50 50 beam splitter using a one-dimensional metal photonic crystal with parabolalike
dispersion
Linfang Shen, Tzong-Jer Yang, and Yuan-Fong Chau
Citation: Applied Physics Letters 90, 251909 (2007); doi: 10.1063/1.2750385
View online: http://dx.doi.org/10.1063/1.2750385
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/90/25?ver=pdfcov Published by the AIP Publishing
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50/ 50 beam splitter using a one-dimensional metal photonic crystal
with parabolalike dispersion
Linfang Shena兲
Department of Information Science and Electronic Engineering, Electromagnetic Academy, Zhejiang University, Hangzhou 310027, China
Tzong-Jer Yang
Department of Electrophysics, National Chiao Tung University, Hsinchu, 30050 Taiwan, Republic of China
Yuan-Fong Chau
Department of Electronic Engineering, Chin Yun University, Jung Li, 320 Taiwan, Republic of China
共Received 12 October 2006; accepted 26 May 2007; published online 19 June 2007兲
The spatial dispersion properties of a one-dimensional metal photonic crystal have been analyzed and five types of dispersion curves have been shown at a normalized frequency less than 1. It is demonstrated that by exploiting a parabolalike dispersion behavior, a metal photonic crystal slab can be used to realize an exactly 50/ 50 beam splitter. © 2007 American Institute of Physics.
关DOI:10.1063/1.2750385兴
The dispersion characteristics of photonic crystals共PCs兲 have recently attracted considerable interest. Various spatial dispersion properties of PCs underlay foundation for a vari-ety of potential applications such as spatial beam routing1 and dispersion-based beam splitter.2,3So far, the known stud-ies of the spatial dispersion phenomena of PCs are mainly devoted to dielectrics PCs共DPCs兲,4–6while very few refer to metal PCs共MPCs兲.7,8This can be explained by the fact that metals in the visible/infrared range are often accompanied by absorption loss, which creates an obvious obstacle in utiliz-ing MPCs in practical devices. However, if the fillutiliz-ing frac-tion of metallic material in the MPC is small enough, it is possible to reduce the influence of absorption loss to a toler-able level. Compared to DPCs, MPCs comprised of materials with permittivities of opposite signs provide more opportu-nities of exhibiting unusual dispersion behaviors. We also notice some recent research works on PCs containing left-handed materials that have simultaneously negative permit-tivity and permeability.9–11 In this letter, we will study the dispersion characteristic of a one-dimensional 共1D兲 MPC with a small metal filling fraction and demonstrate a MPC slab with special dispersion property to realize a 50/ 50 beam splitter.
Consider a 1D MPC consisting of silver films and SiO2
layers stacked alternatively with a period a along the x di-rection. The silver films have a thickness dⰆa, i.e., its filling fraction fm= d / aⰆ1. The interaction of an electromagnetic wave inside such MPC with the structure can be effectively interpreted through a dispersion diagram. For TE-polarized waves共with the magnetic field in the y direction兲 propagat-ing in the xz plane共k=xˆkx+ zˆkz兲, the dispersion relation is given by
cos共kxa兲 = cos关共1 − fm兲pxa兴cosh共fmqxa兲 −1 2
冉
mpx rqx − rqx mpx冊
⫻sin关共1 − fm兲pxa兴sinh共fmqxa兲, 共1兲wherem andr are the relative permittivities of silver and SiO2, respectively; px=
冑
rk02− kz2 and qx=冑
kz2−mk02 withk0=/ c, whereis the angular frequency of the wave and c the speed of light in free space. Here, we taker= 2.1, while the dispersivemis taken from Ref.12. kx is limited to the first Brillouin zone, i.e., −/ a艋kx艋/ a.
To distinguish propagating waves from evanescent waves in the MPC and find the dispersion properties of the structure, we first neglect the imaginary part ofmin Eq.共1兲. To illustrate typical dispersion behaviors, the dispersion curves, namely, the equifrequency contours, for a MPC with a filling fraction of fm= 0.07 at a normalized frequency of a / 2c = 1 / 3 are shown in Fig. 1. Since m is frequency dependent, the dispersion properties of such a PC cannot be characterized simply by the normalized frequency. Thus, as seen from Fig.1, there exist five types of shapes of disper-sion curves at the same normalized frequency for this PC. Here we choose a fixed normalized frequency 共a / 2c
= 1 / 3兲 only to ensure that no higher-order Bragg coupling occurs in our subsequent application of this MPC.6The cor-responding wavelengths of waves in free space共=/ 2c兲
in the five cases are 1.94, 1.24, 0.76, 0.66, and 0.41m, and
a兲Electronic mail: [email protected]
FIG. 1. 共Color online兲 Dispersion curves for TE waves at various wave-lengths for MPCs with a filling fraction fm= 0.07. Normalized frequency
a / 2c is 1 / 3 for all cases. The numbers marked indicate the values of
wavelengths.
APPLIED PHYSICS LETTERS 90, 251909共2007兲
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the respective MPC periods are 0.65, 0.41, 0.25, 0.22, and 0.14m, respectively. The four types of the dispersion curves for the wavelengths=1.94, 1.24, 0.66, and 0.41m 共corresponding to the silver permittivities m= −198.2, −78, −20.1, and −5.2, respectively12兲 are similar to those found for the TM wave in a structure formed by a periodic array of uniaxial magnetic resonant scatters:13They are hyperbolalike curve, ellipselike curve, or hyperbolalike 共ellipselike兲 curve combined with ellipselike 共hyperbolalike兲 curve which passes through the vicinities of kz= ±
冑
rk0 at kx= 0. How-ever, of particular interest is the fifth type of the dispersion curves corresponding to the wavelength of 0.76m 共i.e., m= −27.5兲, the two branches of the dispersion curves cross at kx= 0, for which kz⬇ ±冑
r+ fm2共r−m兲/2k0. Though thisMPC cannot be approximated by an effectively homoge-neous medium, we may use the quantities of the effective permittivities, which are defined byt= fmm+共1− fm兲rand n=mr/关fmr+共1− fm兲m兴,14 instead of the parameters m and fm, to characterize the structure, and we find that the special dispersion behavior occurs at t⬇0, which never happens in a DPC. Note that the parameter n⬇r for all cases since fmⰆ1.
When兩t兩 is much less than r and兩m兩, in the limiting case of k0aⰆ1 and 兩k兩aⰆ1, the dispersion equation 关Eq.共1兲兴
reduces to kx 2 =
冋
t+ 1 12共1 − fm兲 2 r 2冉
k02− kz 2 n冊
a2册
冉
k02− kz 2 n冊
. 共2兲When t= 0, the dispersion curve becomes a pair of parabolas, i.e., kx= ±共1− fm兲r共k0
2
− kz 2/
n兲a/2
冑
3. Note thatfm=r/共r−m兲 in this case. So the special dispersion curves for the aforementioned MPC should be parabolalike. By tak-ing appropriate filltak-ing fraction at each wavelength such that t= 0, our numerical analysis indicates that the parabolalike dispersion behavior is preserved, at least for the case of
a / 2c艋0.5. In what follows, we focus on the case of fm= 0.07 and=0.76m, as a typical example, to demon-strate that such a MPC with parabolalike dispersion has the potential of application in beam splitting.
The accurate dispersion diagram for a MPC with fm = 0.07 at =0.76m is shown in Fig. 2, where a /=1/3
and the silver loss is taken into account,m= −27.5+ 0.3i.12 For a given real kx, kz now becomes a complex number. However, the imaginary part of kz共for propagating waves兲 is much less than the real part 共see Fig. 2兲, indicating that
propagating waves attenuate slowly in the MPC. Consider a plane wave incident from air with wave vector ki on a boundary of such a MPC along the x direction共i.e., the pe-riodic direction兲. Two Bloch waves corresponding to the wave vector points A and B will be excited inside the MPC,4 as illustrated in Fig. 2共a兲. The Poynting vectors of the two Bloch waves SAand SB, which point away from the source, are opposite in the x direction, indicating that one wave is negatively refracted and the other positively refracted. Inter-estingly, as kx= ki· xˆ→0 共i.e., approaching to the case of nor-mal incidence兲, the wave vector points A and B fall onto the same point A*, but the Poynting vectors SA and SB will not become identical, and they only tend to be symmetric with respect to the surface normal 关see Fig. 2共a兲兴, implying the existence of two different共but symmetric兲 refracted waves in the MPC. The reasoning behind this is the symmetry of the crossing dispersion curves with respect to the line kx= 0 which is normal to the boundary, as illustrated in Fig.2共a兲. Evidently, for a normally incident Gaussian beam with a cer-tain width, which has a finite symmetric angle span around its propagation direction, i.e., the direction of the surface normal, two symmetric beams, one in each side of the sur-face normal, will be excited inside the MPC due to the sym-metry of the dispersion curves. The spatial separation of the two beams in the MPC will increase as they propagate away from the interface, so a MPC slab with a certain width can realize an exactly 50/ 50 beam splitter.
To verify the beam splitting behavior in the MPC, we simulate a Gaussian beam of width 5 incident normally on a MPC slab with a thickness of w = 10 in both the lossy 共m= −27.5+ 0.3i兲 and lossless 共m= −27.5兲 cases, using the finite-difference time-domain technique with uniaxial per-fectly matched layers.15The distribution of the amplitude of the magnetic field共Hy兲 共obtained by a Fourier transform for the given frequency兲 for the lossless case is shown in Fig.
3共a兲 and the field pattern for the lossy case is found to be similar. As seen from Fig. 3共a兲, the incident beam is split symmetrically when it enters the MPC slab between z = 0 and
z = 10, and the split beams separate by nearly 12 at the exit
surface. Both output beams are parallel to the incident beam, which follows directly from the conservation of kxacross the interfaces for each plane wave component of the beam. Since
a / 2c = 1 / 3⬍0.5,6 no higher-order Bragg propagating beams exit from this MPC slab. Our numerical analysis shows that the output beams have approximately the same width as the incident beam. The total power of both output beams is nearly 60%共with respect to the incident power兲 in the lossless case, but it reduces to 40% in the lossy case.
Figure3共b兲shows the total power of the output beams as a function of the thickness of the MPC slab, where the lines with open and solid circles represent the results for the loss-less and lossy cases, respectively. While the spatial separa-tion of the exit beams increases linearly with the thickness w, the total output power varies almost periodically in both cases, as seen from Fig. 3共b兲, where a slow decaying of power peaks is also observed for the lossy case. The maxi-mum of total output power is found to be 42% in the lossy case, whereas up to nearly 80% in the lossless case. FIG. 2.共Color online兲 Wave vector diagram 共for TE waves兲 for a MPC with
fm= 0.07 andm= −27.5+ 0.3i.共a兲 Re共kz兲, 共b兲 Im共kz兲. The red circle in 共a兲 is the equifrequency contour for air, and the dotted line indicates the direction of the interface between the MPC and air. The dot-dashed lines are the construction lines for three cases of wave incidence from air. kiand ki⬘are symmetric about the line kx= 0, so are the intersections A共B兲 and A⬘共B⬘兲. The Poynting vectors of refracted waves are indicated by bold plain arrows. Note that k0a /= 2 / 3.
251909-2 Shen, Yang, and Chau Appl. Phys. Lett. 90, 251909共2007兲
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The MPC splitter can also output asymmetrically split beams as long as a Gaussian beam is incident obliquely on the MPC slab. For example, when a Gaussian beam of width 5 is incident on the MPC slab with w=10 at an incidence angle of 30°, the output beams at the exit surface are shifted asymmetrically in the positive or negative x directions, but they are still parallel to the incident beam共see Fig.4兲. The
power ratio of the output beams is 1.7 and the total output power is 25% in the lossy case, while they become 2.2 and 60% in the lossless case, respectively. From Fig. 2共a兲, it seems evidently that the asymmetry of two split beams en-hances with the increase of incidence angle.
To summarize, we have shown five types of dispersion curves for a 1D MPC at a normalized frequency less than 1. Among these, a parabolalike dispersion behavior has been found for the first time in PCs. By exploiting this special dispersion property, a MPC slab can be used to realize an exactly 50/ 50 beam splitter by normal incidence. Asymmet-ric beams with different output powers can also be obtained
in a MPC splitter by oblique incidence. This MPC splitter works only for TE-polarized waves. Based on a similar mechanism, however, a splitter for TM-polarized waves can also be realized by using a photonic crystal formed by an array of magnetic resonant components such as nanowire pairs.
The authors would like to acknowledge the National Sci-ence Council of ROC for the financial support through the Grant the NSC 95-2119-M-009-029. One of the authors 共L.S.兲 sincerely thanks the hospitality of NCTU when he visited there.
1D. W. Prather, S. Shi, D. Pustai, C. Chen, S. Venkataraman, A. Sharkawy,
G. Schneider, and J. Murakowski, Opt. Lett. 29, 50共2004兲.
2X. Yu and S. Fan, Appl. Phys. Lett. 83, 3251共2003兲.
3S. Shi, A. Sharkawy, C. Chen, D. Pustai, and D. W. Prather, Opt. Lett. 29,
617共2004兲.
4P. St. J. Russell, Appl. Phys. B: Photophys. Laser Chem. 39, 231共1986兲. 5M. Notomi, Phys. Rev. B 62, 10696共2000兲.
6S. Foteinopoulou and C. M. Soukoulis, Phys. Rev. B 72, 165112共2005兲. 7X. Y. Ao and S. He, Opt. Lett. 30, 2152共2005兲.
8X. Fan, G. P. Wang, J. C. Lee, and C. T. Chan, Phys. Rev. Lett. 97,
073901共2006兲.
9J. Li, L. Zhou, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 90, 083901
共2003兲.
10H. Daninthe, S. Foteinopoulou, and C. M. Soukoulis, Photonics
Nano-struct. Fundam. Appl. 4, 123共2006兲.
11D. R. Fredkin and A. Ron, Appl. Phys. Lett. 81, 1753共2002兲. 12P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370共1972兲. 13P. A. Belov and C. R. Simovski, Phys. Rev. E 72, 026615共2005兲. 14D. E. Aspnes, Am. J. Phys. 50, 704共1982兲.
15A. Taflove and S. C. Hagness, Computational Electrodynamics-The finite
Difference Time-Domain Method共Artech, Boston, 2000兲.
FIG. 3. 共Color online兲 共a兲 Magnetic-field distribution for a TE-polarized Gaussian beam normally incident on a MPC slab with parabolalike disper-sion.共b兲 Percentage of total output power versus the slab thickness w. The line with solid共open兲 circles corresponds to the lossy 共lossless兲 case.
FIG. 4.共Color online兲 Magnetic-field distribution for a TE-polarized Gauss-ian beam incident on a MPC slab with fm= 0.07 andm= −27.5+ 0.3i at an incidence angle of 30°.
251909-3 Shen, Yang, and Chau Appl. Phys. Lett. 90, 251909共2007兲
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