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Low-pass spatial filters with small angle-domain bandwidth based on one-dimensional metamaterial photonic crystals
Optik 127 (2016) 259–262
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Low-pass spatial filters with small angle-domain bandwidth based on one-dimensional metamaterial photonic crystals Zhaoming Luo a,b,∗ , Min Chen a , Jiyuan Deng a , Ying Chen b , Jing Liu a a
College of Information and Communication Engineering, Hunan Institute of Science and Technology, Yueyang 414006, China Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education, College of Physics and Microelectronic Science, Hunan University, Changsha 410082, China b
a r t i c l e
i n f o
Article history: Received 18 December 2014 Accepted 10 October 2015 Keywords: Spatial filter Metamaterial Zero-index gap
a b s t r a c t By using the spatial properties of zero-index gaps in one-dimensional metamaterial photonic crystals, we have designed low-pass spatial filters with small angle-domain bandwidth less than 10 degrees. It is demonstrated by transfer matrix method that the small angle-domain bandwidth and roll-off factor of the spatial filters can be adjusted by changing the structure parameters of the metamaterial photonic crystals. In addition, the spatial filters have absolute polarization-independent properties compared with previous photonic crystal spatial filters. These spatial filters with small angle-domain bandwidth may be applied to the laser science in the future, and decrease the space occupied by the traditional spatial filters. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Low-pass spatial filters have been widely applied to image enhancement and information processing in several regions of the electromagnetic spectrum, particularly in laser science [1–3]. A simple and traditional low-pass spatial filter consists of two focusing lenses in a confocal arrangement and a pinhole in the focus plane, and its angle-domain bandwidth can be tuned by changing the size of the pinhole. The setup, although widely used, has a considerable size (at least four focal lengths) and high standard for the quality and installation of the lens. To overcome the deficiencies of traditional spatial filters, some modern spatial filters are performed based on conventional photonic crystals (PCs) with positive-index materials (PIMs) [4–10]. The application of the spatial filters such as beam smoothing has been demonstrated [7,9,10], but the angledomain bandwidth of the PC spatial filters is very big compared to the requirements of the laser science [11,12]. Recently, metamaterials have been realized including singlenegative (SNG) materials, negative-index materials (NIMs) and zero-index materials (ZIMs), and attract intensive studies due to their unique electromagnetic properties and potential applications [13–24]. With metamaterials being introduced into PCs, three new kinds of gaps have been identified: the zero-average-index
∗ Corresponding author at: College of Information and Communication Engineering, Hunan Institute of Science and Technology, Yueyang 414006, China. Tel.: +86 15273033069. E-mail address: [email protected] (Z. Luo). http://dx.doi.org/10.1016/j.ijleo.2015.10.034 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
gap, the zero-effective-phase gap and the zero-index gap [22–24]. In contrast to Bragg gap, originating from interference mechanisms, it is the material dispersion of the metamaterials that mainly determines the appearances of these new gaps [25]. The zero-average-index gap appears in one-dimensional (1D) PC structures combining PIMs and NIMs, and the zero-effective-phase gap emerges in 1D PC structures containing two different SNG materials. Based on the unique properties of the two gaps, the performances of some frequency filters and spatial filters are improved [22–29]. The zero-index gap is more special and appears near frequencies where the index equals to zero. Different from the former two gaps, it possesses many excellent properties. Thus, we have reason to expect to design a new spatial filter based on the zeroindex gap to improve the performance of the PC spatial filters. In this paper, we aim to test the feasibility of designing lowpass spatial filters with small angle-domain bandwidth based on 1D metamaterial PC structures. Firstly, we investigate the properties of the zero-index gap in detail. Secondly, a practical design is presented based on the gap properties of the zero-index gap. Finally, we summarize the properties of the spatial filters including polarization, angle-domain bandwidth and roll-off factor. 2. Computations model and numerical method The 1D metamaterial PC structure (AB)N is shown in Fig. 1 with A and B indicating metamaterial layers. We use a transmission line model to describe the metamaterials that are [17,22,28] εa = ε1 ,
a = 1 −
2 ωmp
ω2
,
(1)
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Z. Luo et al. / Optik 127 (2016) 259–262
Fig. 1. Schematic of the 1D metamaterial PC structure of (AB)N . A and B indicate metamaterials, and the physical thicknesses of A and B are da and db .
In metamaterial layer A and εb = ε2 −
2 ωep
ω2
,
b = 2 ,
(2)
In metamaterial layer B, where ωmp and ωep are the magnetic plasma frequency and electronic plasma frequency respectively. In Eqs. (1) and (2), ω is the angular frequency measured in GHz. The model is introduced to study the metamaterial properties and applications [22–29]. It is noticed that A is the ZIM at specified ω ω frequency ω = √mp and B at √εep . 1
2
We use the transfer-matrix method [22–29] to analyze the transmittance properties of the 1D metamaterial PC structures. If we inject a plane wave into the 1D metamaterial PC structures at an angle in the +z direction, the electric component and magnetic component for the qth layer can be related via a transfer matrix,
⎛
⎞
−
cos ˇq
Mq = ⎝
−ipq cos ˇq
i cos ˇq pq ⎠
(3)
cos ˇq
√ √ where ˇq = (2/) εq q dq , pq = εq /q
1 − sin2 /(εq q )
2
q /εq 1 − sin /(εq q ) for TM for TE polarization and pq = polarization with εq , q , dq , and being, respectively, the relative permittivity, the relative permeability, the physical thickness of the qth layer and the wavelength of the incident wave. The transmission matrix of the whole structure can be written as M=
Mj (dj , ω) =
x11
x12
x21
x22
(4)
and the reflectivity and transmissivity are derived out to be [28]
(x11 + ps x12 )p0 − (x21 + ps x22 ) 2
, (x + p x )p + (x + p x )
R=
T=
11
s 12
0
21
s 22
(5)
ps
2p0
2
, p0 (x11 + ps x12 )p0 + (x21 + ps x22 )
Fig. 2. Transmittance spectra of the 1D metamaterial PC structure of (AB)8 at the different incident angles = 0◦ , 30◦ , 60◦ and 85◦ . (b) is the part of (a). The blue solid (red dashed) curves are for TE (TM) polarization. Structure parameters: ε1 = 2, 1 = 1, ε2 = 1, 2 = 2, ωep = ωmp = ωp = 10/(2) GHz. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(6)
where p0 = ps = 1 − sin2 /(ε0 0 ) = cos for the vacuum (ε0 = 0 = 1) at the left- and right-hand side of the structure. 3. Results and discussion We first investigate the gap properties of the metamaterial PCs (AB)N as shown in Fig. 1, and choose the structure parameters as follows: ε1 = 2, 1 = 1, ε2 = 1, 2 = 2, ωep = ωmp = ωp = 10/(2) GHz, N = 8. According to the structure parameters, A and B are both ZIMs at specified frequency ω = ωp . In the frequency range of ω > ωp , all ε and are positive. That is to say, A and B are both PIMs. The SNG
materials including the mu-negative (MNG) and epsilon-negative (ENG) materials correspond to the frequency range of ω < ωp , with A being MNG materials and B ENG materials. The transmittance spectra of the 1D metamaterial PC structure with the incident angles = 0◦ , 30◦ , 60◦ and 85◦ are shown in Fig. 2(a). It can be seen that the Bragg gap emerges in the double-positive frequency range and the zero-effect-phase gap in the single negative frequency range. Meanwhile, the zero-index gap also arises near the zero-index frequency. In order to observe the zero-index gap distinctly, Fig. 2(a) is redrawn in the frequency range from 0.5 GHz to 2.5 GHz (see Fig. 2(b)). It is clearly shown that there is no gap beside the zeroindex frequency (ωp ≈ 1.592 GHz) at normal incidence, and the zero-index gap emerges and becomes wider and wider with the increase of the incident angle. Thus we can conclude that the zeroindex gap is very sensitive to the incident angles. Such a result is due to the breaking of Snell’s law when the refractive index n satisfies the condition of 0 ≤ n ≤ 1, that is to say, no real solution for any refraction angle r of the equation sin = nr sin r (r = A or B) under the inclined incident angle ( = / 0) [23]. In the above discussion, we have considered the gap properties of the structure for TE polarization. We also calculate the transmittance spectra of the structure for TM polarization, as shown in the dashed curve of Fig. 2(b). It is seen that the two curves for TE and TM polarization are overlapped. Thus the gap is polarization-independent, which is the reason that the structure has good electromagnetic symmetry [30]. The conclusion from the above analysis that the zero-index gap is sensitive to the incident angle is a prerequisite for the realization of spatial filters with small angle-domain bandwidth. We next design a new low-pass spatial filter based on the spatial properties
Z. Luo et al. / Optik 127 (2016) 259–262
261
Fig. 3. Transmittance as a function of the incident angle for the low-pass spatial filters with different polarization. The blue solid (red dashed) curves are for TE (TM) polarization. The center frequency of the spatial filter is zero-index frequency (ω0 = 1.592 GHz), and the other parameters are the same as those in Fig. 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Transmittance as a function of the incident angle for the low-pass spatial filter (AB)8 with the different center frequencies. The blue solid, red dashed, and green doted curves correspond to the center frequencies ω0 = 1.592 GHz, 1.594 GHz and 1.598 GHz, respectively. The other parameters are the same as those in Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Transmittance as a function of the incident angle for the low-pass spatial filters with different period number N. The blue solid, red dashed and green dotted curves correspond to the structures (AB)8 , (AB)4 and (AB)14 , respectively. The other parameters are the same as those in Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Transmittance as a function of the incident angle for low-pass spatial filter (AB)8 with different damping factors. The blue solid, red dashed, and green dotted curves correspond to damping factors = 0, = 5 × 107 , and = 1 × 108 Hz, respectively. The other parameters are the same as those in Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
of the zero-index gap. Usually, the frequency bandwidth of spatial filters is very narrow or even to the monofrequency. In our work (experiment), the center frequency of the low-pass spatial filter is near zero-index frequency (ω0 = 1.592 GHz) as shown in Fig. 2(b). The angular dependence of transmittance of the spatial filter with different polarization is shown in Fig. 3. It is clearly observed that the angle-domain bandwidth is = 9◦ . Compared with that of the PC spatial filters [3–10,28], the angle domain bandwidth is smaller, which is precisely the application required in laser system. Furthermore, we can draw a conclusion that the spatial filter is absolutely polarization-independent owing that the two curves in Fig. 3 with different polarization are overlapped. Therefore the previous polarization-independent spatial filters [28] can only be described as quasi-independent. Then, we investigate the properties of the spatial filter based on the zero-index gap including the roll-off factor and angle-domain bandwidth. The study reveals that the roll-off factor is closely related to the period number of structure, as shown in Fig. 4. It can be seen that the band edge becomes steeper as the period number increases, so that the roll-off factor is getting smaller and smaller and the rectangular characteristics get better and better. Thus we can improve the rectangular characteristics of the spatial filters through increasing the period number of structure. What’s more,
the angle-domain bandwidth of the spatial filters can be adjusted in the range of 10 degrees, which is also the requirement in laser science. Fig. 5 is the transmittance spectra of the spatial filters with different center frequencies. Comparing with the three curves in Fig. 5, we can conclude that the higher the center frequency, the larger the angle-domain bandwidth. Besides, according to structural scalable properties of PCs, angle-domain bandwidth of spatial filters with constant center frequency can be tuned as the structure parameters of PCs change. Finally, we briefly discuss the effect of losses on spatial filters owing that losses in the metamaterials are inevitable. When losses are involved, the relative permittivity and permeability of the metamaterials in Eqs. (1) and (2) should be modified as εa = ε1 , a = 1 −
2 ωmp
ω2 +jω m
, and εb = ε2 −
2 ωep
ω2 +jω e
, b = 2 . Here, e and m
denote the respective electric and magnetic damping factors that contribute to the losses and we assume that e = m = . Fig. 6 is the transmittance spectra of the structure (AB)8 with different damping factors = 0, 5 × 107 , and 1 × 108 Hz. It is found that the losses will decrease the transmission intensity and increase the bandwidth of spatial filters, which is consistent with previous conclusions regarding spatial filters based on zero-average-index gap [28].
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4. Conclusion In summary, we have demonstrated the feasibility of designing spatial filters based on 1D PCs containing the zero-index metamaterials. Firstly, we investigated the properties of zero-index gap and find the gap is highly sensitive to the incident angle. Then we propose the PC structure as a design for low-pass spatial filters with small angle-domain bandwidth. The spatial filter is polarization-independent in comparison with the spatial filters based on zero-average-index gap, and the roller-off factor and angle-domain bandwidth of the filter can be controlled by adjusting the structure parameters of the metamaterial PCs. Moreover, the center frequency of our structures may extend from microwave band to optical band through changing structure parameters, and the structures will have broad prospects of application in electromagnetic spectrum, high-power lasers and radar data processing because of its peculiar properties and easy fabrication. Acknowledgment This work is supported by the National Natural Science Foundation of China (Grant Nos. 61205126, 61308005), the Natural Science Foundation of Hunan Province (Grant No. 14JJ3131) and this Project was supported by China Postdoctoral Science Foundation (2013M542106). References [1] J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1988. [2] S.P. Almeida, G. Indebetouw, Applications of Optical Fourier Transforms, Academic Press, San Diego, 1982. [3] D. Schurig, D.R. Smith, Spatial filtering using media with indefinite permittivity and permeability tensors, Appl. Phys. Lett. 82 (2003) 2215–2217. [4] Z. Tang, D. Fan, S. Wen, Y. Ye, C. Zhao, Low-pass spatial filtering using a two-dimensional self-collimating photonic crystal, Chin. Opt. Lett. 5 (2007) S211–S220. [5] A.E. Serebryannikov, T. Magath, Transmission through photonic crystals with multiple line defects at oblique incidence, J. Opt. Soc. Am. B 25 (2008) 286–296. [6] A.E. Serebryannikov, A.Y. Petrov, E. Ozbay, Toward photonic crystal based spatial filters with wide angle ranges of total transmission, Appl. Phys. Lett. 94 (2009) 181101. [7] K. Staliunas, V.J. Sánchez-Morcillo, Spatial filtering of light by chirped photonic crystals, Phys. Rev. A 79 (2009) 053807. [8] E. Colak, A.O. Cakmak, A.E. Serebryannikov, E. Ozbay, Spatial filtering using dielectric photonic crystals at beam-type excitation, J. Appl. Phys. 108 (2010) 113106. [9] Z. Luo, S. Wen, Z. Tang, H. Luo, Y. Xiang, D. Song, Low-pass rugate spatial filters for beam smoothing, Opt. Commun. 283 (2010) 2665–2668. [10] V. Purlys, L. Maigyte, D. Gaileviˇcius, M. Peckus, M. Malinauskas, K. Staliunas, Spatial filtering by chirped photonic crystals, Phys. Rev. A 87 (2013) 033805.
[11] E.S. Bliss, D.R. Speck, J.F. Holzrichter, J.H. Erkkila, A.J. Glass, Propagation of a high-intensity laser pulse with small-scale intensity modulation, Appl. Phys. Lett. 25 (1974) 448–450. [12] S. Wen, D. Fan, Spatiotemporal instabilities in nonlinear Kerr media in the presence of arbitrary higher-order dispersions, J. Opt. Soc. Am. B 19 (2002) 1653–1659. [13] J.B. Pendry, A.J. Holden, W.J. Stewart, I. Youngs, Extremely low frequency plasmons in metallic mesostructures, Phys. Rev. Lett. 76 (1996) 4773–4776. [14] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microw. Theory Technol. 47 (1999) 2075–2084. [15] D.R. Fredkin, A. Ron, Effectively left-handed (negative index) composite material, Appl. Phys. Lett. 81 (2002) 1753–1755. [16] A. Alù, N. Engheta, Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency, IEEE Trans. Antennas Propag. 51 (2003) 2558–2571. [17] G.V. Eleftheriades, A.K. Iyer, P.C. Kremer, Planar negative refractive index media using periodically LC loaded transmission lines, IEEE Trans. Microw. Theory Technol. 50 (2002) 2702–2712. [18] R.J. Pollard, A. Murphy, W.R. Hendren, P.R. Evans, R. Atkinson, G.A. Wurtz, A.V. Zayats, V.A. Podolskiy, Optical nonlocalities and additional waves in epsilonnear-zero metamaterials, Phys. Rev. Lett. 102 (2009) 127405. [19] M. Silveirinha, N. Engheta, Tunneling of electromagnetic energy through subwavelength channels and bends using -near-zero materials, Phys. Rev. Lett. 97 (2006) 157403. [20] B. Edwards, A. Alù, M.E. Young, M. Silveirinha, N. Engheta, Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide, Phys. Rev. Lett. 100 (2008) 033903. [21] R.W. Ziolkowski, Propagation in and scattering from a matched metamaterial having a zero index of refraction, Phys. Rev. E 70 (2004) 046608. [22] H.T. Jiang, H. Chen, H.Q. Li, Y.W. Zhang, S.Y. Zhu, Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials, Appl. Phys. Lett. 83 (2003) 5386–5388. [23] L.G. Wang, H. Chen, S.Y. Zhu, Omnidirectional gap and defect mode of onedimensional photonic crystals with single-negative materials, Phys. Rev. B 70 (2004) 245102. [24] R.A. Depine, M.L. Martínez-Ricci, J.A. Monsoriu, E. Silvestre, P. Andrés, Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystals, Phys. Lett. A 364 (2007) 352–355. [25] E. Silvestre, R.A. Depine, M.L. Martínez-Ricci, J.A. Monsoriu, Role of dispersion on zero-average-index bandgaps, J. Opt. Soc. Am. B 26 (2009) 581–586. [26] Y. Chen, Tunable omnidirectional multichannel filters based on dual-defective photonic crystals containing negative-index materials, J. Phys. D 42 (2009) 075106. [27] Y. Xiang, X. Dai, S. Wen, Z. Tang, D. Fan, Zero-effective-phase bandgaps in photonic multilayers: analytic expressions for band-edge frequencies and broadband omnidirectional reflection, J. Opt. Soc. Am. B 28 (2011) 1187–1193. [28] Z. Luo, Z. Tang, Y. Xiang, H. Luo, S. Wen, Polarization-independent low-pass spatial filters based on one-dimensional photonic crystals containing negativeindex materials, Appl. Phys. B 94 (2009) 641–646. [29] Z. Luo, S. Wen, H. Luo, Z. Tang, Y. Xiang, J. Liu, Omnidirectional and tunable symmetrical confined states in photonic quantum-well structures with singlenegative materials, Optik 122 (2011) 724–727. [30] Z. Luo, S. Qu, J. Liu, P. Tian, General conditions of polarization-independent transmissions in one-dimensional magnetic photonic crystals, J. Mod. Opt. 60 (2013) 171–176.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Low-pass spatial filters with small angle-domain bandwidth based on one-dimensional metamaterial photonic crystals Zhaoming Luo a,b,∗ , Min Chen a , Jiyuan Deng a , Ying Chen b , Jing Liu a a
College of Information and Communication Engineering, Hunan Institute of Science and Technology, Yueyang 414006, China Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education, College of Physics and Microelectronic Science, Hunan University, Changsha 410082, China b
a r t i c l e
i n f o
Article history: Received 18 December 2014 Accepted 10 October 2015 Keywords: Spatial filter Metamaterial Zero-index gap
a b s t r a c t By using the spatial properties of zero-index gaps in one-dimensional metamaterial photonic crystals, we have designed low-pass spatial filters with small angle-domain bandwidth less than 10 degrees. It is demonstrated by transfer matrix method that the small angle-domain bandwidth and roll-off factor of the spatial filters can be adjusted by changing the structure parameters of the metamaterial photonic crystals. In addition, the spatial filters have absolute polarization-independent properties compared with previous photonic crystal spatial filters. These spatial filters with small angle-domain bandwidth may be applied to the laser science in the future, and decrease the space occupied by the traditional spatial filters. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Low-pass spatial filters have been widely applied to image enhancement and information processing in several regions of the electromagnetic spectrum, particularly in laser science [1–3]. A simple and traditional low-pass spatial filter consists of two focusing lenses in a confocal arrangement and a pinhole in the focus plane, and its angle-domain bandwidth can be tuned by changing the size of the pinhole. The setup, although widely used, has a considerable size (at least four focal lengths) and high standard for the quality and installation of the lens. To overcome the deficiencies of traditional spatial filters, some modern spatial filters are performed based on conventional photonic crystals (PCs) with positive-index materials (PIMs) [4–10]. The application of the spatial filters such as beam smoothing has been demonstrated [7,9,10], but the angledomain bandwidth of the PC spatial filters is very big compared to the requirements of the laser science [11,12]. Recently, metamaterials have been realized including singlenegative (SNG) materials, negative-index materials (NIMs) and zero-index materials (ZIMs), and attract intensive studies due to their unique electromagnetic properties and potential applications [13–24]. With metamaterials being introduced into PCs, three new kinds of gaps have been identified: the zero-average-index
∗ Corresponding author at: College of Information and Communication Engineering, Hunan Institute of Science and Technology, Yueyang 414006, China. Tel.: +86 15273033069. E-mail address: [email protected] (Z. Luo). http://dx.doi.org/10.1016/j.ijleo.2015.10.034 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
gap, the zero-effective-phase gap and the zero-index gap [22–24]. In contrast to Bragg gap, originating from interference mechanisms, it is the material dispersion of the metamaterials that mainly determines the appearances of these new gaps [25]. The zero-average-index gap appears in one-dimensional (1D) PC structures combining PIMs and NIMs, and the zero-effective-phase gap emerges in 1D PC structures containing two different SNG materials. Based on the unique properties of the two gaps, the performances of some frequency filters and spatial filters are improved [22–29]. The zero-index gap is more special and appears near frequencies where the index equals to zero. Different from the former two gaps, it possesses many excellent properties. Thus, we have reason to expect to design a new spatial filter based on the zeroindex gap to improve the performance of the PC spatial filters. In this paper, we aim to test the feasibility of designing lowpass spatial filters with small angle-domain bandwidth based on 1D metamaterial PC structures. Firstly, we investigate the properties of the zero-index gap in detail. Secondly, a practical design is presented based on the gap properties of the zero-index gap. Finally, we summarize the properties of the spatial filters including polarization, angle-domain bandwidth and roll-off factor. 2. Computations model and numerical method The 1D metamaterial PC structure (AB)N is shown in Fig. 1 with A and B indicating metamaterial layers. We use a transmission line model to describe the metamaterials that are [17,22,28] εa = ε1 ,
a = 1 −
2 ωmp
ω2
,
(1)
260
Z. Luo et al. / Optik 127 (2016) 259–262
Fig. 1. Schematic of the 1D metamaterial PC structure of (AB)N . A and B indicate metamaterials, and the physical thicknesses of A and B are da and db .
In metamaterial layer A and εb = ε2 −
2 ωep
ω2
,
b = 2 ,
(2)
In metamaterial layer B, where ωmp and ωep are the magnetic plasma frequency and electronic plasma frequency respectively. In Eqs. (1) and (2), ω is the angular frequency measured in GHz. The model is introduced to study the metamaterial properties and applications [22–29]. It is noticed that A is the ZIM at specified ω ω frequency ω = √mp and B at √εep . 1
2
We use the transfer-matrix method [22–29] to analyze the transmittance properties of the 1D metamaterial PC structures. If we inject a plane wave into the 1D metamaterial PC structures at an angle in the +z direction, the electric component and magnetic component for the qth layer can be related via a transfer matrix,
⎛
⎞
−
cos ˇq
Mq = ⎝
−ipq cos ˇq
i cos ˇq pq ⎠
(3)
cos ˇq
√ √ where ˇq = (2/) εq q dq , pq = εq /q
1 − sin2 /(εq q )
2
q /εq 1 − sin /(εq q ) for TM for TE polarization and pq = polarization with εq , q , dq , and being, respectively, the relative permittivity, the relative permeability, the physical thickness of the qth layer and the wavelength of the incident wave. The transmission matrix of the whole structure can be written as M=
Mj (dj , ω) =
x11
x12
x21
x22
(4)
and the reflectivity and transmissivity are derived out to be [28]
(x11 + ps x12 )p0 − (x21 + ps x22 ) 2
, (x + p x )p + (x + p x )
R=
T=
11
s 12
0
21
s 22
(5)
ps
2p0
2
, p0 (x11 + ps x12 )p0 + (x21 + ps x22 )
Fig. 2. Transmittance spectra of the 1D metamaterial PC structure of (AB)8 at the different incident angles = 0◦ , 30◦ , 60◦ and 85◦ . (b) is the part of (a). The blue solid (red dashed) curves are for TE (TM) polarization. Structure parameters: ε1 = 2, 1 = 1, ε2 = 1, 2 = 2, ωep = ωmp = ωp = 10/(2) GHz. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(6)
where p0 = ps = 1 − sin2 /(ε0 0 ) = cos for the vacuum (ε0 = 0 = 1) at the left- and right-hand side of the structure. 3. Results and discussion We first investigate the gap properties of the metamaterial PCs (AB)N as shown in Fig. 1, and choose the structure parameters as follows: ε1 = 2, 1 = 1, ε2 = 1, 2 = 2, ωep = ωmp = ωp = 10/(2) GHz, N = 8. According to the structure parameters, A and B are both ZIMs at specified frequency ω = ωp . In the frequency range of ω > ωp , all ε and are positive. That is to say, A and B are both PIMs. The SNG
materials including the mu-negative (MNG) and epsilon-negative (ENG) materials correspond to the frequency range of ω < ωp , with A being MNG materials and B ENG materials. The transmittance spectra of the 1D metamaterial PC structure with the incident angles = 0◦ , 30◦ , 60◦ and 85◦ are shown in Fig. 2(a). It can be seen that the Bragg gap emerges in the double-positive frequency range and the zero-effect-phase gap in the single negative frequency range. Meanwhile, the zero-index gap also arises near the zero-index frequency. In order to observe the zero-index gap distinctly, Fig. 2(a) is redrawn in the frequency range from 0.5 GHz to 2.5 GHz (see Fig. 2(b)). It is clearly shown that there is no gap beside the zeroindex frequency (ωp ≈ 1.592 GHz) at normal incidence, and the zero-index gap emerges and becomes wider and wider with the increase of the incident angle. Thus we can conclude that the zeroindex gap is very sensitive to the incident angles. Such a result is due to the breaking of Snell’s law when the refractive index n satisfies the condition of 0 ≤ n ≤ 1, that is to say, no real solution for any refraction angle r of the equation sin = nr sin r (r = A or B) under the inclined incident angle ( = / 0) [23]. In the above discussion, we have considered the gap properties of the structure for TE polarization. We also calculate the transmittance spectra of the structure for TM polarization, as shown in the dashed curve of Fig. 2(b). It is seen that the two curves for TE and TM polarization are overlapped. Thus the gap is polarization-independent, which is the reason that the structure has good electromagnetic symmetry [30]. The conclusion from the above analysis that the zero-index gap is sensitive to the incident angle is a prerequisite for the realization of spatial filters with small angle-domain bandwidth. We next design a new low-pass spatial filter based on the spatial properties
Z. Luo et al. / Optik 127 (2016) 259–262
261
Fig. 3. Transmittance as a function of the incident angle for the low-pass spatial filters with different polarization. The blue solid (red dashed) curves are for TE (TM) polarization. The center frequency of the spatial filter is zero-index frequency (ω0 = 1.592 GHz), and the other parameters are the same as those in Fig. 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Transmittance as a function of the incident angle for the low-pass spatial filter (AB)8 with the different center frequencies. The blue solid, red dashed, and green doted curves correspond to the center frequencies ω0 = 1.592 GHz, 1.594 GHz and 1.598 GHz, respectively. The other parameters are the same as those in Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Transmittance as a function of the incident angle for the low-pass spatial filters with different period number N. The blue solid, red dashed and green dotted curves correspond to the structures (AB)8 , (AB)4 and (AB)14 , respectively. The other parameters are the same as those in Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Transmittance as a function of the incident angle for low-pass spatial filter (AB)8 with different damping factors. The blue solid, red dashed, and green dotted curves correspond to damping factors = 0, = 5 × 107 , and = 1 × 108 Hz, respectively. The other parameters are the same as those in Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
of the zero-index gap. Usually, the frequency bandwidth of spatial filters is very narrow or even to the monofrequency. In our work (experiment), the center frequency of the low-pass spatial filter is near zero-index frequency (ω0 = 1.592 GHz) as shown in Fig. 2(b). The angular dependence of transmittance of the spatial filter with different polarization is shown in Fig. 3. It is clearly observed that the angle-domain bandwidth is = 9◦ . Compared with that of the PC spatial filters [3–10,28], the angle domain bandwidth is smaller, which is precisely the application required in laser system. Furthermore, we can draw a conclusion that the spatial filter is absolutely polarization-independent owing that the two curves in Fig. 3 with different polarization are overlapped. Therefore the previous polarization-independent spatial filters [28] can only be described as quasi-independent. Then, we investigate the properties of the spatial filter based on the zero-index gap including the roll-off factor and angle-domain bandwidth. The study reveals that the roll-off factor is closely related to the period number of structure, as shown in Fig. 4. It can be seen that the band edge becomes steeper as the period number increases, so that the roll-off factor is getting smaller and smaller and the rectangular characteristics get better and better. Thus we can improve the rectangular characteristics of the spatial filters through increasing the period number of structure. What’s more,
the angle-domain bandwidth of the spatial filters can be adjusted in the range of 10 degrees, which is also the requirement in laser science. Fig. 5 is the transmittance spectra of the spatial filters with different center frequencies. Comparing with the three curves in Fig. 5, we can conclude that the higher the center frequency, the larger the angle-domain bandwidth. Besides, according to structural scalable properties of PCs, angle-domain bandwidth of spatial filters with constant center frequency can be tuned as the structure parameters of PCs change. Finally, we briefly discuss the effect of losses on spatial filters owing that losses in the metamaterials are inevitable. When losses are involved, the relative permittivity and permeability of the metamaterials in Eqs. (1) and (2) should be modified as εa = ε1 , a = 1 −
2 ωmp
ω2 +jω m
, and εb = ε2 −
2 ωep
ω2 +jω e
, b = 2 . Here, e and m
denote the respective electric and magnetic damping factors that contribute to the losses and we assume that e = m = . Fig. 6 is the transmittance spectra of the structure (AB)8 with different damping factors = 0, 5 × 107 , and 1 × 108 Hz. It is found that the losses will decrease the transmission intensity and increase the bandwidth of spatial filters, which is consistent with previous conclusions regarding spatial filters based on zero-average-index gap [28].
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4. Conclusion In summary, we have demonstrated the feasibility of designing spatial filters based on 1D PCs containing the zero-index metamaterials. Firstly, we investigated the properties of zero-index gap and find the gap is highly sensitive to the incident angle. Then we propose the PC structure as a design for low-pass spatial filters with small angle-domain bandwidth. The spatial filter is polarization-independent in comparison with the spatial filters based on zero-average-index gap, and the roller-off factor and angle-domain bandwidth of the filter can be controlled by adjusting the structure parameters of the metamaterial PCs. Moreover, the center frequency of our structures may extend from microwave band to optical band through changing structure parameters, and the structures will have broad prospects of application in electromagnetic spectrum, high-power lasers and radar data processing because of its peculiar properties and easy fabrication. Acknowledgment This work is supported by the National Natural Science Foundation of China (Grant Nos. 61205126, 61308005), the Natural Science Foundation of Hunan Province (Grant No. 14JJ3131) and this Project was supported by China Postdoctoral Science Foundation (2013M542106). References [1] J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1988. [2] S.P. Almeida, G. Indebetouw, Applications of Optical Fourier Transforms, Academic Press, San Diego, 1982. [3] D. Schurig, D.R. Smith, Spatial filtering using media with indefinite permittivity and permeability tensors, Appl. Phys. Lett. 82 (2003) 2215–2217. [4] Z. Tang, D. Fan, S. Wen, Y. Ye, C. Zhao, Low-pass spatial filtering using a two-dimensional self-collimating photonic crystal, Chin. Opt. Lett. 5 (2007) S211–S220. [5] A.E. Serebryannikov, T. Magath, Transmission through photonic crystals with multiple line defects at oblique incidence, J. Opt. Soc. Am. B 25 (2008) 286–296. [6] A.E. Serebryannikov, A.Y. Petrov, E. Ozbay, Toward photonic crystal based spatial filters with wide angle ranges of total transmission, Appl. Phys. Lett. 94 (2009) 181101. [7] K. Staliunas, V.J. Sánchez-Morcillo, Spatial filtering of light by chirped photonic crystals, Phys. Rev. A 79 (2009) 053807. [8] E. Colak, A.O. Cakmak, A.E. Serebryannikov, E. Ozbay, Spatial filtering using dielectric photonic crystals at beam-type excitation, J. Appl. Phys. 108 (2010) 113106. [9] Z. Luo, S. Wen, Z. Tang, H. Luo, Y. Xiang, D. Song, Low-pass rugate spatial filters for beam smoothing, Opt. Commun. 283 (2010) 2665–2668. [10] V. Purlys, L. Maigyte, D. Gaileviˇcius, M. Peckus, M. Malinauskas, K. Staliunas, Spatial filtering by chirped photonic crystals, Phys. Rev. A 87 (2013) 033805.
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