- Email: [email protected]
Photonic cloak made of subwavelength dielectric elliptical rod arrays
Optics Communications 284 (2011) 4820–4823
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Photonic cloak made of subwavelength dielectric elliptical rod arrays Hanhong Gao a,⁎, Baile Zhang b, George Barbastathis b, c a b c
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Singapore-MIT Alliance for Research and Technology (SMART) Centre, 3 Science Drive 2, Singapore 117543, Singapore Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
a r t i c l e
i n f o
Article history: Received 1 May 2011 Received in revised form 14 June 2011 Accepted 15 June 2011 Available online 30 June 2011 Keywords: Photonic device Subwavelength structure Elliptical rod arrays
a b s t r a c t Modern photonic devices require more and more compact integration of photonic components and mechanical components, which, however, will cause interference and degrade the photonic performance. Here we propose a method that can be used to reduce the interference caused by non-photonic parts in a photonic device. It employs a photonic cloak made of subwavelength dielectric elliptical rod arrays in a photonic waveguide environment. This cloak design is able to preserve the device's photonic performance with the mechanical components penetrating through, and thus provide a more compact platform. Cloaks that suppress the interference from the periphery and the inner region of the waveguide are designed and verified through numerical simulation. They can be implemented through standard photonic crystal fabrication. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Photonic devices need mechanical components to support, connect and stack functional photonic components. Most previous device designs simply separate the mechanical components from the photonic components beyond a sufficient distance at which the interference is negligible. With increasing demand for smaller device sizes, it is desirable to integrate mechanical components into photonic components directly, thus making photonic devices more compact with even potential extension to the multilayered layout analogous to integrated circuits. In this paper, we propose a cloaking method to embed mechanical connections inside photonic components of integrated photonic devices. Different from previous cloak designs [1–13] generally aiming to make objects free from external electromagnetic detection, here this idea is applied to accommodating mechanical components into a photonic device in a more compact way. Our design candidate is a photonic subwavelength cloak made of elliptical rod arrays. Similar structures have been utilized in photonic crystals for both imaging [14] and nonlinear second harmonic generation [15]. Our scheme consists of uniform square unit cells with elliptical silicon rods immersed in air, which achieves anisotropy and homogeneity simultaneously, and will greatly facilitate the fabrication process [16]. This method is compatible with previous photonic design strategies since only slight modification is needed. As examples of this cloaking scheme, cloaks that suppress the interference from the periphery and the inner region of the waveguide are designed and numerically verified with the finite-difference timedomain (FDTD) method. It is the first practical design for accommodating
non-photonic components in integrated photonic components, which will provide more flexibility for designing future photonic devices. 2. Anisotropy implementation with elliptical rods We use elliptical rod arrays to implement the material anisotropy. Fig. 1(a) shows the isofrequency curves from the dispersion diagram of a two dimensional (2D) unit cell with silicon elliptical rod (permittivity = 12) immersed in air ( = 1). The 2D assumption is generally harmless, but our recent research has shown that exceptions do exist in extremely thin slabs [17]. A complete analysis of finite thickness effects is beyond the scope of the present paper. The free space wavelength is λ = 8a, where a is the lattice constant; it is indicated as the bold blue line in Fig. 1(a). The effective refractive index for a particular wave vector k is determined by neff = ck/ω based on the isofrequency diagram, where c is the velocity of light in free space and ω is the frequency. From the isofrequency line, we determine the effective index surface, which is approximately an ellipse with effective principal indices nx = 1.30 and ny = 1.80. Note that this significant anisotropy exists only for transverse magnetic (TM) illumination, whose polarization will be utilized in this paper. As a comparison, the isofrequency diagram of a lattice with circular rods of radius r = 0.294a is illustrated in Fig. 1(b), where the effective index surface is a sphere with effective index n0 = 1.30. 3. Photonic cloak examples 3.1. Accommodation of peripheral non-photonic components
⁎ Corresponding author. Tel.: + 1 617 258 0649. E-mail address: [email protected] (H. Gao). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.06.028
Now we proceed to design a cloak using the elliptical rod array for accommodating more peripheral non-photonic components. We start
H. Gao et al. / Optics Communications 284 (2011) 4820–4823
4821
Fig. 1. Isofrequency diagrams of silicon elliptical (a) and circular (b) rod unit cells, where only the first TM band is shown. The size of the unit cell is a × a. The long and short axes of the elliptical rod in (a) are 0.95a and 0.5a, respectively; the radius of circular rod in (b) was chosen as 0.294a to match the effective index of the isotropic circular rod case with the effective value of the index along the x axis in the anisotropic elliptical rod case. Labels on the lines denote the corresponding normalized frequency ωa/2πc. The bold blue lines correspond to the free space wavelength λ = 8a used in this paper.
the boundary to appear virtually at its original position [11,18], thus maintaining the photonic performance. Assume that the size of the medium along the y direction has been reduced from unit size 1 to 1 − δ, the relative permittivity and permeability tensors become [11]
with a uniform medium above a perfect electric conductor (PEC) where the PEC is assumed to be the boundary of the photonic functional region. It is worth mentioning that although we use PEC to serve as a reflective surface for the sake of simplicity, this design also works for total internal reflection at the boundary in absence of the PEC. We then squeeze the medium immediately above the boundary along the y direction to create an empty area for placing more peripheral mechanical components, as shown in Fig. 2. The squeezed medium should be anisotropic in order for
0 =μ=@
1 = ð1−δÞ 0 0
1 0 0 A: 1−δ 0 0 1 = ð1−δÞ
ð1Þ
Fig. 2. Space transformation of the first design example. More peripheral area (green) for accommodating mechanical components is created by squeezing the uniform medium into an anisotropic medium. Blue arrows are ray trajectories, and red lines illustrate wavefronts.
a
b
d 80
80
60
60
y (a)
y (a)
c
40 20 0
0
40 20
50
100
x (a)
150
0
0
50
100
150
x (a)
Fig. 3. (a) and (b) Original structure and the structure of the first design example. Blue boxes are the zoomed-in view of the lattices. Red lines denote the interfaces between different media: PEC, isotropic and anisotropic medium. (c) and (d) FDTD results of original structure and accommodating design. Illumination is TM Gaussian source with incident angle of 45°. Black stripe is PEC and green stripe is area for accommodating mechanical components. Color shading denotes the magnetic field (Hz) distribution. Red is positive and blue is negative.
H. Gao et al. / Optics Communications 284 (2011) 4820–4823
In terms of practical implementation, it is preferable to use only non-magnetic materials. Thus, under TM illumination, while keeping the z component of μ equal to 1, the permittivity tensor should be [11] 0
1 = ð1−δÞ2 = 0 @ 0 0
1 0 0 A: 1 0 0 1 = ð1−δÞ2
Optical axis
ð2Þ
α
Therefore, the squeezed medium exhibits anisotropy with nx = n0 and ny = γn0, where γ = 1/(1 − δ). Fig. 3(a) illustrates the original structure where a uniform photonic medium is terminated by a PEC boundary. The uniform medium is implemented with an array of circular silicon rods with radius r = 0.294a with isotropic effective refractive index n0 = 1.30, as previously calculated in Fig. 1(b). The heights of the rod array and the PEC are 96a and 5a, respectively. To achieve more space for peripheral mechanical components, the lower part of the uniform layer, with thickness of 54a immediately above the PEC is squeezed to an anisotropic medium with thickness 39a. The corresponding squeezing factor is δ = 5/18 resulting in anisotropy with nx = 1.30 and ny = 1.80, as we previously calculated from Fig. 1(a). The resulting structure is shown in Fig. 3(b). From top to bottom, it contains four layers: (1) circular rod arrays of height 42a, (2) elliptical rod arrays of height 39a, (3) PEC layer with height 5a and (4) region for more mechanical components with height 15a. FDTD results of both the structures are illustrated and compared at Fig. 3(c) and (d). It can be observed that the resulting output beam profile is the same as that of the original circularly rod structure; hence the photonic medium maintains its performance although its size is reduced and more mechanical components can be accommodated around the periphery. Some reflection appears at the boundary of the circular and elliptical rod arrays, but it can be suppressed by using anti-reflection techniques. Besides accommodating mechanical components, this design can also be applied to the isolation layer between two neighboring photonic components for more compact layout design in the future.
As another example, we show how to use elliptical rod arrays for accommodating non-photonic components inside a waveguide directly. The non-photonic components are placed in a parallelogram region under TM illumination as shown in Fig. 4(a). The original isotropic media above and below the parallelogram region are squeezed, resulting in four triangular uniform anisotropic regions. The transformation from virtual coordinates (x, y) to physical coordinates (x′, y′) is [12,13,19] y′ = κy + τða− jxj Þ;
ð3Þ
where κ = [tan(α + β) − tanβ]/tan(α + β) and τ = tanβ. To make the materials non-magnetic while maintaining the required refractive indices, the permittivity tensor in the x–y plane becomes [19] ′ = 0
1 = κ2 −τ = κ2
−τ = κ2 1 + τ2 = κ2
! :
ð4Þ
The corresponding optical axes are all aligned to the vertical axis with angle [12] θ=
1 2τ : arctan 2 2 κ + τ2 −1
β
1
D 2 1 2
b
c
150
150
100
100
50
50
0 0
50
100
0 0
50
d
100
x (a)
x (a)
150
e 150
100
100
50
50
y (a)
3.2. Accommodation of internal non-photonic components
x′ = x;
a
y (a)
4822
ð5Þ
Here θ = 30 ∘ with effective refractive indices nx = 1.30 and ny = 1.80 as in the design of Fig. 1(a). The ambient refractive index is 1.37 which corresponds to an array of circular rods with radius r = 0.33a. This design was also verified with FDTD under TM plane wave illumina-
0 0
50
100
x (a)
0 0
50
100
x (a)
Fig. 4. (a) Structure of the second design example, composed of uniform elliptical and circular silicon rod arrays. Grey parallelogram in the middle is the region for mechanical components and magenta arrows are the optical axes of the anisotropic media. The dimensions are D = 50a, α = 38.5∘ and β = 13.3∘. (b)–(e) FDTD simulation results of this design illuminated by plane wave with λ = 8a at time t = 6.25λ/c, 12.5λ/c, 18.75λ/c, 25λ/c, respectively. Color shading denotes the magnetic field (Hz) distribution, where red is positive and blue is negative.
tion. The results are shown in Fig. 4(b)–(e). A plane wave with little distortion at the output to the right of the cloak can be observed. Imperfections are mainly due to scattering at the sharp edges and reflections at the boundaries between the isotropic and anisotropic media. Since the reflection mainly occurs to the left of the parallelogram region, the overall wave profile is preserved to a large degree, with negligible energy loss. The reflection can be further suppressed by using anti-reflection techniques. Since the propagation of the wave avoids the parallelogram shape region in the photonic device, the mechanical components can be accommodated in a photonic waveguide directly while the performance is still preserved to a large extent. A mechanical connector can penetrate through this waveguide, connect with other parts of the device and stack a large number of layers of “photonic boards” analogous to “printed circuit
H. Gao et al. / Optics Communications 284 (2011) 4820–4823
board” (PCB). Alternatively, through this empty region, an optical fiber can deliver optical signals across this waveguide along the third dimension similar to the “jumper” in circuits. This can be useful in future photonic industry, being able to provide more design flexibility in large scale photonic integration. In sum, we presented the first practical design for integrating mechanical components into photonic devices. Elliptical rod arrays are used as anisotropic materials. Designs that can be used for accommodating mechanical components from the periphery and inner region of a waveguide are designed and verified through the FDTD simulation. This design can be an extension and supplement of current photonic designs and provide more flexible design strategy for future photonic integration in a more compact form. Acknowledgements The authors thank Lei Tian and Satoshi Takahashi for useful discussions. Financial support was provided by Singapore's National Research Foundation through the Singapore-MIT Alliance for Research and Technology (SMART) Centre and the Air Force Office of Scientific Research MURI program on Nanomembranes under contract No. FA9550-08-1-0379.
4823
References [1] U. Leonhardt, Science 312 (5781) (2006) 1777. [2] J.B. Pendry, D. Schurig, D.R. Smith, Science 312 (5781) (2006) 1780. [3] D. Schurig, J.J. Mock, B.J. Justice, S.A. Cummer, J.B. Pendry, A.F. Starr, D.R. Smith, Science 314 (5801) (2006) 977. [4] R. Liu, C. Ji, J.J. Mock, J.Y. Chin, T.J. Cui, D.R. Smith, Science 323 (5912) (2009) 366. [5] T. Ergin, N. Stenger, P. Brenner, J.B. Pendry, M. Wegener, Science 328 (5976) (2010) 337. [6] J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, Nat. Mater. 8 (7) (2009) 568. [7] L.H. Gabrielli, J. Cardenas, C.B. Poitras, M. Lipson, Nat. Photonics 3 (8) (2009) 461. [8] C. Qiu, L. Hu, B. Zhang, B.I. Wu, S.G. Johnson, J.D. Joannopoulos, Opt. Express 17 (16) (2009) 13467. [9] J. Li, J.B. Pendry, Phys. Rev. Lett. 101 (20) (2008) 203901–1. [10] W. Cai, U.K. Chettiar, A.V. Kildishev, V.M. Shalaev, Nat. Photonics 1 (4) (2007) 224. [11] B. Zhang, T. Chan, B.I. Wu, Phys. Rev. Lett. 104 (23) (2010) 233903. [12] B. Zhang, Y. Luo, X. Liu, G. Barbastathis, Phys. Rev. Lett. 106 (3) (2011) 033901. [13] X. Chen, Y. Luo, J. Zhang, K. Jiang, J.B. Pendry, S. Zhang, Nat. Commun. 2 (2011) 176. [14] S. Feng, Z.Y. Li, Z.F. Feng, B.Y. Cheng, D.Z. Zhang, Phys. Rev. B 72 (7) (2005) 075101. [15] J. Li, Z. Li, D. Zhang, J. Appl. Phys. 102 (9) (2007) 093101. [16] S. Takahashi, C. Chang, S.Y. Yang, G. Barbastathis, Optical MEMS and Nanophotonics, 2010 International Conference on, IEEE, 2010, p. 179. [17] H. Gao, B. Zhang, S.G. Johnson, G. Barbastathis, Guidance condition correction into the design of two dimensional nanophotonic devices, submitted to Optical MEMS and Nanophotonics, 2011 International Conference on (2011), 2011. [18] T. Han, C.-W. Qiu, Opt. Express 18 (12) (2010) 13038. [19] S. Xi, H. Chen, B.I. Wu, J.A. Kong, IEEE Microwave Wireless Compon. Lett. 19 (3) (2009) 131.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Photonic cloak made of subwavelength dielectric elliptical rod arrays Hanhong Gao a,⁎, Baile Zhang b, George Barbastathis b, c a b c
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Singapore-MIT Alliance for Research and Technology (SMART) Centre, 3 Science Drive 2, Singapore 117543, Singapore Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
a r t i c l e
i n f o
Article history: Received 1 May 2011 Received in revised form 14 June 2011 Accepted 15 June 2011 Available online 30 June 2011 Keywords: Photonic device Subwavelength structure Elliptical rod arrays
a b s t r a c t Modern photonic devices require more and more compact integration of photonic components and mechanical components, which, however, will cause interference and degrade the photonic performance. Here we propose a method that can be used to reduce the interference caused by non-photonic parts in a photonic device. It employs a photonic cloak made of subwavelength dielectric elliptical rod arrays in a photonic waveguide environment. This cloak design is able to preserve the device's photonic performance with the mechanical components penetrating through, and thus provide a more compact platform. Cloaks that suppress the interference from the periphery and the inner region of the waveguide are designed and verified through numerical simulation. They can be implemented through standard photonic crystal fabrication. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Photonic devices need mechanical components to support, connect and stack functional photonic components. Most previous device designs simply separate the mechanical components from the photonic components beyond a sufficient distance at which the interference is negligible. With increasing demand for smaller device sizes, it is desirable to integrate mechanical components into photonic components directly, thus making photonic devices more compact with even potential extension to the multilayered layout analogous to integrated circuits. In this paper, we propose a cloaking method to embed mechanical connections inside photonic components of integrated photonic devices. Different from previous cloak designs [1–13] generally aiming to make objects free from external electromagnetic detection, here this idea is applied to accommodating mechanical components into a photonic device in a more compact way. Our design candidate is a photonic subwavelength cloak made of elliptical rod arrays. Similar structures have been utilized in photonic crystals for both imaging [14] and nonlinear second harmonic generation [15]. Our scheme consists of uniform square unit cells with elliptical silicon rods immersed in air, which achieves anisotropy and homogeneity simultaneously, and will greatly facilitate the fabrication process [16]. This method is compatible with previous photonic design strategies since only slight modification is needed. As examples of this cloaking scheme, cloaks that suppress the interference from the periphery and the inner region of the waveguide are designed and numerically verified with the finite-difference timedomain (FDTD) method. It is the first practical design for accommodating
non-photonic components in integrated photonic components, which will provide more flexibility for designing future photonic devices. 2. Anisotropy implementation with elliptical rods We use elliptical rod arrays to implement the material anisotropy. Fig. 1(a) shows the isofrequency curves from the dispersion diagram of a two dimensional (2D) unit cell with silicon elliptical rod (permittivity = 12) immersed in air ( = 1). The 2D assumption is generally harmless, but our recent research has shown that exceptions do exist in extremely thin slabs [17]. A complete analysis of finite thickness effects is beyond the scope of the present paper. The free space wavelength is λ = 8a, where a is the lattice constant; it is indicated as the bold blue line in Fig. 1(a). The effective refractive index for a particular wave vector k is determined by neff = ck/ω based on the isofrequency diagram, where c is the velocity of light in free space and ω is the frequency. From the isofrequency line, we determine the effective index surface, which is approximately an ellipse with effective principal indices nx = 1.30 and ny = 1.80. Note that this significant anisotropy exists only for transverse magnetic (TM) illumination, whose polarization will be utilized in this paper. As a comparison, the isofrequency diagram of a lattice with circular rods of radius r = 0.294a is illustrated in Fig. 1(b), where the effective index surface is a sphere with effective index n0 = 1.30. 3. Photonic cloak examples 3.1. Accommodation of peripheral non-photonic components
⁎ Corresponding author. Tel.: + 1 617 258 0649. E-mail address: [email protected] (H. Gao). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.06.028
Now we proceed to design a cloak using the elliptical rod array for accommodating more peripheral non-photonic components. We start
H. Gao et al. / Optics Communications 284 (2011) 4820–4823
4821
Fig. 1. Isofrequency diagrams of silicon elliptical (a) and circular (b) rod unit cells, where only the first TM band is shown. The size of the unit cell is a × a. The long and short axes of the elliptical rod in (a) are 0.95a and 0.5a, respectively; the radius of circular rod in (b) was chosen as 0.294a to match the effective index of the isotropic circular rod case with the effective value of the index along the x axis in the anisotropic elliptical rod case. Labels on the lines denote the corresponding normalized frequency ωa/2πc. The bold blue lines correspond to the free space wavelength λ = 8a used in this paper.
the boundary to appear virtually at its original position [11,18], thus maintaining the photonic performance. Assume that the size of the medium along the y direction has been reduced from unit size 1 to 1 − δ, the relative permittivity and permeability tensors become [11]
with a uniform medium above a perfect electric conductor (PEC) where the PEC is assumed to be the boundary of the photonic functional region. It is worth mentioning that although we use PEC to serve as a reflective surface for the sake of simplicity, this design also works for total internal reflection at the boundary in absence of the PEC. We then squeeze the medium immediately above the boundary along the y direction to create an empty area for placing more peripheral mechanical components, as shown in Fig. 2. The squeezed medium should be anisotropic in order for
0 =μ=@
1 = ð1−δÞ 0 0
1 0 0 A: 1−δ 0 0 1 = ð1−δÞ
ð1Þ
Fig. 2. Space transformation of the first design example. More peripheral area (green) for accommodating mechanical components is created by squeezing the uniform medium into an anisotropic medium. Blue arrows are ray trajectories, and red lines illustrate wavefronts.
a
b
d 80
80
60
60
y (a)
y (a)
c
40 20 0
0
40 20
50
100
x (a)
150
0
0
50
100
150
x (a)
Fig. 3. (a) and (b) Original structure and the structure of the first design example. Blue boxes are the zoomed-in view of the lattices. Red lines denote the interfaces between different media: PEC, isotropic and anisotropic medium. (c) and (d) FDTD results of original structure and accommodating design. Illumination is TM Gaussian source with incident angle of 45°. Black stripe is PEC and green stripe is area for accommodating mechanical components. Color shading denotes the magnetic field (Hz) distribution. Red is positive and blue is negative.
H. Gao et al. / Optics Communications 284 (2011) 4820–4823
In terms of practical implementation, it is preferable to use only non-magnetic materials. Thus, under TM illumination, while keeping the z component of μ equal to 1, the permittivity tensor should be [11] 0
1 = ð1−δÞ2 = 0 @ 0 0
1 0 0 A: 1 0 0 1 = ð1−δÞ2
Optical axis
ð2Þ
α
Therefore, the squeezed medium exhibits anisotropy with nx = n0 and ny = γn0, where γ = 1/(1 − δ). Fig. 3(a) illustrates the original structure where a uniform photonic medium is terminated by a PEC boundary. The uniform medium is implemented with an array of circular silicon rods with radius r = 0.294a with isotropic effective refractive index n0 = 1.30, as previously calculated in Fig. 1(b). The heights of the rod array and the PEC are 96a and 5a, respectively. To achieve more space for peripheral mechanical components, the lower part of the uniform layer, with thickness of 54a immediately above the PEC is squeezed to an anisotropic medium with thickness 39a. The corresponding squeezing factor is δ = 5/18 resulting in anisotropy with nx = 1.30 and ny = 1.80, as we previously calculated from Fig. 1(a). The resulting structure is shown in Fig. 3(b). From top to bottom, it contains four layers: (1) circular rod arrays of height 42a, (2) elliptical rod arrays of height 39a, (3) PEC layer with height 5a and (4) region for more mechanical components with height 15a. FDTD results of both the structures are illustrated and compared at Fig. 3(c) and (d). It can be observed that the resulting output beam profile is the same as that of the original circularly rod structure; hence the photonic medium maintains its performance although its size is reduced and more mechanical components can be accommodated around the periphery. Some reflection appears at the boundary of the circular and elliptical rod arrays, but it can be suppressed by using anti-reflection techniques. Besides accommodating mechanical components, this design can also be applied to the isolation layer between two neighboring photonic components for more compact layout design in the future.
As another example, we show how to use elliptical rod arrays for accommodating non-photonic components inside a waveguide directly. The non-photonic components are placed in a parallelogram region under TM illumination as shown in Fig. 4(a). The original isotropic media above and below the parallelogram region are squeezed, resulting in four triangular uniform anisotropic regions. The transformation from virtual coordinates (x, y) to physical coordinates (x′, y′) is [12,13,19] y′ = κy + τða− jxj Þ;
ð3Þ
where κ = [tan(α + β) − tanβ]/tan(α + β) and τ = tanβ. To make the materials non-magnetic while maintaining the required refractive indices, the permittivity tensor in the x–y plane becomes [19] ′ = 0
1 = κ2 −τ = κ2
−τ = κ2 1 + τ2 = κ2
! :
ð4Þ
The corresponding optical axes are all aligned to the vertical axis with angle [12] θ=
1 2τ : arctan 2 2 κ + τ2 −1
β
1
D 2 1 2
b
c
150
150
100
100
50
50
0 0
50
100
0 0
50
d
100
x (a)
x (a)
150
e 150
100
100
50
50
y (a)
3.2. Accommodation of internal non-photonic components
x′ = x;
a
y (a)
4822
ð5Þ
Here θ = 30 ∘ with effective refractive indices nx = 1.30 and ny = 1.80 as in the design of Fig. 1(a). The ambient refractive index is 1.37 which corresponds to an array of circular rods with radius r = 0.33a. This design was also verified with FDTD under TM plane wave illumina-
0 0
50
100
x (a)
0 0
50
100
x (a)
Fig. 4. (a) Structure of the second design example, composed of uniform elliptical and circular silicon rod arrays. Grey parallelogram in the middle is the region for mechanical components and magenta arrows are the optical axes of the anisotropic media. The dimensions are D = 50a, α = 38.5∘ and β = 13.3∘. (b)–(e) FDTD simulation results of this design illuminated by plane wave with λ = 8a at time t = 6.25λ/c, 12.5λ/c, 18.75λ/c, 25λ/c, respectively. Color shading denotes the magnetic field (Hz) distribution, where red is positive and blue is negative.
tion. The results are shown in Fig. 4(b)–(e). A plane wave with little distortion at the output to the right of the cloak can be observed. Imperfections are mainly due to scattering at the sharp edges and reflections at the boundaries between the isotropic and anisotropic media. Since the reflection mainly occurs to the left of the parallelogram region, the overall wave profile is preserved to a large degree, with negligible energy loss. The reflection can be further suppressed by using anti-reflection techniques. Since the propagation of the wave avoids the parallelogram shape region in the photonic device, the mechanical components can be accommodated in a photonic waveguide directly while the performance is still preserved to a large extent. A mechanical connector can penetrate through this waveguide, connect with other parts of the device and stack a large number of layers of “photonic boards” analogous to “printed circuit
H. Gao et al. / Optics Communications 284 (2011) 4820–4823
board” (PCB). Alternatively, through this empty region, an optical fiber can deliver optical signals across this waveguide along the third dimension similar to the “jumper” in circuits. This can be useful in future photonic industry, being able to provide more design flexibility in large scale photonic integration. In sum, we presented the first practical design for integrating mechanical components into photonic devices. Elliptical rod arrays are used as anisotropic materials. Designs that can be used for accommodating mechanical components from the periphery and inner region of a waveguide are designed and verified through the FDTD simulation. This design can be an extension and supplement of current photonic designs and provide more flexible design strategy for future photonic integration in a more compact form. Acknowledgements The authors thank Lei Tian and Satoshi Takahashi for useful discussions. Financial support was provided by Singapore's National Research Foundation through the Singapore-MIT Alliance for Research and Technology (SMART) Centre and the Air Force Office of Scientific Research MURI program on Nanomembranes under contract No. FA9550-08-1-0379.
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