- Email: [email protected]
Directional field enhancement of dielectric nano optical disc antenna arrays
Optical Materials 34 (2011) 126–130
Contents lists available at SciVerse ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Directional field enhancement of dielectric nano optical disc antenna arrays Ivan Wang ⇑, Y. Du Dept. BSE, The HongKong Polytechnic University, Hong Kong
a r t i c l e
i n f o
Article history: Received 28 December 2010 Received in revised form 7 July 2011 Accepted 21 July 2011 Available online 9 September 2011 Keywords: Optical Disc antenna Nano antenna Directivity Field enhancement
a b s t r a c t This paper presents a discussion on the directive field enhancement of dielectric disc antenna arrays in optical band. The property of dielectric material is addressed, and field modes in a cylindrical resonator are discussed. It is identified that the fundamental mode of HE11d generates the far field with a higher directivity than other modes. More effective field enhancement in the radiation direction could be achieved by using multiple-disc antenna arrays. Simulation examples indicate that the directivity of a disc antenna array varies with the disc spacing. The maximum directivity is observed when the disc spacing is approximately equal to the half of the vacuum wavelength. The maximum directivity can be improved significantly when the disc number is increased. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Recently nano antennas have received increasing attention in nanotechnology research. Such antennas can be applied in microscopy, spectroscopy, data-communication, and even solar energy harvesting [1–3]. It is known that antennas can be made with a metal or a dielectric material. In the upper microwave band dielectric antennas are favored as they have some advantages such as wider bandwidth, less loss and avoidance of surface waves, compared to metal antennas [4]. In optical band, both metallic and dielectric antennas are utilized practically as unique material properties exhibit in the metallic and dielectric materials [5–8]. The metal in optical frequency actually works as solid plasma, having its own plasma frequency, collision frequency, damping and so on, which result in a complex permittivity with a negative real part. The dielectric, which cannot be described by the Drude model [9], has a frequency-dependent complex permittivity in optical frequency. Nano antennas can be analyzed using the classical electromagnetic (EM) theory with the parameters of permittivity, wave number and so on; the quantum theory is not used here. The EM equations are considered sufficient in the analysis of electromagnetic fields in nano-material which is characterized with the parameters mentioned previously. In [10], the dielectric waveguide theory was used to discuss the effective length of rod optical antennas. It was found that the effective length was shorter than the physical length of the rods. The equivalent circuit theory
⇑ Corresponding author. Tel.: +852 3400 3602; fax: +852 2765 7198. E-mail address: [email protected] (I. Wang). 0925-3467/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2011.07.027
derived from the classic EM theory was also used to analyze the electromagnetic fields in nano devices [11–13]. In this paper, the electromagnetic property of dielectric materials in optical band is first described. Using the classical electromagnetic theory, the excited field modes in nano-optical disc antennas are then discussed. The mode with the maximal direct field enhancement is identified. Simulation examples are given to illustrate the coupling effect of discs in disc antenna arrays. An investigation into the effect of field enhancement in disc antenna arrays is presented. Optimal disc spacing for the maximal direct field enhancement in the disc antenna array is discussed. 2. Dielectric material in optical band Dielectric materials show unique characteristics in optical band. Note that a dielectric material is characterized with relative permittivity er(x) = er1 + ier2. The real part er1 and the imaginary er2 are generally expressed by [6]
er1 ðxÞ ¼ 1 þ v þ K er2 ðxÞ ¼ K
x20 x2 ðx x2 Þ2 þ ðcxÞ2 2 0
cx ðx20 x2 Þ2 þ ðcxÞ2
ð1Þ
ð2Þ
where v is electric susceptibility, K is a constant determined by physical parameters of atoms and electrons in the material, and c is a damping rate. In (2) and (3) x0 is the resonant frequency of electrons within the material, and is generally greater than the frequency of visible light. It is noted from (2) and (3) that the relative permittivity in microwave band has an effective real part with a
127
I. Wang, Y. Du / Optical Materials 34 (2011) 126–130
nearly zero imaginary part. When the frequency is increased to the infrared, optical or ultraviolet band, the imaginary part is non-trivial as the frequency is close to the resonant frequency of the material. The relative permittivity tends to be a frequency-dependent complex number with both real and imaginary parts being positive. This has been verified in the measurement [7], It is known that the real part er1 represents the capability that of the material holds energy, and the imaginary part er2 represents the capability that of the material generates the loss of energy. Clearly a dielectric material in optical band loses some energy when it is excited by an EM field source. As the real part of relative permittivity is positive, the dielectric material in optical band can also be used as a resonator. In contrast, plasmonic material (metal in optical frequency) which is characterized by the Drude model has a negative real part. Therefore, an EM wave cannot penetrate into the metal. The following mode analysis for a dielectric waveguide is then inapplicable to metal in optical frequency. Note again that the real part er1 in optical frequency is larger than that in microwave frequency. The effective wavelength of a dielectric antenna is usually shorter in optical frequency. 3. Analysis of electric field distribution In microwave band, antennas made of dielectric can effectively radiate EM waves [14]. These antennas work as a resonator. As a resonator is a section of a waveguide, the electromagnetic field around the resonator can be analyzed using the dielectric waveguide theory. Similarly, a dielectric disc antenna in optical band works as a radiative cylindrical dielectric resonator, as illustrated in Fig. 1. The wave propagation within a dielectric waveguide can be supported. The electric field distribution within the dielectric waveguide is determined by the boundary conditions. The equivalent magnetic current yields on the dielectric material as a result of the inner electric field. This current radiates the electromagnetic field from the disc antenna like a traditional metal antenna [11,12]. It is noted that in a dielectric waveguide propagation constant c is determined from [6],
c2 ¼ k2c k2
ð3Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where wavenumber k ¼ 2pf l0 er e0 and cutoff wavenumber kc = (k/kc)k. Cutoff wavelength kc can be computed by enforcing the boundary conditions on the waveguide surface [16]. It is determined by geometry and material properties of the waveguide, and field mode in the waveguide. The complex propagation constant can be expressed by c = a + jb. Both wave attenuation constant a and wave phase constant b are given by
b ¼ ð2pf Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi le0 er1 1 ðk=kc Þ2
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a ¼ pðer2 =er1 Þ= k 1 ðk=kc Þ2
ð5Þ
L εr Waveguide εout (a) Cylindrical Rod
(a)HE11
E-field
H-field
Fig. 2. EM field distributions on the cross-section area of a cylindrical dielectric waveguide or resonator: (a) the fondamental mode of HE11 (b) the 2nd mode of TM01, and (c) the 3rd mode of TE01.
For the dielectric material in optical band, both real part er1 and imaginary part er2 are positive. Wave propagation in the dielectric waveguide is supported if the wavelength of the electromagnetic (EM) field is less than the cutoff wavelength of the waveguide, as shown in (4). The fundamental field mode in a cylindrical dielectric waveguide is found to be the HE11 mode, not the TM01 mode appearing in a traditional metallic waveguide in microwave band. Fig. 2 illustrates the field distributions of first three field modes in the cross-section area of a cylindrical waveguide or resonator, which are also applicable to a dielectric circular rod [15]. The numerical analysis in [15] indicates that E-field lines for the mode HE11 run in parallel on the cross-section area. The field reaches the maximum value at the central point of the cross-section, and declines when moving away from the central point. For the mode TM01 or TE01, the E-field distribution is rotationally symmetric on the cross-section area. In a circular dielectric waveguide [16], if the attenuation conqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi stant of a cladding medium q ¼ b2 ð2pf Þ2 leout has a complex value (eout is the permittivity of the medium cladding around the dielectric material, Fig. 1), the EM field can be effectively radiated from the dielectric material to the outer medium. When parameter q has a positive value, the EM wave is confined within the waveguide. No radiation is generated from the waveguide or resonator in this case. Normally, a cylindrical resonator is considered as a segment of the cylindrical waveguide with the length of k/2, as illustrated in Fig. 1. When the cylindrical resonator works as a disc antenna for radiating an EM wave, segment length, which is often defined as d, is much smaller than k/2. The E-field distribution on the cross section area is similar to those given in Fig. 2. The field distribution b on the surdetermines the equivalent magnetic current M ¼ E n face of a disc antenna [4], which is treated as the radiating source. The disc antenna made of a dielectric material may support the mode HE11d [16] while the other modes (TE01d and TM01d) may also be generated. As seen in Fig. 2, the E field has quasi-straight parallel field lines for the mode HE11d, radial field lines for the mode TM01d, or circular field lines for the mode TE01d. The field peak appears in the center for the mode HE11d, and appears at
1-disc L
(c)TE01
(b)TM01
2-disc
4-disc
9-disc
εr εout
(b) Disc Resonator/Antenna (a section of cylindrical rod)
Fig. 1. Cylindrical rod and disc resonator/antenna.
16-disc
25-disc
Fig. 3. Configurations of dielectric nano optical disc antennas arrays.
128
I. Wang, Y. Du / Optical Materials 34 (2011) 126–130
Fig. 4. Directivity in the E-plane and H plane of a single disc antenna.
the half of disc radius for the mode TE01d or TM01d. For a single disc antenna, the mode HE11d can effectively radiate directionally in the direction perpendicular to the top surface of the disc. As the field lines are rotationally symmetric for the modes TM01d and TE01d, the disc antenna operating at these modes has the maximum radiation in radial directions on a plane. The radiation pattern, therefore, is less directive. When several disc antennas are assembled to form an antenna array, the effect of mutual coupling between elements should be taken into consideration [17]. The coupling between adjacent discs determines scattering of the field which finally has an effect on the directivity of the array. The directivity of the antenna array at the mode of HE11d can be further enhanced.
4. FDTD simulation of dielectric disc antenna arrays FDTD (finite-time finite-difference) simulations have been performed for several examples of dielectric disc antenna arrays to illustrate their directional field enhancement. Fig. 3 shows the configurations of disc antenna arrays with different disc number. These discs are made of Silicon with the relative permittivity of 15.598 + j0.21464 at 500 THz [6]. The disc radius is set to be the half wavelength k/2 of the source field, which is given by
k ¼ k0 =
pffiffiffiffiffiffi
er1
ð6Þ
where k0 is the vacuum wavelength. For dielectric material with complex permittivity, the real part is used only in (6). Disc spacing, however, varies from one tenth of the wavelength to the full
Table 1 Directivity of disc arrays with different disc number and spacing (unit: dBi). Spacing (s0)
k0/10
k0/4
k0/3
k0/2
2k0/3
3k0/4
k0
2-disc 4-disc 9-disc 16-disc 25-disc
4.3 5.2 9.1 14.5 22.1
5.3 4.1 14.1 19.4 24.3
5.9 4.7 15.3 27.4 37.9
6.3 5.7 12.4 28.9 59.5
6.1 6.1 11.7 27.9 59.9
5.8 5.7 12.7 22.8 41.7
5.3 5.4 14.9 22.1 45.9
wavelength for the discussion of maximal coupling among discs in the antenna array. In the simulation the disc array is excited by an exciting source at one circular end of the discs, and radiates the electromagnetic field from another circular end (top surface) of the discs. The far field pattern of the disc antenna array is computed for discussion. To illustrate the directional field enhancement, a single mode is excited in the antenna array at the frequency of 500 THz. As a result, the discs have the radius of 76 nm. The disc thickness is set to be 50 nm in all cases. The far field pattern generated by a single disc antenna is computed first using the FDTD method. The disc antenna is excited to create three basic modes HE11d, TE01d and TM0d1 in turn for examining the effect of directional field enhancement. The simulation results of directivity in the E- and H-planes are shown in Fig. 4. It is noted that the directional field enhancement at the mode HE11d is the best. The directivity of the disc antenna is 2.82dBi for the mode HE11, 1.41dBi for the mode TE01 and 1.28dBi for the mode TM01. The radiation pattern for the mode TE01d or TM0d1 is less directive as shown in Fig. 4.
Fig. 5. Comparison of the effect of mutual coupling in 2-disc antenna array.
I. Wang, Y. Du / Optical Materials 34 (2011) 126–130
(a.1) n=4 s0= 2λ0/3, E-Plane
(a.2) n=4 s0= 2λ0 /3, H-Plane
(b.1) n=9 s0= λ 0/3, E-Plane
(b.2) n=9 s0= λ0/3, H-Plane
(c.1) n=16 s0= λ0/2, E-Plane
(c.2) n=16 s0= λ0/2, H-Plane
(d.1) n=25 s0= 2λ0/3, E-Plane
129
(d.2) n=25 s0= 2λ 0/3, H-Plane
Fig. 6. Directional field enhancement of the n-element array at the mode HE11d.
Fig. 5 shows the simulation results for a two-disc antenna array with the disc spacing of k0/2 at the mode HE11d. To investigate the
coupling effect of two discs, the E- and H-planes of the antenna array without any coupling are also presented in the figure. It is
130
I. Wang, Y. Du / Optical Materials 34 (2011) 126–130
Table 2 Comparison of directivity in different modes (unit: dBi).
5. Conclusions
Mode
HE11
TM01
TE01
2-disc@k0/2 4-disc@2k0/3 9-disc@k0/3 16-disc@k0/2 25-disc@2k0/3
6.3 6.1 15.3 28.9 59.9
5.12 5.06 3.75 8.25 23.1
2.7 2.53 4.68 11.4 8.37
noted that there is a difference of the field pattern in these two cases. The directivity is 6.3dBi for the case with mutual coupling, and 5.64dBi for the case without mutual coupling. This is mainly caused by the interaction of the electromagnetic fields within two discs. This coupling enhances the directivity of the far field pattern, and leads to a narrower main lobe and a higher directivity, as illustrated in Fig. 5. The directional field enhancement is a multiple-disc antenna array is then investigated numerically by varying the disc number and spacing. In all cases the mode HE11d is excited in the discs. Fig. 2 shows the disc array with the number n of 2, 4, 9, 16 or 25. The disc spacing between any two adjacent discs (from edge to edge) remains same, but its value is taken to be k0/10, k0/4, k0/3, k0/2, 2k0/3, 3k0/4 or k0 in the stimulation. k0 is the vacuum wavelength of the field at 500 THz, and is equal to 600 nm. Table 1 shows the directivity of disc antenna arrays at the mode HE11d with different disc number and spacing. It is noted that the directivity of the disc array generally increases with increasing disc number. The maximum directivity of an n-element array is 6.1dBi for n = 4, and reaches 59.9dBi for n = 4. Fig. 6 shows the far-field pattern of the n-element array (n = 4, 9, 16 and 25) with the spacing for maximum directivity being achieved. Note that the main lobe becomes narrow when the element number increases. When the disc number is fixed, the directivity of the disc array varies with the disc spacing. The maximal directivity is observed if the disc spacing is approximately equal to half of the free-space wavelength k0. This indicates that there is an effective mutual coupling among the disc elements. When the disc number increases the disc spacing for the maximum directivity varies in the range of k0/3–2k0/3. A comparison of the directivity against other modes is also conducted. Table 2 shows the directivity of disc antenna arrays with fixed spacing at three different modes HE11d, TE01d and TM0d. The disc spacing is selected in such a way that the directivity at the mode HE11d reaches the maximal value. It is noted that the directive field enhancement at other modes is not as strong as that at the mode HE11d. The mode HE11d radiates the electromagnetic field with the maximal directivity. .
This paper presented a discussion on the directive field enhancement in a disc antenna array in optical band. The property of dielectric materials was addressed, and field modes in a cylindrical resonator were discussed. It was identified that the fundamental mode of HE11d generated the far field pattern with higher directivity than other modes. More effective field enhancement in the radiation direction could be achieved by using a multipledisc antenna array. Simulation examples indicated that the directivity of the square array varied with the disc spacing, and the maximum directivity was observed when the disc spacing was approximately equal to the half of the vacuum wavelength. The maximum directivity could be improved significantly when the disc number was increased. Acknowledgments The work leading to this paper was supported by grants from the Research Committee of the Hong Kong Polytechnic University, and the Research Grants Council of the Hong Kong Special Administrative Region (Project No. 516008). References [1] H.A. Atwater, The Promising of Plasmonics, Scientific American, 2007. [2] A. Ahmadi, S. Saadat, H. Mosallaei, Resonance and Q performance of ellipsoidal ENG subwavelength radiators, IEEE Trans. Antennas Propagat. 59 (3) (2011) 706–713. [3] A. Ahmadi, S. Ghadarghadr, H. Mosallaei, An optical reflectarray nanoantenna: the concept and design, Opt. Express 18 (1) (2010) 123–133. [4] K.M. Luk, K.W. Leung, Dielectric Resonator Antennas, Research Studies Press, 2002. [5] P.B. Johnson, R.W. Christy, Optical constants of the noble metals, Phys. Rev. B 6 (1972) 4370–4379. [6] Mark Fox, Optical Properties of Solids, Oxford University Press, 2002. [7] R. Hull, Properties of Crystalline Silicon, The Institution of Engineering and Technology, 1999. [8] A. Ahmadi, H. Mosallaei, Physical configuration and performance modeling of all-dielectric metamaterials, Phys. Rev. B 77 (2008) 045104. [9] Stefan A. Maier, Plasmonics: Fundamentals and Applications, Springer, 2007. [10] Lukas Novotny, Effective wavelength scaling for optical antennas, Phys. Rev. Lett. 98 (2007) 266802. [11] Nader Engheta et al., Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors, Phys. Rev. Lett. 95 (2005) 095504. [12] Andrea Alù, Nader Engheta, Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas, Phys. Rev. Lett. 101 (2008) 043901. [13] S. Ghadarghadr, Z. Hao, H. Mosallaei, Plasmonic array nanoantennas on layered substrates: modeling and radiation characteristics, Opt. Express 17 (21) (2009) 18556–18570. [14] D. Hondros, P. Debye, Elektromagnetische wellen an dielektrischen Drähten, Annalen der Physik 337 (8) (1910). [15] E. Snitzer, Cylindrical dielectric waveguide modes, JOSA 51 (5) (1961) 491–498. [16] Darko Kajfez, Pierre Guillon, Dielectric Resonators, second ed., Artech House, 1986. [17] A.A. Kishk, Dielectric resonator antenna elements for array applications, in: Proc. of IEEE International Symposium on Phased Array Systems and Technology, 2003.
Contents lists available at SciVerse ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Directional field enhancement of dielectric nano optical disc antenna arrays Ivan Wang ⇑, Y. Du Dept. BSE, The HongKong Polytechnic University, Hong Kong
a r t i c l e
i n f o
Article history: Received 28 December 2010 Received in revised form 7 July 2011 Accepted 21 July 2011 Available online 9 September 2011 Keywords: Optical Disc antenna Nano antenna Directivity Field enhancement
a b s t r a c t This paper presents a discussion on the directive field enhancement of dielectric disc antenna arrays in optical band. The property of dielectric material is addressed, and field modes in a cylindrical resonator are discussed. It is identified that the fundamental mode of HE11d generates the far field with a higher directivity than other modes. More effective field enhancement in the radiation direction could be achieved by using multiple-disc antenna arrays. Simulation examples indicate that the directivity of a disc antenna array varies with the disc spacing. The maximum directivity is observed when the disc spacing is approximately equal to the half of the vacuum wavelength. The maximum directivity can be improved significantly when the disc number is increased. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Recently nano antennas have received increasing attention in nanotechnology research. Such antennas can be applied in microscopy, spectroscopy, data-communication, and even solar energy harvesting [1–3]. It is known that antennas can be made with a metal or a dielectric material. In the upper microwave band dielectric antennas are favored as they have some advantages such as wider bandwidth, less loss and avoidance of surface waves, compared to metal antennas [4]. In optical band, both metallic and dielectric antennas are utilized practically as unique material properties exhibit in the metallic and dielectric materials [5–8]. The metal in optical frequency actually works as solid plasma, having its own plasma frequency, collision frequency, damping and so on, which result in a complex permittivity with a negative real part. The dielectric, which cannot be described by the Drude model [9], has a frequency-dependent complex permittivity in optical frequency. Nano antennas can be analyzed using the classical electromagnetic (EM) theory with the parameters of permittivity, wave number and so on; the quantum theory is not used here. The EM equations are considered sufficient in the analysis of electromagnetic fields in nano-material which is characterized with the parameters mentioned previously. In [10], the dielectric waveguide theory was used to discuss the effective length of rod optical antennas. It was found that the effective length was shorter than the physical length of the rods. The equivalent circuit theory
⇑ Corresponding author. Tel.: +852 3400 3602; fax: +852 2765 7198. E-mail address: [email protected] (I. Wang). 0925-3467/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2011.07.027
derived from the classic EM theory was also used to analyze the electromagnetic fields in nano devices [11–13]. In this paper, the electromagnetic property of dielectric materials in optical band is first described. Using the classical electromagnetic theory, the excited field modes in nano-optical disc antennas are then discussed. The mode with the maximal direct field enhancement is identified. Simulation examples are given to illustrate the coupling effect of discs in disc antenna arrays. An investigation into the effect of field enhancement in disc antenna arrays is presented. Optimal disc spacing for the maximal direct field enhancement in the disc antenna array is discussed. 2. Dielectric material in optical band Dielectric materials show unique characteristics in optical band. Note that a dielectric material is characterized with relative permittivity er(x) = er1 + ier2. The real part er1 and the imaginary er2 are generally expressed by [6]
er1 ðxÞ ¼ 1 þ v þ K er2 ðxÞ ¼ K
x20 x2 ðx x2 Þ2 þ ðcxÞ2 2 0
cx ðx20 x2 Þ2 þ ðcxÞ2
ð1Þ
ð2Þ
where v is electric susceptibility, K is a constant determined by physical parameters of atoms and electrons in the material, and c is a damping rate. In (2) and (3) x0 is the resonant frequency of electrons within the material, and is generally greater than the frequency of visible light. It is noted from (2) and (3) that the relative permittivity in microwave band has an effective real part with a
127
I. Wang, Y. Du / Optical Materials 34 (2011) 126–130
nearly zero imaginary part. When the frequency is increased to the infrared, optical or ultraviolet band, the imaginary part is non-trivial as the frequency is close to the resonant frequency of the material. The relative permittivity tends to be a frequency-dependent complex number with both real and imaginary parts being positive. This has been verified in the measurement [7], It is known that the real part er1 represents the capability that of the material holds energy, and the imaginary part er2 represents the capability that of the material generates the loss of energy. Clearly a dielectric material in optical band loses some energy when it is excited by an EM field source. As the real part of relative permittivity is positive, the dielectric material in optical band can also be used as a resonator. In contrast, plasmonic material (metal in optical frequency) which is characterized by the Drude model has a negative real part. Therefore, an EM wave cannot penetrate into the metal. The following mode analysis for a dielectric waveguide is then inapplicable to metal in optical frequency. Note again that the real part er1 in optical frequency is larger than that in microwave frequency. The effective wavelength of a dielectric antenna is usually shorter in optical frequency. 3. Analysis of electric field distribution In microwave band, antennas made of dielectric can effectively radiate EM waves [14]. These antennas work as a resonator. As a resonator is a section of a waveguide, the electromagnetic field around the resonator can be analyzed using the dielectric waveguide theory. Similarly, a dielectric disc antenna in optical band works as a radiative cylindrical dielectric resonator, as illustrated in Fig. 1. The wave propagation within a dielectric waveguide can be supported. The electric field distribution within the dielectric waveguide is determined by the boundary conditions. The equivalent magnetic current yields on the dielectric material as a result of the inner electric field. This current radiates the electromagnetic field from the disc antenna like a traditional metal antenna [11,12]. It is noted that in a dielectric waveguide propagation constant c is determined from [6],
c2 ¼ k2c k2
ð3Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where wavenumber k ¼ 2pf l0 er e0 and cutoff wavenumber kc = (k/kc)k. Cutoff wavelength kc can be computed by enforcing the boundary conditions on the waveguide surface [16]. It is determined by geometry and material properties of the waveguide, and field mode in the waveguide. The complex propagation constant can be expressed by c = a + jb. Both wave attenuation constant a and wave phase constant b are given by
b ¼ ð2pf Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi le0 er1 1 ðk=kc Þ2
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a ¼ pðer2 =er1 Þ= k 1 ðk=kc Þ2
ð5Þ
L εr Waveguide εout (a) Cylindrical Rod
(a)HE11
E-field
H-field
Fig. 2. EM field distributions on the cross-section area of a cylindrical dielectric waveguide or resonator: (a) the fondamental mode of HE11 (b) the 2nd mode of TM01, and (c) the 3rd mode of TE01.
For the dielectric material in optical band, both real part er1 and imaginary part er2 are positive. Wave propagation in the dielectric waveguide is supported if the wavelength of the electromagnetic (EM) field is less than the cutoff wavelength of the waveguide, as shown in (4). The fundamental field mode in a cylindrical dielectric waveguide is found to be the HE11 mode, not the TM01 mode appearing in a traditional metallic waveguide in microwave band. Fig. 2 illustrates the field distributions of first three field modes in the cross-section area of a cylindrical waveguide or resonator, which are also applicable to a dielectric circular rod [15]. The numerical analysis in [15] indicates that E-field lines for the mode HE11 run in parallel on the cross-section area. The field reaches the maximum value at the central point of the cross-section, and declines when moving away from the central point. For the mode TM01 or TE01, the E-field distribution is rotationally symmetric on the cross-section area. In a circular dielectric waveguide [16], if the attenuation conqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi stant of a cladding medium q ¼ b2 ð2pf Þ2 leout has a complex value (eout is the permittivity of the medium cladding around the dielectric material, Fig. 1), the EM field can be effectively radiated from the dielectric material to the outer medium. When parameter q has a positive value, the EM wave is confined within the waveguide. No radiation is generated from the waveguide or resonator in this case. Normally, a cylindrical resonator is considered as a segment of the cylindrical waveguide with the length of k/2, as illustrated in Fig. 1. When the cylindrical resonator works as a disc antenna for radiating an EM wave, segment length, which is often defined as d, is much smaller than k/2. The E-field distribution on the cross section area is similar to those given in Fig. 2. The field distribution b on the surdetermines the equivalent magnetic current M ¼ E n face of a disc antenna [4], which is treated as the radiating source. The disc antenna made of a dielectric material may support the mode HE11d [16] while the other modes (TE01d and TM01d) may also be generated. As seen in Fig. 2, the E field has quasi-straight parallel field lines for the mode HE11d, radial field lines for the mode TM01d, or circular field lines for the mode TE01d. The field peak appears in the center for the mode HE11d, and appears at
1-disc L
(c)TE01
(b)TM01
2-disc
4-disc
9-disc
εr εout
(b) Disc Resonator/Antenna (a section of cylindrical rod)
Fig. 1. Cylindrical rod and disc resonator/antenna.
16-disc
25-disc
Fig. 3. Configurations of dielectric nano optical disc antennas arrays.
128
I. Wang, Y. Du / Optical Materials 34 (2011) 126–130
Fig. 4. Directivity in the E-plane and H plane of a single disc antenna.
the half of disc radius for the mode TE01d or TM01d. For a single disc antenna, the mode HE11d can effectively radiate directionally in the direction perpendicular to the top surface of the disc. As the field lines are rotationally symmetric for the modes TM01d and TE01d, the disc antenna operating at these modes has the maximum radiation in radial directions on a plane. The radiation pattern, therefore, is less directive. When several disc antennas are assembled to form an antenna array, the effect of mutual coupling between elements should be taken into consideration [17]. The coupling between adjacent discs determines scattering of the field which finally has an effect on the directivity of the array. The directivity of the antenna array at the mode of HE11d can be further enhanced.
4. FDTD simulation of dielectric disc antenna arrays FDTD (finite-time finite-difference) simulations have been performed for several examples of dielectric disc antenna arrays to illustrate their directional field enhancement. Fig. 3 shows the configurations of disc antenna arrays with different disc number. These discs are made of Silicon with the relative permittivity of 15.598 + j0.21464 at 500 THz [6]. The disc radius is set to be the half wavelength k/2 of the source field, which is given by
k ¼ k0 =
pffiffiffiffiffiffi
er1
ð6Þ
where k0 is the vacuum wavelength. For dielectric material with complex permittivity, the real part is used only in (6). Disc spacing, however, varies from one tenth of the wavelength to the full
Table 1 Directivity of disc arrays with different disc number and spacing (unit: dBi). Spacing (s0)
k0/10
k0/4
k0/3
k0/2
2k0/3
3k0/4
k0
2-disc 4-disc 9-disc 16-disc 25-disc
4.3 5.2 9.1 14.5 22.1
5.3 4.1 14.1 19.4 24.3
5.9 4.7 15.3 27.4 37.9
6.3 5.7 12.4 28.9 59.5
6.1 6.1 11.7 27.9 59.9
5.8 5.7 12.7 22.8 41.7
5.3 5.4 14.9 22.1 45.9
wavelength for the discussion of maximal coupling among discs in the antenna array. In the simulation the disc array is excited by an exciting source at one circular end of the discs, and radiates the electromagnetic field from another circular end (top surface) of the discs. The far field pattern of the disc antenna array is computed for discussion. To illustrate the directional field enhancement, a single mode is excited in the antenna array at the frequency of 500 THz. As a result, the discs have the radius of 76 nm. The disc thickness is set to be 50 nm in all cases. The far field pattern generated by a single disc antenna is computed first using the FDTD method. The disc antenna is excited to create three basic modes HE11d, TE01d and TM0d1 in turn for examining the effect of directional field enhancement. The simulation results of directivity in the E- and H-planes are shown in Fig. 4. It is noted that the directional field enhancement at the mode HE11d is the best. The directivity of the disc antenna is 2.82dBi for the mode HE11, 1.41dBi for the mode TE01 and 1.28dBi for the mode TM01. The radiation pattern for the mode TE01d or TM0d1 is less directive as shown in Fig. 4.
Fig. 5. Comparison of the effect of mutual coupling in 2-disc antenna array.
I. Wang, Y. Du / Optical Materials 34 (2011) 126–130
(a.1) n=4 s0= 2λ0/3, E-Plane
(a.2) n=4 s0= 2λ0 /3, H-Plane
(b.1) n=9 s0= λ 0/3, E-Plane
(b.2) n=9 s0= λ0/3, H-Plane
(c.1) n=16 s0= λ0/2, E-Plane
(c.2) n=16 s0= λ0/2, H-Plane
(d.1) n=25 s0= 2λ0/3, E-Plane
129
(d.2) n=25 s0= 2λ 0/3, H-Plane
Fig. 6. Directional field enhancement of the n-element array at the mode HE11d.
Fig. 5 shows the simulation results for a two-disc antenna array with the disc spacing of k0/2 at the mode HE11d. To investigate the
coupling effect of two discs, the E- and H-planes of the antenna array without any coupling are also presented in the figure. It is
130
I. Wang, Y. Du / Optical Materials 34 (2011) 126–130
Table 2 Comparison of directivity in different modes (unit: dBi).
5. Conclusions
Mode
HE11
TM01
TE01
2-disc@k0/2 4-disc@2k0/3 9-disc@k0/3 16-disc@k0/2 25-disc@2k0/3
6.3 6.1 15.3 28.9 59.9
5.12 5.06 3.75 8.25 23.1
2.7 2.53 4.68 11.4 8.37
noted that there is a difference of the field pattern in these two cases. The directivity is 6.3dBi for the case with mutual coupling, and 5.64dBi for the case without mutual coupling. This is mainly caused by the interaction of the electromagnetic fields within two discs. This coupling enhances the directivity of the far field pattern, and leads to a narrower main lobe and a higher directivity, as illustrated in Fig. 5. The directional field enhancement is a multiple-disc antenna array is then investigated numerically by varying the disc number and spacing. In all cases the mode HE11d is excited in the discs. Fig. 2 shows the disc array with the number n of 2, 4, 9, 16 or 25. The disc spacing between any two adjacent discs (from edge to edge) remains same, but its value is taken to be k0/10, k0/4, k0/3, k0/2, 2k0/3, 3k0/4 or k0 in the stimulation. k0 is the vacuum wavelength of the field at 500 THz, and is equal to 600 nm. Table 1 shows the directivity of disc antenna arrays at the mode HE11d with different disc number and spacing. It is noted that the directivity of the disc array generally increases with increasing disc number. The maximum directivity of an n-element array is 6.1dBi for n = 4, and reaches 59.9dBi for n = 4. Fig. 6 shows the far-field pattern of the n-element array (n = 4, 9, 16 and 25) with the spacing for maximum directivity being achieved. Note that the main lobe becomes narrow when the element number increases. When the disc number is fixed, the directivity of the disc array varies with the disc spacing. The maximal directivity is observed if the disc spacing is approximately equal to half of the free-space wavelength k0. This indicates that there is an effective mutual coupling among the disc elements. When the disc number increases the disc spacing for the maximum directivity varies in the range of k0/3–2k0/3. A comparison of the directivity against other modes is also conducted. Table 2 shows the directivity of disc antenna arrays with fixed spacing at three different modes HE11d, TE01d and TM0d. The disc spacing is selected in such a way that the directivity at the mode HE11d reaches the maximal value. It is noted that the directive field enhancement at other modes is not as strong as that at the mode HE11d. The mode HE11d radiates the electromagnetic field with the maximal directivity. .
This paper presented a discussion on the directive field enhancement in a disc antenna array in optical band. The property of dielectric materials was addressed, and field modes in a cylindrical resonator were discussed. It was identified that the fundamental mode of HE11d generated the far field pattern with higher directivity than other modes. More effective field enhancement in the radiation direction could be achieved by using a multipledisc antenna array. Simulation examples indicated that the directivity of the square array varied with the disc spacing, and the maximum directivity was observed when the disc spacing was approximately equal to the half of the vacuum wavelength. The maximum directivity could be improved significantly when the disc number was increased. Acknowledgments The work leading to this paper was supported by grants from the Research Committee of the Hong Kong Polytechnic University, and the Research Grants Council of the Hong Kong Special Administrative Region (Project No. 516008). References [1] H.A. Atwater, The Promising of Plasmonics, Scientific American, 2007. [2] A. Ahmadi, S. Saadat, H. Mosallaei, Resonance and Q performance of ellipsoidal ENG subwavelength radiators, IEEE Trans. Antennas Propagat. 59 (3) (2011) 706–713. [3] A. Ahmadi, S. Ghadarghadr, H. Mosallaei, An optical reflectarray nanoantenna: the concept and design, Opt. Express 18 (1) (2010) 123–133. [4] K.M. Luk, K.W. Leung, Dielectric Resonator Antennas, Research Studies Press, 2002. [5] P.B. Johnson, R.W. Christy, Optical constants of the noble metals, Phys. Rev. B 6 (1972) 4370–4379. [6] Mark Fox, Optical Properties of Solids, Oxford University Press, 2002. [7] R. Hull, Properties of Crystalline Silicon, The Institution of Engineering and Technology, 1999. [8] A. Ahmadi, H. Mosallaei, Physical configuration and performance modeling of all-dielectric metamaterials, Phys. Rev. B 77 (2008) 045104. [9] Stefan A. Maier, Plasmonics: Fundamentals and Applications, Springer, 2007. [10] Lukas Novotny, Effective wavelength scaling for optical antennas, Phys. Rev. Lett. 98 (2007) 266802. [11] Nader Engheta et al., Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors, Phys. Rev. Lett. 95 (2005) 095504. [12] Andrea Alù, Nader Engheta, Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas, Phys. Rev. Lett. 101 (2008) 043901. [13] S. Ghadarghadr, Z. Hao, H. Mosallaei, Plasmonic array nanoantennas on layered substrates: modeling and radiation characteristics, Opt. Express 17 (21) (2009) 18556–18570. [14] D. Hondros, P. Debye, Elektromagnetische wellen an dielektrischen Drähten, Annalen der Physik 337 (8) (1910). [15] E. Snitzer, Cylindrical dielectric waveguide modes, JOSA 51 (5) (1961) 491–498. [16] Darko Kajfez, Pierre Guillon, Dielectric Resonators, second ed., Artech House, 1986. [17] A.A. Kishk, Dielectric resonator antenna elements for array applications, in: Proc. of IEEE International Symposium on Phased Array Systems and Technology, 2003.