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Image transfer properties by photonic crystal slab with negative refractive index
Solid State Communications 146 (2008) 192–196 www.elsevier.com/locate/ssc
Image transfer properties by photonic crystal slab with negative refractive index Hongbo Chen, Xiaoshuang Chen ∗ , Renlong Zhou, Wei Lu ∗ National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, 200083 Shanghai, People’s Republic of China Received 10 August 2007; received in revised form 28 November 2007; accepted 17 January 2008 by X.C. Shen Available online 31 January 2008
Abstract We have studied the properties of image transferred by photonic crystal (PhC) slab with negative refractive index n = −1 and confirmed the negative refractive phonomenon, but not found the saturated image properties as expected. It is found that real images will not be formed when the source distance larger than the thickness of PhC, and the transferred images are virtual images. Furthermore, comparing the quality of images transferred by a PhC slab and a cascaded stack of photonic crystal slab (CSPS), we found that the transferred images are distorted in both situations. The image resolution is good along the direction parallel to the slab interface, but bad along the direction normal to the slab interface. Simulation results show that the image formed by a CSPS is no better than a PhC slab. c 2008 Elsevier Ltd. All rights reserved.
PACS: 41.20.Jb; 42.25.Bs; 78.20.-e; 42.70.Qs Keywords: A. Photonic crystal; A. Negative refractive; A. Virtual image; A. Transferred image
1. Instruction As was shown by Vesselago over 30 years ago [1], the left-handed materials (LHM) or negative refractive-index materials (NIMs), characterized by simultaneously negative permittivity and permeability, have recently attracted renewed interest because of the experimental progress [2–5]. Unusual electromagnetic phenomena, such as negative refraction, reversed Doppler shift and reversed Cerenkov radiation have been shown. Thus, the existence of LHM or NIM opens the door for new approaches to a variety of applications, such as a perfect lens that overcomes the diffraction limit by regenerating the entire spectrum of the source at the image plane [6,7]. Due to the absence of these materials in nature, various approaches have been proposed to fabricate the equivalent metamaterial. In one approach, a metamaterial with negative refraction, made of a periodic array of split ring resonators and thin wires, was first demonstrated experimentally at microwave frequency [2]. ∗ Corresponding author. Tel.: +86 21 65420850x24309; fax: +86 21 65830734. E-mail addresses: [email protected] (X. Chen), [email protected] (W. Lu).
c 2008 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2008.01.019
In 2000, Notomi has shown theoretically that photonic crystals, periodic dielectric composite structures with an electric permittivity ε > 0 and a magnetic permeability µ = 1, at the frequency near the bandgap behave as if they have an effective refractive index. At some frequencies of the bands, the effective refractive index can be negative, and the wave vector of the incident waves and the group velocity of the transmitted waves fall into the opposite sides of the interface by analysing the equifrequency-surface contours (EFS) of the band structures [8]. As a result, in these frequencies, PhCs are considered as an alternative for the realization of LHMs. Imaging effects have been demonstrated experimentally [9–12] as well as theoretically [8,13–17] for triangular-lattice and square-lattice PhCs [18]. It has been shown that the photonic crystal-based slab has ability to focus light. Veselago pointed out that a flat plate made of a LHM of thickness D with refractive index n = −1 and situated in vacuum (n = 1) can focus radiation of a point source P positioned at a distance L < D from one side of the slab to a point P 0 located at a distance L 0 = D − L from the other side of the slab [1]. Therefore, an image was transferred with a distance being not greater than 2D, due to the Veselago’s relationship. However, recently Guilin et al. have referred that a
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is determined by its parameters, such as the air holes, lattice constant and interface. Ref. [20] shows the existence of surface modes in a 2D PhC slab is one of the reasons for the subwavelength lens. Some results have shown the transmission of light is strongly depends on the orientation of the PhC and on the terminations of the air/slab interfaces. Transmission is maximal if the surface normal of the slabs is along the Γ M direction [21] and the air holes at the air/slab interfaces are cut [14,15]. So all the PhC slab considered here have a cut air/slab interface and the interface normal of slabs are along Γ M direction. Numerical simulations were performed using finite-difference time-domain (FDTD) method [22]. Fig. 1. Equifrequency surface plot of TM modes for a triangular lattice photonic crystal.
real image can still be formed from source distance L > D until to L = 2D [17]. And in this regime the image distance changes very little with changing the source distance L. Therefore, they have described it as the saturated image phenomenon. In addition, it can form a better image applying the principle of image transferred by a CSPS than using a PhC slab [13]. To our knowledge, the intrinsic reason is not still understood about the images transferred by PhC slabs with the source distance L > D. Thus, in this paper, we discuss the properties of the images transferred by PhC slabs. It is found that with the source distance increasing to L > D, there are not real images being formed, and the transferred images are virtual images. Furthermore, comparing the quality of images transferred by a PhC slab and a CSPS, we found that the transferred images are distorted in both situations. When comparing the widening of the intensity distributions for the transferred images of a PhC slab and a CSPS in the direction normal to the slab interface, we find that the image formed by a CSPS is no better than a PhC slab. 2. Parameters for the calculation In our study, we work on a 2D photonic-crystal consisting of a hexagonal lattice of circular air-holes in a dielectric with refractive index of 3.6, and r = 0.4a, here r and a are the radius of the air hole and the lattice constant, respectively. For simplicity, only the TM-polarized point source is considered. It has been found that the structure exhibits negative refraction at the second band of the photonic bands. To visualize and analyse the propagation behaviour of light in a photonic crystal, an EFS in kspace of the photonic bands can be introduced, where gradient vectors give the group velocities of the photonic modes. The shape of the EFS corresponding to a frequency in the second band for the TM mode is depicted in Fig. 1. One can then define an effective refractive index from the radius of the EFS with Snell’s law [8,19] and use it to describe the light refraction in the photonic crystal. For ω = 0.30(a/λ) with λ being the wavelength, the circle is almost circle, so the effective refractive index of the PhC is hardly dependent of θ and is almost equal to −1. A LHM with n = −1 is thought to be a perfect lens. However, a PhC is a model of LHM, the behaviour of which
3. Image transferred by a PhC slab Sun et al. have investigated the properties of the image transferred by a PhC slab. They have found that a real image can be formed when the source distance D < L < 2D (D is the slab thickness), and in this regime the image is saturated, namely the image distance L 0 changes very little with changing the source distance L [17]. In order to form a clear image we choose a thick PhC slab with 20 layers (the thickness D is about 17.72a). Firstly, we consider a near-field image. The source distance is L = 8.86a, which is about 0.5D. A typical result of the E z field pattern is shown in Fig. 2a. One can find a quite high quality image formed in the opposite side of the slab. It is found that the image distance is L 0 = 7.94a, which is about 0.5D. The sum of source distance L and image distance L 0 , L + L 0 = 8.86a + 7.94a = 16.8a, is slightly smaller than D. It may be a slight deviation of the effective refractive index from −1. Then, we investigate the case for which the source distance L is equal to the slab thickness D. Fig. 2b shows the calculated field pattern. A clear point source is formed almost at the air/slab interface. Next, we continue increasing the source distance to 1.5D and 2D. The electric fields patterns are shown in Fig. 2c and d. On the image field, one can find that the wavefront almost spread from the virtual image point A and B, respectively. The imaging behaviour is different from that as Guilin et al. have declared. We also have changed the source distance L from 0.1D to 3D to see the relation between L and L 0 , as shown in Fig. 3. With increasing L from 0.1D to 3D, L 0 decreases from 0.82D to −2.06D and the correlativity between L 0 and L is about L 0 + L ≈ 1.02D in accordance with the Veselago’s relationship. Changing the slab thickness into 16, 13, 8, and 5 layers, we still find the same relations between L and L 0 . Obviously the PhCs have the imaging properties as LHM and are a candidate for the LHM in the applications. The saturated imaging properties are not found by changing the source distance L. Image quality is an important issue for evaluating the negative refractive-index materials. In previous studies, many authors use the Full-Width-Half-Maximum (FWHM) of the image intensity along the direction that is parallel to the slab interface, to investigate image resolution [17]. The image intensity distribution of Fig. 2a is plotted not only parallel to the interface but also along the normal direction of the slab interface, as shown of curve 1 and curve 2 in Fig. 4. We find
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Fig. 2. E z field patterns of a point source placed at 0.5D, 1D, 1.5D and 2D in (a), (b), (c), and (d).
Fig. 3. The relation between normalized source and image distances for a PhC slabs with 20-layer.
that the image is distorted along the propagation direction. In all of our calculations, the image intensities are normalized to the source intensities. The spatial resolution of the image, defined as the ratio of the full width at half maximum to λ, is different in the two directions. One can say that the image is high quality from curve 1 because the image resolution is only 0.46λ, well below the conventional diffraction limit. We also can say the image is awful from curve 2 because the amplitude of the side peak is very strong, almost equivalent to the intensity of the image centre. Therefore, to analysis the resolution of an image one should show the image intensity distribution along the directions, not only parallel but also normal to the PhC slab interface. For different thickness of PhC slabs, 8, 12, 60, and 120 layers, the image intensity distribution, parallel and normal to the interface of PhC slab, are plotted in Fig. 5a, and b, respectively. With increasing the thickness of the slab, the intensity of the image decreases rapidly, but the image resolutions, parallel to the PhC slab interface, change very little. The resolution of the worst image is 0.76λ (curve 4) in Fig. 5a, which is still below the conventional diffraction limit. From Fig. 5b, it can be seen that the resolution along the normal
Fig. 4. The calculated radiation intensity across the image centre of Fig. 2a: curve 1 is along the direction parallel to the interface of PhC slab, curve 2 is along the normal direction of the interface.
direction becomes worse with increasing the slab thickness. One could not almost distinguish which of the imaging peaks is related to the main peak of the source when the thickness is larger than 60 layers. It means the electric intensity along the normal direction is elongated. In such a criterion, the image formed by a PhC slab does not possess high quality, because the image is elongated in the direction of propagation. 4. Image transferred by cascaded PhC slabs To increase the distance between source P and image P 0 there are two possible ways: increasing the slab thickness D or applying CSPS. In this section we discuss the quality of the image transferred by CSPS and compare the image quality between the two situations. The calculation parameters are the same as mentioned previously. Firstly, we consider a cascaded stack composed of two PhC slabs with each slab being 20 layers. The two slabs are separated by an air layer of thickness 18.28a. The source is
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Fig. 5. The calculated radiation intensity across the image centre, curve 1, 2, 3, and 4 corresponding to 8, 12, 60 and 120 layer PhC slabs, (a) parallel to the slab interface; (b) normal to the slab interface.
Fig. 6. (a) Schematic picture depicting the transfer of a source by two stack PhC slabs. (b) E z field pattern of a point source transferred by two stack Phc slabs.
Fig. 7. The calculated radiation intensity across the image centre of Fig. 5b: curve 1 is along the direction parallel to the interface of PhC slab, curve 2 is along the normal direction of the interface.
placed at a distance L = 9.14a from the left surface of the first PhC slab. According to Veselago relation, the source should be focused inside the two PhC slabs, between the two slabs and at the right of the second slabs as shown in Fig. 6a. Applying FDTD method, we calculate the field pattern as shown in Fig. 6b. One can see that there are four focusing images as predicted, except that the images inside PhC slabs are not very obvious. The distance between P and P 0 is about 70.0a, which is approximately equal to double the thickness of the two slabs D1 + D2 = 35.4a. The image intensity distribution, parallel and normal to the PhC slab interface is plotted in Fig. 7. The quality of the image is similar to that of a PhC slab: elongated along the normal direction. The resolution along the direction parallel to the interface is 0.46λ, and the intensity of image is only 8.2% of the source.
In addition, we double the thickness of one PhC slab to realize the same focusing image at the same distance, as shown in Fig. 8a. A 40 layers PhC slab with cutting interface is used and it is about double thickness of that in Fig. 5b. The source is located at a distance L = 18.5a from the left surface of the slab. The simulation result of electric field pattern is shown in Fig. 8b. One can find that a focused image is formed on the other hand of the slab. The sum of L and L 0 is about 34.5a that is also approximately equal to the thickness of PhC slab D = 35.0a. And the distance between P and P 0 is about 69.5a. Therefore, the source can form an image at the same distance in the two situations. The image in Fig. 8b is also elongated in the direction of propagation, which has been discussed in Section 3, and the intensity is 19.1%, which is stronger than that in Fig. 6b, of the source. Increasing the number of slabs in the stack, we consider a cascaded stack composed of three PhC slabs with 20 layers and cutting interface. Simulation result of electric field pattern is shown in Fig. 9. The resolution of image along the normal direction deteriorated more seriously than that in Fig. 6b. And the intensity of image is much weak (2.6% of the source). Using a 60 layers PhC slab, we also make a comparison. The resolution is shown at curve 3 in Fig. 5 and the imaging intensity is 12.6% of the source. As a result, the transferred image does not possess high quality when applying CSPS because the image is elongated in the normal direction and the intensity is weaken. With increasing the number of slabs in the stack, the resolution along the normal direction become worse and the intensity become weaker. The image formed by a CSPS is no better than a PhC slab.
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Fig. 8. (a) Schematic picture depicting the transfer of a source by a PhC slab. (b) E z field pattern of a point transferred by a Phc slab.
References
Fig. 9. E z field patterns of a point source transferred by three stack PhC slabs.
5. Conclusions We have studied the properties of image transferred by PhC slab with negative refractive index n = −1 and confirmed the negative refractive phonomenon. But the saturated image properties are not found. In PhC slab, the resolution of image is good along the direction parallel to the slab interface, but awful along the direction normal to the slab interface. With increasing the slab thickness, the distortion becomes more seriously and the intensity of the image becomes weaker. The images transferred by CSPS are also distorted similar to that transferred by PhC slab. The intensity of image from CSPS is weaker than that formed by a PhC slab because of the reflection of many air/slab interfaces. The images formed by a CSPS are no better than that formed by a PhC slab. They are both elongated in the propagating direction. Acknowledgements This work is supported in part by Chinese National Key Basic Research Special Fund (2007CB613206 and 2006CB921704), Chinese National Science Fund (60576068, 10725418), and the computational support from Shanghai Super-computer Center.
[1] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [2] R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77. [3] Hongsheng Chen, Lixin Ran, Jiantao Huangfu, Xianmin Zhang, Kangsheng Chen, Tomasz M. Grzegorczyk, Jin Au Kong, Phys. Rev. E 70 (2004) 057605. [4] Ertugrul Cubukcu, Koray Aydin, Ekmel Ozbay, Stavroula Foteinopoulou, Costas M. Soukoulis, Nature 423 (2003) 604. [5] Patanjali V. Parimi, Wentao T. Lu, Plarenta Vodo and Srinivas Sridhar, Nature 426 (2003) 404. [6] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. [7] Nicholas Fang, Hyesog Lee, Cheng Sun, Xiang Zhang, Science 308 (2005) 534. [8] M. Notomi, Phys. Rev. B 62 (2000) 10696. [9] A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thyl’en, A. Talneau, S. Anand, Phys. Rev. Lett. 93 (2004) 073902. [10] A. Martinez, H. Miguez, A. Griol, J. Marti, Phys. Rev. B 69 (2004) 165119. [11] K. Guven, K. Aydin, K.B. Alici, C.M. Soukoulis, E. Ozbay, Phys. Rev. B 70 (2004) 205125. [12] E. Ozbay, I. Bulu, K. Aydin, H. Caglayan, K.B. Alici, K. Guven, Laser Phys. 15 (2005) 217. [13] C. Shen, K. Michielsen, H. De Raedt, Opt. Express 14 (2006) 879. [14] S. Xiao, M. Qiu, Z. Ruan, S. He, Appl. Phys. Lett. 85 (2004) 4269. [15] X. Wang, K. Kempa, Phys. Rev. B 71 (2005) 085101. [16] X. Wang, Z.F. Ren, K. Kempa, Appl. Phys. Lett. 86 (2005) 061105. [17] G. Sun, A.S. Jugessur, A.G. Kirk, Opt. Express 14 (2005) 6755. [18] C. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 65 (2002) 201104. [19] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, C.M. Soukoulis, Nature 423 (2003) 604. [20] C. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 68 (2003) 045115. [21] Z. Ruan, M. Qiu, S. Xiao, S. He, L. Thyl’en, Phys. Rev. B 71 (2005) 045111. [22] Ardavan Farjadpour, David Roundy, Alejandro Rodriguez, Mihai Ibanescu, Peter Bermel, J.D. Joannopoulos, Steven G. Johnson, Geoffrey Burr, Opt. Lett. 31 (20) (2006) 2972.
Image transfer properties by photonic crystal slab with negative refractive index Hongbo Chen, Xiaoshuang Chen ∗ , Renlong Zhou, Wei Lu ∗ National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, 200083 Shanghai, People’s Republic of China Received 10 August 2007; received in revised form 28 November 2007; accepted 17 January 2008 by X.C. Shen Available online 31 January 2008
Abstract We have studied the properties of image transferred by photonic crystal (PhC) slab with negative refractive index n = −1 and confirmed the negative refractive phonomenon, but not found the saturated image properties as expected. It is found that real images will not be formed when the source distance larger than the thickness of PhC, and the transferred images are virtual images. Furthermore, comparing the quality of images transferred by a PhC slab and a cascaded stack of photonic crystal slab (CSPS), we found that the transferred images are distorted in both situations. The image resolution is good along the direction parallel to the slab interface, but bad along the direction normal to the slab interface. Simulation results show that the image formed by a CSPS is no better than a PhC slab. c 2008 Elsevier Ltd. All rights reserved.
PACS: 41.20.Jb; 42.25.Bs; 78.20.-e; 42.70.Qs Keywords: A. Photonic crystal; A. Negative refractive; A. Virtual image; A. Transferred image
1. Instruction As was shown by Vesselago over 30 years ago [1], the left-handed materials (LHM) or negative refractive-index materials (NIMs), characterized by simultaneously negative permittivity and permeability, have recently attracted renewed interest because of the experimental progress [2–5]. Unusual electromagnetic phenomena, such as negative refraction, reversed Doppler shift and reversed Cerenkov radiation have been shown. Thus, the existence of LHM or NIM opens the door for new approaches to a variety of applications, such as a perfect lens that overcomes the diffraction limit by regenerating the entire spectrum of the source at the image plane [6,7]. Due to the absence of these materials in nature, various approaches have been proposed to fabricate the equivalent metamaterial. In one approach, a metamaterial with negative refraction, made of a periodic array of split ring resonators and thin wires, was first demonstrated experimentally at microwave frequency [2]. ∗ Corresponding author. Tel.: +86 21 65420850x24309; fax: +86 21 65830734. E-mail addresses: [email protected] (X. Chen), [email protected] (W. Lu).
c 2008 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2008.01.019
In 2000, Notomi has shown theoretically that photonic crystals, periodic dielectric composite structures with an electric permittivity ε > 0 and a magnetic permeability µ = 1, at the frequency near the bandgap behave as if they have an effective refractive index. At some frequencies of the bands, the effective refractive index can be negative, and the wave vector of the incident waves and the group velocity of the transmitted waves fall into the opposite sides of the interface by analysing the equifrequency-surface contours (EFS) of the band structures [8]. As a result, in these frequencies, PhCs are considered as an alternative for the realization of LHMs. Imaging effects have been demonstrated experimentally [9–12] as well as theoretically [8,13–17] for triangular-lattice and square-lattice PhCs [18]. It has been shown that the photonic crystal-based slab has ability to focus light. Veselago pointed out that a flat plate made of a LHM of thickness D with refractive index n = −1 and situated in vacuum (n = 1) can focus radiation of a point source P positioned at a distance L < D from one side of the slab to a point P 0 located at a distance L 0 = D − L from the other side of the slab [1]. Therefore, an image was transferred with a distance being not greater than 2D, due to the Veselago’s relationship. However, recently Guilin et al. have referred that a
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is determined by its parameters, such as the air holes, lattice constant and interface. Ref. [20] shows the existence of surface modes in a 2D PhC slab is one of the reasons for the subwavelength lens. Some results have shown the transmission of light is strongly depends on the orientation of the PhC and on the terminations of the air/slab interfaces. Transmission is maximal if the surface normal of the slabs is along the Γ M direction [21] and the air holes at the air/slab interfaces are cut [14,15]. So all the PhC slab considered here have a cut air/slab interface and the interface normal of slabs are along Γ M direction. Numerical simulations were performed using finite-difference time-domain (FDTD) method [22]. Fig. 1. Equifrequency surface plot of TM modes for a triangular lattice photonic crystal.
real image can still be formed from source distance L > D until to L = 2D [17]. And in this regime the image distance changes very little with changing the source distance L. Therefore, they have described it as the saturated image phenomenon. In addition, it can form a better image applying the principle of image transferred by a CSPS than using a PhC slab [13]. To our knowledge, the intrinsic reason is not still understood about the images transferred by PhC slabs with the source distance L > D. Thus, in this paper, we discuss the properties of the images transferred by PhC slabs. It is found that with the source distance increasing to L > D, there are not real images being formed, and the transferred images are virtual images. Furthermore, comparing the quality of images transferred by a PhC slab and a CSPS, we found that the transferred images are distorted in both situations. When comparing the widening of the intensity distributions for the transferred images of a PhC slab and a CSPS in the direction normal to the slab interface, we find that the image formed by a CSPS is no better than a PhC slab. 2. Parameters for the calculation In our study, we work on a 2D photonic-crystal consisting of a hexagonal lattice of circular air-holes in a dielectric with refractive index of 3.6, and r = 0.4a, here r and a are the radius of the air hole and the lattice constant, respectively. For simplicity, only the TM-polarized point source is considered. It has been found that the structure exhibits negative refraction at the second band of the photonic bands. To visualize and analyse the propagation behaviour of light in a photonic crystal, an EFS in kspace of the photonic bands can be introduced, where gradient vectors give the group velocities of the photonic modes. The shape of the EFS corresponding to a frequency in the second band for the TM mode is depicted in Fig. 1. One can then define an effective refractive index from the radius of the EFS with Snell’s law [8,19] and use it to describe the light refraction in the photonic crystal. For ω = 0.30(a/λ) with λ being the wavelength, the circle is almost circle, so the effective refractive index of the PhC is hardly dependent of θ and is almost equal to −1. A LHM with n = −1 is thought to be a perfect lens. However, a PhC is a model of LHM, the behaviour of which
3. Image transferred by a PhC slab Sun et al. have investigated the properties of the image transferred by a PhC slab. They have found that a real image can be formed when the source distance D < L < 2D (D is the slab thickness), and in this regime the image is saturated, namely the image distance L 0 changes very little with changing the source distance L [17]. In order to form a clear image we choose a thick PhC slab with 20 layers (the thickness D is about 17.72a). Firstly, we consider a near-field image. The source distance is L = 8.86a, which is about 0.5D. A typical result of the E z field pattern is shown in Fig. 2a. One can find a quite high quality image formed in the opposite side of the slab. It is found that the image distance is L 0 = 7.94a, which is about 0.5D. The sum of source distance L and image distance L 0 , L + L 0 = 8.86a + 7.94a = 16.8a, is slightly smaller than D. It may be a slight deviation of the effective refractive index from −1. Then, we investigate the case for which the source distance L is equal to the slab thickness D. Fig. 2b shows the calculated field pattern. A clear point source is formed almost at the air/slab interface. Next, we continue increasing the source distance to 1.5D and 2D. The electric fields patterns are shown in Fig. 2c and d. On the image field, one can find that the wavefront almost spread from the virtual image point A and B, respectively. The imaging behaviour is different from that as Guilin et al. have declared. We also have changed the source distance L from 0.1D to 3D to see the relation between L and L 0 , as shown in Fig. 3. With increasing L from 0.1D to 3D, L 0 decreases from 0.82D to −2.06D and the correlativity between L 0 and L is about L 0 + L ≈ 1.02D in accordance with the Veselago’s relationship. Changing the slab thickness into 16, 13, 8, and 5 layers, we still find the same relations between L and L 0 . Obviously the PhCs have the imaging properties as LHM and are a candidate for the LHM in the applications. The saturated imaging properties are not found by changing the source distance L. Image quality is an important issue for evaluating the negative refractive-index materials. In previous studies, many authors use the Full-Width-Half-Maximum (FWHM) of the image intensity along the direction that is parallel to the slab interface, to investigate image resolution [17]. The image intensity distribution of Fig. 2a is plotted not only parallel to the interface but also along the normal direction of the slab interface, as shown of curve 1 and curve 2 in Fig. 4. We find
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Fig. 2. E z field patterns of a point source placed at 0.5D, 1D, 1.5D and 2D in (a), (b), (c), and (d).
Fig. 3. The relation between normalized source and image distances for a PhC slabs with 20-layer.
that the image is distorted along the propagation direction. In all of our calculations, the image intensities are normalized to the source intensities. The spatial resolution of the image, defined as the ratio of the full width at half maximum to λ, is different in the two directions. One can say that the image is high quality from curve 1 because the image resolution is only 0.46λ, well below the conventional diffraction limit. We also can say the image is awful from curve 2 because the amplitude of the side peak is very strong, almost equivalent to the intensity of the image centre. Therefore, to analysis the resolution of an image one should show the image intensity distribution along the directions, not only parallel but also normal to the PhC slab interface. For different thickness of PhC slabs, 8, 12, 60, and 120 layers, the image intensity distribution, parallel and normal to the interface of PhC slab, are plotted in Fig. 5a, and b, respectively. With increasing the thickness of the slab, the intensity of the image decreases rapidly, but the image resolutions, parallel to the PhC slab interface, change very little. The resolution of the worst image is 0.76λ (curve 4) in Fig. 5a, which is still below the conventional diffraction limit. From Fig. 5b, it can be seen that the resolution along the normal
Fig. 4. The calculated radiation intensity across the image centre of Fig. 2a: curve 1 is along the direction parallel to the interface of PhC slab, curve 2 is along the normal direction of the interface.
direction becomes worse with increasing the slab thickness. One could not almost distinguish which of the imaging peaks is related to the main peak of the source when the thickness is larger than 60 layers. It means the electric intensity along the normal direction is elongated. In such a criterion, the image formed by a PhC slab does not possess high quality, because the image is elongated in the direction of propagation. 4. Image transferred by cascaded PhC slabs To increase the distance between source P and image P 0 there are two possible ways: increasing the slab thickness D or applying CSPS. In this section we discuss the quality of the image transferred by CSPS and compare the image quality between the two situations. The calculation parameters are the same as mentioned previously. Firstly, we consider a cascaded stack composed of two PhC slabs with each slab being 20 layers. The two slabs are separated by an air layer of thickness 18.28a. The source is
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Fig. 5. The calculated radiation intensity across the image centre, curve 1, 2, 3, and 4 corresponding to 8, 12, 60 and 120 layer PhC slabs, (a) parallel to the slab interface; (b) normal to the slab interface.
Fig. 6. (a) Schematic picture depicting the transfer of a source by two stack PhC slabs. (b) E z field pattern of a point source transferred by two stack Phc slabs.
Fig. 7. The calculated radiation intensity across the image centre of Fig. 5b: curve 1 is along the direction parallel to the interface of PhC slab, curve 2 is along the normal direction of the interface.
placed at a distance L = 9.14a from the left surface of the first PhC slab. According to Veselago relation, the source should be focused inside the two PhC slabs, between the two slabs and at the right of the second slabs as shown in Fig. 6a. Applying FDTD method, we calculate the field pattern as shown in Fig. 6b. One can see that there are four focusing images as predicted, except that the images inside PhC slabs are not very obvious. The distance between P and P 0 is about 70.0a, which is approximately equal to double the thickness of the two slabs D1 + D2 = 35.4a. The image intensity distribution, parallel and normal to the PhC slab interface is plotted in Fig. 7. The quality of the image is similar to that of a PhC slab: elongated along the normal direction. The resolution along the direction parallel to the interface is 0.46λ, and the intensity of image is only 8.2% of the source.
In addition, we double the thickness of one PhC slab to realize the same focusing image at the same distance, as shown in Fig. 8a. A 40 layers PhC slab with cutting interface is used and it is about double thickness of that in Fig. 5b. The source is located at a distance L = 18.5a from the left surface of the slab. The simulation result of electric field pattern is shown in Fig. 8b. One can find that a focused image is formed on the other hand of the slab. The sum of L and L 0 is about 34.5a that is also approximately equal to the thickness of PhC slab D = 35.0a. And the distance between P and P 0 is about 69.5a. Therefore, the source can form an image at the same distance in the two situations. The image in Fig. 8b is also elongated in the direction of propagation, which has been discussed in Section 3, and the intensity is 19.1%, which is stronger than that in Fig. 6b, of the source. Increasing the number of slabs in the stack, we consider a cascaded stack composed of three PhC slabs with 20 layers and cutting interface. Simulation result of electric field pattern is shown in Fig. 9. The resolution of image along the normal direction deteriorated more seriously than that in Fig. 6b. And the intensity of image is much weak (2.6% of the source). Using a 60 layers PhC slab, we also make a comparison. The resolution is shown at curve 3 in Fig. 5 and the imaging intensity is 12.6% of the source. As a result, the transferred image does not possess high quality when applying CSPS because the image is elongated in the normal direction and the intensity is weaken. With increasing the number of slabs in the stack, the resolution along the normal direction become worse and the intensity become weaker. The image formed by a CSPS is no better than a PhC slab.
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Fig. 8. (a) Schematic picture depicting the transfer of a source by a PhC slab. (b) E z field pattern of a point transferred by a Phc slab.
References
Fig. 9. E z field patterns of a point source transferred by three stack PhC slabs.
5. Conclusions We have studied the properties of image transferred by PhC slab with negative refractive index n = −1 and confirmed the negative refractive phonomenon. But the saturated image properties are not found. In PhC slab, the resolution of image is good along the direction parallel to the slab interface, but awful along the direction normal to the slab interface. With increasing the slab thickness, the distortion becomes more seriously and the intensity of the image becomes weaker. The images transferred by CSPS are also distorted similar to that transferred by PhC slab. The intensity of image from CSPS is weaker than that formed by a PhC slab because of the reflection of many air/slab interfaces. The images formed by a CSPS are no better than that formed by a PhC slab. They are both elongated in the propagating direction. Acknowledgements This work is supported in part by Chinese National Key Basic Research Special Fund (2007CB613206 and 2006CB921704), Chinese National Science Fund (60576068, 10725418), and the computational support from Shanghai Super-computer Center.
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