub-wavelength focusing in the visible range using transparent metallo-dielectric stacks, Opt. Express 15 (2007) 508–523. [6] J.L. Zhang, H.T. Jiang, B. Gralak, S. Enoch, G. Tayeb, M. Lequime, Towards-1 effective index with one-dimensional metal–dielectric metamaterial: a quantitative analysis of the role of absorption losses, Opt. Express 15 (2007) 7720–7729. [7] K.J. Webb, M. Yang, Subwavelength imaging with a multilayer silver film structure, Opt. Lett. 31 (2006) 2130–2132. [8] B. Wood, J.B. Pendry, D.P. Tsai, Directed subwavelength imaging using a layered metal–dielectric system, Phys. Rev. B 74 (2006) 115116. [9] X. Li, S.L. He, Y. Jin, Subwavelength focusing with a multilayered Fabry–Perot structure at optical frequencies, Phys. Rev. B 75 (2007) 045103. [10] M.Y. Li, P.F. Gu, J.L. Zhang, Y.Y. Li, X. Liu, Subwavelength imaging in a onedimensional metal–dielectric structure, Chin. Phys. Lett. 24 (2007) 3539–3542. [11] W. Prather, S. Shi, J. Murakowski, G.J. Schneider, A. Sharkawy, C. Chen, B.L. Miao, R. Martin, Self-collimation in photonic crystal structures: a new paradigm for applications and device development, J. Phys. D 40 (2007) 2635. [12] E. Centeno, D. Cassagne, Graded photonic crystals, Opt. Lett. 30 (2005) 2278. [13] E. Centeno, D. Cassagne, J.P. Albert, Mirage and superbending effect in twodimensional graded photonic crystals, Phys. Rev. B 73 (2006) 235119. [14] E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, J.M. Lourtioz1, Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect, Appl. Phys. Lett. 92 (2008) 133501. [15] P.B. Johnson, R.W. Christy, Optical constants of the noble metals, Phys. Rev. B 6 (1972) 4370–4379.