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Development of curved two-dimensional photonic crystal waveguides
Optics Communications 281 (2008) 5788–5792
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Development of curved two-dimensional photonic crystal waveguides Jun-ichiro Sugisaka a,b,*, Noritsugu Yamamoto a, Makoto Okano a, Kazuhiro Komori a, Toyohiko Yatagai b,c, Masahide Itoh b a
Photonics Research Institute, National Institute of Advanced Industrial Science and Technology, Tsukuba central 2, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan Applied Physics Institute, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8573, Japan c Center for Optical Research and Education, Utsunomiya Univesity, 7-1-2, Yoto, Utsunomiya, Tochigi 321-8585, Japan b
a r t i c l e
i n f o
Article history: Received 10 June 2008 Received in revised form 19 August 2008 Accepted 19 August 2008
a b s t r a c t A two-dimensional photonic crystal waveguide with a novel geometry is introduced. The center line of this waveguide is bent along a free-curve such that the direction of the propagating light can be changed without scattering or reflection losses. The design method is described for a triangular lattice, its optical properties such as transmission spectrum and dispersion relation are calculated, and actual devices are then fabricated and demonstrated that they worked as optical waveguides. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals (PhCs) are optical structures that have a spatially periodic dielectric constant. Important characteristics of photonic crystals include the existence of photonic bandgaps and unusual dispersion relations for the propagating light, thereby supporting applications such as high Q cavities and ultrafine waveguides. A line defect in a two-dimensional (2D) PhC functions as an optical waveguide. The incident light is strongly confined by the photonic bandgap and propagates along the defect. The confinement of light by the bandgap is distinct from total reflection. It enables waveguides having sharp bending points [1]. The 2D-PhC waveguides are mainly fabricated in dielectric slabs [2], and the light propagation through the 2D-PhC slab waveguides have been observed [3]. The 2D-PhC slabs can confine the light in cavities or waveguides by photonic band gaps in planar directions and by total reflection in the direction perpendicular to the slab. In this paper, we point out some of the problems of traditional bent 2D-PhC slab waveguides such as radiation and reflection, and propose free-curve waveguides as a solution. Their transmission characteristics are calculated numerically by 2D-FDTD (two dimensional finite difference time domain) method, indicating that a free-curve waveguide can confine light just as well as a conventional straight waveguide. We also show that the free-curve waveguide can decrease the reflection of propagating light. Since 2DFDTD method cannot simulate the radiation from the surface of the 2D-PhC slab, we fabricate such free-curve waveguides in
* Corresponding author. Address: Applied Physics Institute, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8573, Japan. E-mail address: [email protected] (J.-i. Sugisaka). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.08.039
190 nm thick of GaAs slab, and characterize their transmission spectra, indicating the decreasing of the radiation. 2. Issues with bent waveguides One critical problem with standard bent waveguides is that a significant fraction of the light leaks out of them at the bending points. Until now, this radiation has not been noticed. However the radiation can be observed as a bright spot at the bending point of the 2D-PhC slab. This radiation causes the increase of the propagation loss of the waveguides or substantial decrease in the Q value of the ring cavity [4]. In addition, the radiation can couple neighboring waveguides. For example, a directional coupler based on a 2D-PhC slab [5] has two pairs of sharply bent points near each other. Consequently its transmission properties (such as the extinction ratio) are degraded. A second problem with bent waveguides is that some of the propagating light is reflected backward at the bending points [6]. This effect not only causes losses but also fluctuations in the transmission spectrum due to Fabry–Perot interference [7]. A third problem is the geometrical inflexibility of traditional bent PhC waveguides. For a triangular lattice, one can only change direction by 60° or 120°. Otherwise one could not maintain the periodicity of the PhC, which is required to confine and propagate light through the waveguide. Many approaches have been proposed to deal with these issues. One of the most frequently studied approaches is to modulate the geometry of 2D-PhC at the bending point. For example, a study to displace the position of air holes at the bent point [8], to change the radii of dielectric rods around the bent point [9], to add some small air holes in the bent wavegide [10], to replace the regular geometry (air circular holes) by annular air groove [11], and to apply a topological optimization [12–14]. Another study is to sandwich a
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2D-PhC slab between two 3D-PhCs, and inhibit the radiation from the surface of the slab [15]. A hybrid structures to use both the conventional waveguide and PhCs at bent points has been also proposed [16,17]. These approaches improve the transmission, but geometrical inflexibility and fabrication difficulties such as dielectric rod type of 2D-PhC, complicated shapes of air holes, and 3DPhC clad remain. Our approach is to change the direction of propagation adiabatically, whereby one propagation mode is converted to another without coupling to back-propagating or free-space radiating modes. This approach uses curved PhC waveguides, which was proposed by Arentoft et al. [18]. The confinement of light in the waveguide was, however, weaker than straight waveguide, and radiated toward outside of the waveguide [19]. After that, similar approaches have been proposed to use circular bent photonic crystals [20,21]. These waveguides are designed by using 2D-PhCs with dielectric rod [20,21], which cannot adopt to 2D-PhC slab. The waveguides with curvilinear-lattice [21] have different crystal systems, which can change the dispersion relation and can increase the reflectance. We design and fabricate waveguides bent along general free curves by using 2D-PhC slab with single crystal system, and realize the suppression of radiation and reflection. 3. Design of free-curve waveguides We assume a waveguide is designed in x y plane. First, define a free curve by position vector f ðtÞ in x y plane. As shown in Fig. 1, this free curve is the center line of the waveguide. Anchor points f ðtn Þ are indicated by dots along the curve f ðtÞ in Fig. 1, where tn are given by the recurrence equation
1 a¼ 2
Z
d f ðtÞdt: tn1 dt tn
if n is an even integer, and
1 pffiffiffi d f ðt n Þ ^z rðm; nÞ ¼ f ðt n Þ þ ðm þ Þ 3a dtd f ðt n Þ 2 dt
ð3Þ
if n is an odd integer, where ^ z is a unit vector perpendicular to x y plane, and m is all integers from 10 to 10 except there are no air holes at rð0; 2n0 Þ (where n0 is any integer) so as to make a line defect. We have not considered when the air holes are overlapped each other. In the following sections, we discuss only the free curve waveguide whose no air holes overlapped. This condition provides a limit to the maximum curvature of the waveguide, which is determined by the lattice constant, radii of the air holes and the width of the PhC clad layer. 4. Optical properties of free-curve waveguides The transmission spectra and dispersion relation are quantified for the TE mode of a free-curve waveguide as an example. 4.1. Transmission properties We modeled four waveguides with different curvatures: (a) 0°/a (straight waveguide), (b) 0.5°/a, (c) 1.0°/a, and (d) 2.0°/a. The angles between the input and output ports for (b), (c), and (d) are the same, namely 60°, and thus the length of the waveguide is 120a, 60a, and 30a, respectively. The length of the straight waveguide was 60a. The lattice constant a is set to 390 nm, and the radii of air holes for each waveguide were set to 0.29a. The transmission spectra were calculated with 2D-FDTD method. A lattice constant was discretized by 15 Yee cells. The refractive index of air holes and that of 190 nm thick of GaAs slab were set to 1.0 and 2.65,
ð1Þ
pffiffiffi Air-filled circular holes are arranged with a periodicity of 3a along lines normal to f ðtÞ that pass through the points f ðtn Þ, as shown in Fig. 1 by dashed lines. The center positions (rðm; nÞ) of the air holes are given by
pffiffiffi d f ðt n Þ ^z rðm; nÞ ¼ f ðt n Þ þ m 3a dtd f ðt n Þ
a
1.0 0.5 0 0.26
ð2Þ
0.27
0.28
0.29
0.3
0.27
0.28
0.29
0.3
0.27
0.28
0.29
0.3
0.27
0.28
0.29
0.3
dt
b
1.0
Transmittance
0.5 0 0.26 1.0
c
0.5
y z
t 2n x
t 2n+1
t 2n-1
0 0.26 1.0
d
0.5 0 0.26
Fig. 1. The structure of a PhC free-curve waveguide. A curve f ðtÞ shown as a solid line is defined and anchor points tn are laid out along it. The distance between adjacent anchor points is equal to half of the lattice constant a=2. Air-filledpcircular ffiffiffi holes are then arranged along the dashed normal lines with a spacing of 3a.
ωa 2πc
Fig. 2. Transmission spectra of bent waveguides. The curvature of each waveguide is (a) 0°/a, (b) 0.5°/a, (c) 1.0°/a(60a), and (d) 2.0°/a. The transmittances are normalized to that of a straight waveguide of length 10a.
J.-i. Sugisaka et al. / Optics Communications 281 (2008) 5788–5792
Transmittance
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1.0 0.8 0.6 0.4 0.2 0 0.26
0.27
Transmittance
(a) Waveguide A
1.0 0.8 0.6 0.4 0.2 0 0.26
0.27
0.28
0.29
0.3
0.28
0.29
0.3
0.28
0.29
0.3
ωa 2πc
ωa 2πc
Transmittance
(b) Waveguide B
1.0 0.8 0.6 0.4 0.2 0 0.26
0.27
ωa 2πc
(c) Waveguide C Fig. 3. Transmission spectra of a waveguide with two bent guides at its ends. (a) Waveguide A has 60-degree cornered waveguides. (b) Waveguide B has circular ends with a curvature of 2.0°/a. (c) Waveguide C is a sinusoidal guide that connects smoothly to the straight waveguide. The length of the straight waveguide between the bending points is 100a. The corresponding spectra are shown to the right of each geometrical sketch.
respectively. At the input port of each waveguide, oscillating dipoles were placed, and the component of the Poynting vector parallel to the waveguide was observed at the output port. In order to prevent reflecting at the waveguide terminal, the light output from the waveguide terminals were radiated to GaAs slab, and finally absorbed by PML (Perfectly Matched Layer). Next, the transmission spectra were normalized to that of a straight waveguide of length 10a. The results are shown in Fig. 2. These spectra show that incident light can be confined by and propagate along a waveguide bent into a circular arc. If the curvature of the waveguide is less than 0.5°/a, the transmittance is nearly equal to that of a straight waveguide. For larger curvatures, we see that, although the normalized lower cut-off frequency remains 0.273, the useful transmission band starts at somewhat higher frequencies. This effect results from the positional dependence of photonic bandgap as the crystalline structure of the waveguide is distorted. At this moment, the free-curve waveguides cannot have smaller curvature radius than the ridge waveguides. To increase the curvature limit, we should eliminate the positional dependence of photonic band gap by some geometrical modulations such as radii of air holes. The 2D-FDTD method cannot simulate the radiation from the surface of the slab, thus, the cut-off for higher frequency side cannot appear in the transmission spectra. The higher cut-off frequency can be found from the dispersion diagram, which is discussed in the next subsection. In order to confirm the reduction in the reflectance at the bending points, we calculated the transmittance of a straight waveguide (of
length 100a) with bent waveguides connected to both of its ends. The reflection can be observed as a fluctuation in the transmittance due to Fabry–Perot interferences between two bending points. For this purpose, we prepared three configurations of bent waveguides as illustrated in Fig. 3: A – a sharply cornered waveguide, B – a circular arc waveguide, and C – a sinusoidally curved waveguide. The waveguide C is a free-curve waveguide that can match the waveguide curvature with the straight waveguides at the connecting point. The angle between the input and output ports are 60° in all three cases, and the maximum curvature of waveguide C is equal to the (constant) curvature of waveguide B, namely 1°/a. The graphs in Fig. 3a–c plot the transmittance of waveguide A, B, and C, respectively. The sharp periodic peaks in Fig. 3a are due to Fabry–Perot interferences between the two sharp corners. On the other hand, in Fig. 3b, only a small fluctuation remains due to a weak reflection from the connecting point between the circular and straight waveguides. Finally, for waveguide C, fluctuations cannot be observed in the spectrum in Fig. 3b because it was designed to have matching waveguide curvatures at the connecting points. The result of waveguide C also shows that the shape of our suggested waveguide is not limited to the circular curve. 4.2. Dispersion relation The plane-wave expansion method and 2D-FDTD method with Bloch boundary conditions are standard techniques for calculating dispersion relations for a photonic band. The PhC free-curve wave-
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0.33
0.31
0.29 0.28 0.27
0.20
0.30 0.40 Normalized wavenumber [2π/a]
Fig. 5. Dispersion relation for propagation along a straight waveguide (squares), along a free-curve waveguide with curvature 1.0°/a (open circles and solid line), and along a free-curve waveguide with curvature 2.0°/a (filled circles). The dashed curve is the light line.
Power spectrum [a.u.]
Power spectrum [a.u.] 0
0
60
120
Length [a]
Power spectrum [a.u.]
magnetic field magnetic field [a.u.] magnetic field [a.u.]
c
0.30
Fig. 6. SEM images of PhC circular waveguides with curvatures of (a) 1.0°/a and (b) 2.0°/a.
a
b
Straight waveguide Free curve waveguide (1.0degree/a) Free curve waveguide (2.0degree/a) Light line
0.32
ωa 2πc
guide, however, has no translational symmetry, and thus Bloch boundaries cannot be defined. Instead the following method [22] was adopted, as illustrated in Fig. 4. First, the field distribution of light propagating with a frequency x is calculated. Second, the magnetic field perpendicular to the 2D-PhC slab is determined along the center line of the waveguide. It is Fourier transformed to find the peaks. We obtained four peaks, derived from two peaks folded into the first Brillouin zone. These two peaks correspond to backward and forward traveling waves having the same propagation mode. We thereby find a relation between frequency and wavenumber for the confined light, and so obtain the dispersion relation. To calculate the field distributions, as shown in Fig. 4a, we set the length and angle between the input and output ports to be 120a and 120°, respectively (the curvature becomes 1.0°/a). The normalized wavenumber resolution is therefore 1/120. The resulting dispersion relation is graphed in Fig. 5. In the same way, the dispersion relation of the waveguide with the curvature 2.0°/a was also calculated. The length and the angle between input and output ports were set to 60a and 120°, respectively. The result is also graphed in Fig. 5. The dispersion relation is in excellent agreement with that for a straight waveguide calculated by the 2D-FDTD method. The reason why the wavenumber resolution for the waveguide with the curva-
Wavelength [nm] 1360 1340
1380 150
a
1320 in
100
out
50 0
0.285
0.29
0.295
Wavelength [nm] 1360 1340
1380 300
ωa/2πc
b
1320 in
200 out
100 0
0.285
0.29
0.295
Wavelength [nm] 1360 1340
1380 300
ωa/2πc
c
1320 in
200
out
100 0
0.285
ωa/2πc
0.29
0.295
Fig. 7. Transmission spectra of a waveguide having (a) cornered waveguides, (b) circular waveguides with a curvature of 1.0°/a, and (c) circular waveguides with a curvature of 2.0°/a.
-1
k-1
-k
0
k
-k+1
1
Normalized wavenumber ka/2π Fig. 4. Procedure to obtain the dispersion relation for propagation in a free-curve waveguide: (a) The field distribution is calculated by the 2D-FDTD method. (b) Its value is extracted along the center line of the waveguide. (c) That value is Fourier transformed to obtain the peak location.
ture of 2.0°/a is low is because the length of the calculated waveguide is half of that of 1.0°/a, and the normalized wavenumber resolution is 1/60. The dispersion relation of our waveguides, therefore, does not depend on the curvature, and this result shows
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J.-i. Sugisaka et al. / Optics Communications 281 (2008) 5788–5792
Next, we imaged the waveguides from above to look for light leakage at the bending points, as shown in Fig. 8. The incident light for the wavelength of 1335.5 nm propagates rightward. One sees bright spots at the sharp corners indicated by the arrows in Fig. 8a, demonstrating that light is radiated at those bends. On the other hand, in the case of the free-curve bends in Fig. 8b and c, no bright spots are observed. The radiation was not observed over the entire transmission band in free-curve waveguide, indicating a considerable reduction in lost light from free-curve waveguides. Fig. 8. The top view of (a) sharply bent waveguide, (b) circular bent waveguide (curvature of 1.0°/a) and (c) circular bent waveguide (curvature of 2.0°/a). Each image is the top view taken with an infrared vidicon. In the case of (a), bright spots (radiation) can be observed (shown by the arrows). On the other hand in the case of free-curve bent waveguide ((b) and (c)), radiations were not observed.
that the dispersion relation of a PhC waveguide is conserved even if the guide is bent along a free-curve. By finding the frequency of the point of intersection of the dispersion curve with the light line (dashed in Fig. 5), a normalized cut-off wavelength of 0.294 is found, in agreement with that of a straight waveguide. 5. Fabrication and testing of free-curve waveguides To investigate the effect of free-curve waveguides on the radiation loss, waveguides were fabricated on GaAs epitaxial wafers, with a 1-lm AlGaAs sacrificial layer on the substrate and 190 nm of GaAs on top of that. We lithographically fabricated PhC waveguides using an electron beam (EB) followed by dry etching of the GaAs top layer using an inductively coupled plasma. Then the sacrificial layer was removed using hydrofluoric acid. Scanning electron microscope (SEM) images of the resulting waveguides are shown in Fig. 6. The lattice constant a was set to 390 nm, and radii of air holes are set to 0.29a. The both ends of all waveguides have planar solid immersion lenses (SIL) [23], which can decrease the insertion loss and Fabry–Perot resonance in the waveguide. The curvatures of the waveguides in panels (a) and (b) are 1.0°/a and 2.0°/a, respectively. Next we measured the transmission spectra of these fabricated waveguides. The output waveguide needs to be parallel to the input waveguides due to the alignment of the measuring system. Therefore, the devices under test had two oppositely bent waveguides. A straight waveguide with a length of 100a was inserted between these bent waveguides, as shown in the insets in Fig. 7. Fig. 7a is a spectrum of a PhC waveguide with sharp corners, while Fig. 7b and c are for free-curve waveguides with curvatures of 1.0°/ a and 2.0°/a, respectively. The results confirm that light is confined and propagates through the waveguides. However, we see that the transmittance at long wavelengths decreases as the curvature of a waveguide increases. The flat transmittance at longer wavelength side in Fig. 7b and c often appear in the spectra of the waveguides which contain the SIL, and that is due to the fabrication error of the shapes of SILs. The spectrum of the waveguide with sharp bends in Fig. 7a oscillates with a period of about 4 nm at short wavelengths and with a period of about 2 nm at long wavelengths. These periods agree with the peak interval for Fabry–Perot resonances across a length of 100a. We conclude that the light reflects at the corners. On the other hand, the spectra of the free-curve waveguides in Fig. 7b and c exhibit minimal oscillations.
6. Conclusion A conventional PhC waveguide with sharp corners exhibits problems such as radiation and reflection at the bending points. As a solution to those problems, we developed a method to fabricate a PhC waveguide that is bent along a free-curve. Their transmittances and dispersion relation were calculated, showing that a free-curve waveguide can confine and propagate light and that the reflection at the bends is diminished. It was found that a free-curve waveguide has nearly the same dispersion relation as a straight waveguide. Free-curve waveguides were fabricated and it was experimentally demonstrated that radiation and reflection at the bending points were negligible. Such free-curve waveguides afford flexibility for the design of nanowire circuits and could help with the development of optical integrated PhC circuits. References [1] A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, Phys. Rev. Lett. 77 (1996) 3787. [2] P.L. Gourley, J.R. Wendt, G.A. Vawter, T.M. Brennan, B.E. Hammons, Appl. Phys. Lett. 64 (1994) 687. [3] T. Baba, N. Fukaya, J. Yonekura, Electron. Lett. 35 (1999) 654. [4] S.-H. Jeong, N. Yamamoto, J. Sugisaka, M. Okano, K. Komori, J. Opt. Soc. Am. 24 (2007) 1951. [5] N. Yamamoto, T. Ogawa, K. Komori, Opt. Express 14 (2006) 1223. [6] Alongkarn Chutinan, Susumu Noda, Jpn. J. Appl. Phys. 39 (2000) L595. [7] M.H. Shih, W.J. Kim, W. Kuang, J.R. Cao, S.-J. Choi, J.D. OfBrien, P.D. Dapkus, Appl. Phys. Lett. 86 (2005) 191104. [8] A. Talneau, M. Kafesaki, C.M. Soukoulis, M. Agio, Appl. Phys. Lett. 80 (2002) 547. [9] J. Smajic, C. Hafner, D. Erni, Opt. Express 11 (2003) 1378. [10] A. Chutinan, M. Okano, S. Noda, Appl. Phys. Lett. 80 (2002) 1698. [11] Y. Zhang, B. Li, Opt. Lett. 32 (2007) 787. [12] P.I. Borel, A. Harpoth, L.H. Frandsen, M. Kristensen, Opt. Express 12 (2004) 1996. [13] L.H. Frandsen, A. Harpoth, P.I. Borel, M. Kristensen, J.S. Jensen, O. Sigmund, Opt. Express 12 (2004) 5916. [14] J.S. Jensen, O. Sigmund, Appl. Phys. Lett. 84 (2004) 2022. [15] Alongkarn Chutinan, Sajeev John, Ovidiu Toader, Phys. Rev. Lett. 90 (2003) 123901. [16] J. Cai, G.P. Nordin, S. Kim, J. Jiang Appl. Opt. 43 (2004) 4244. [17] S.H. Tao, M.B. Yu, J.F. Song, Q. Fang, R. Yang, G.Q. Lo, D.L. Kwong, Appl. Phys. Lett. 92 (2008) 031113. [18] J. Arentoft, T. Sondergaard, M. Kristensen, A. Boltasseva, M. Thorhauge, L. Frandsen, Electron. Lett. 38 (2002) 274. [19] S.I. Bozhevolnyi, V.S. Volkov, T. Sondergaard, A. Boltasseva, P.I. Borel, M. Kristensen, Phys. Rev. B 66 (2002) 235204. [20] N. Horiuchi, Y. Segawa, T. Nozokido, K. Mizuno, H. Miyazaki, Opt. Lett. 30 (2005) 973. [21] J. Zarbakhsh, F. Hagmann, S.F. Mingaleev, K. Busch, K. Hingerl, Appl. Phys. Lett. 84 (2004) 4687. [22] B. Dastmalchi, A. Mohtashami, K. Hingerl, J. Zarbakhsh, Opt. Lett. 32 (2007) 2915. [23] N. Ikeda, H. Kawashima1, Y. Sugimoto, T. Hasama1, K. Asakawa, H. Ishikawa, in: 19th International Conference on Indium Phosphide and Related Materials, 2007, p. 484.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Development of curved two-dimensional photonic crystal waveguides Jun-ichiro Sugisaka a,b,*, Noritsugu Yamamoto a, Makoto Okano a, Kazuhiro Komori a, Toyohiko Yatagai b,c, Masahide Itoh b a
Photonics Research Institute, National Institute of Advanced Industrial Science and Technology, Tsukuba central 2, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan Applied Physics Institute, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8573, Japan c Center for Optical Research and Education, Utsunomiya Univesity, 7-1-2, Yoto, Utsunomiya, Tochigi 321-8585, Japan b
a r t i c l e
i n f o
Article history: Received 10 June 2008 Received in revised form 19 August 2008 Accepted 19 August 2008
a b s t r a c t A two-dimensional photonic crystal waveguide with a novel geometry is introduced. The center line of this waveguide is bent along a free-curve such that the direction of the propagating light can be changed without scattering or reflection losses. The design method is described for a triangular lattice, its optical properties such as transmission spectrum and dispersion relation are calculated, and actual devices are then fabricated and demonstrated that they worked as optical waveguides. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals (PhCs) are optical structures that have a spatially periodic dielectric constant. Important characteristics of photonic crystals include the existence of photonic bandgaps and unusual dispersion relations for the propagating light, thereby supporting applications such as high Q cavities and ultrafine waveguides. A line defect in a two-dimensional (2D) PhC functions as an optical waveguide. The incident light is strongly confined by the photonic bandgap and propagates along the defect. The confinement of light by the bandgap is distinct from total reflection. It enables waveguides having sharp bending points [1]. The 2D-PhC waveguides are mainly fabricated in dielectric slabs [2], and the light propagation through the 2D-PhC slab waveguides have been observed [3]. The 2D-PhC slabs can confine the light in cavities or waveguides by photonic band gaps in planar directions and by total reflection in the direction perpendicular to the slab. In this paper, we point out some of the problems of traditional bent 2D-PhC slab waveguides such as radiation and reflection, and propose free-curve waveguides as a solution. Their transmission characteristics are calculated numerically by 2D-FDTD (two dimensional finite difference time domain) method, indicating that a free-curve waveguide can confine light just as well as a conventional straight waveguide. We also show that the free-curve waveguide can decrease the reflection of propagating light. Since 2DFDTD method cannot simulate the radiation from the surface of the 2D-PhC slab, we fabricate such free-curve waveguides in
* Corresponding author. Address: Applied Physics Institute, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8573, Japan. E-mail address: [email protected] (J.-i. Sugisaka). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.08.039
190 nm thick of GaAs slab, and characterize their transmission spectra, indicating the decreasing of the radiation. 2. Issues with bent waveguides One critical problem with standard bent waveguides is that a significant fraction of the light leaks out of them at the bending points. Until now, this radiation has not been noticed. However the radiation can be observed as a bright spot at the bending point of the 2D-PhC slab. This radiation causes the increase of the propagation loss of the waveguides or substantial decrease in the Q value of the ring cavity [4]. In addition, the radiation can couple neighboring waveguides. For example, a directional coupler based on a 2D-PhC slab [5] has two pairs of sharply bent points near each other. Consequently its transmission properties (such as the extinction ratio) are degraded. A second problem with bent waveguides is that some of the propagating light is reflected backward at the bending points [6]. This effect not only causes losses but also fluctuations in the transmission spectrum due to Fabry–Perot interference [7]. A third problem is the geometrical inflexibility of traditional bent PhC waveguides. For a triangular lattice, one can only change direction by 60° or 120°. Otherwise one could not maintain the periodicity of the PhC, which is required to confine and propagate light through the waveguide. Many approaches have been proposed to deal with these issues. One of the most frequently studied approaches is to modulate the geometry of 2D-PhC at the bending point. For example, a study to displace the position of air holes at the bent point [8], to change the radii of dielectric rods around the bent point [9], to add some small air holes in the bent wavegide [10], to replace the regular geometry (air circular holes) by annular air groove [11], and to apply a topological optimization [12–14]. Another study is to sandwich a
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2D-PhC slab between two 3D-PhCs, and inhibit the radiation from the surface of the slab [15]. A hybrid structures to use both the conventional waveguide and PhCs at bent points has been also proposed [16,17]. These approaches improve the transmission, but geometrical inflexibility and fabrication difficulties such as dielectric rod type of 2D-PhC, complicated shapes of air holes, and 3DPhC clad remain. Our approach is to change the direction of propagation adiabatically, whereby one propagation mode is converted to another without coupling to back-propagating or free-space radiating modes. This approach uses curved PhC waveguides, which was proposed by Arentoft et al. [18]. The confinement of light in the waveguide was, however, weaker than straight waveguide, and radiated toward outside of the waveguide [19]. After that, similar approaches have been proposed to use circular bent photonic crystals [20,21]. These waveguides are designed by using 2D-PhCs with dielectric rod [20,21], which cannot adopt to 2D-PhC slab. The waveguides with curvilinear-lattice [21] have different crystal systems, which can change the dispersion relation and can increase the reflectance. We design and fabricate waveguides bent along general free curves by using 2D-PhC slab with single crystal system, and realize the suppression of radiation and reflection. 3. Design of free-curve waveguides We assume a waveguide is designed in x y plane. First, define a free curve by position vector f ðtÞ in x y plane. As shown in Fig. 1, this free curve is the center line of the waveguide. Anchor points f ðtn Þ are indicated by dots along the curve f ðtÞ in Fig. 1, where tn are given by the recurrence equation
1 a¼ 2
Z
d f ðtÞdt: tn1 dt tn
if n is an even integer, and
1 pffiffiffi d f ðt n Þ ^z rðm; nÞ ¼ f ðt n Þ þ ðm þ Þ 3a dtd f ðt n Þ 2 dt
ð3Þ
if n is an odd integer, where ^ z is a unit vector perpendicular to x y plane, and m is all integers from 10 to 10 except there are no air holes at rð0; 2n0 Þ (where n0 is any integer) so as to make a line defect. We have not considered when the air holes are overlapped each other. In the following sections, we discuss only the free curve waveguide whose no air holes overlapped. This condition provides a limit to the maximum curvature of the waveguide, which is determined by the lattice constant, radii of the air holes and the width of the PhC clad layer. 4. Optical properties of free-curve waveguides The transmission spectra and dispersion relation are quantified for the TE mode of a free-curve waveguide as an example. 4.1. Transmission properties We modeled four waveguides with different curvatures: (a) 0°/a (straight waveguide), (b) 0.5°/a, (c) 1.0°/a, and (d) 2.0°/a. The angles between the input and output ports for (b), (c), and (d) are the same, namely 60°, and thus the length of the waveguide is 120a, 60a, and 30a, respectively. The length of the straight waveguide was 60a. The lattice constant a is set to 390 nm, and the radii of air holes for each waveguide were set to 0.29a. The transmission spectra were calculated with 2D-FDTD method. A lattice constant was discretized by 15 Yee cells. The refractive index of air holes and that of 190 nm thick of GaAs slab were set to 1.0 and 2.65,
ð1Þ
pffiffiffi Air-filled circular holes are arranged with a periodicity of 3a along lines normal to f ðtÞ that pass through the points f ðtn Þ, as shown in Fig. 1 by dashed lines. The center positions (rðm; nÞ) of the air holes are given by
pffiffiffi d f ðt n Þ ^z rðm; nÞ ¼ f ðt n Þ þ m 3a dtd f ðt n Þ
a
1.0 0.5 0 0.26
ð2Þ
0.27
0.28
0.29
0.3
0.27
0.28
0.29
0.3
0.27
0.28
0.29
0.3
0.27
0.28
0.29
0.3
dt
b
1.0
Transmittance
0.5 0 0.26 1.0
c
0.5
y z
t 2n x
t 2n+1
t 2n-1
0 0.26 1.0
d
0.5 0 0.26
Fig. 1. The structure of a PhC free-curve waveguide. A curve f ðtÞ shown as a solid line is defined and anchor points tn are laid out along it. The distance between adjacent anchor points is equal to half of the lattice constant a=2. Air-filledpcircular ffiffiffi holes are then arranged along the dashed normal lines with a spacing of 3a.
ωa 2πc
Fig. 2. Transmission spectra of bent waveguides. The curvature of each waveguide is (a) 0°/a, (b) 0.5°/a, (c) 1.0°/a(60a), and (d) 2.0°/a. The transmittances are normalized to that of a straight waveguide of length 10a.
J.-i. Sugisaka et al. / Optics Communications 281 (2008) 5788–5792
Transmittance
5790
1.0 0.8 0.6 0.4 0.2 0 0.26
0.27
Transmittance
(a) Waveguide A
1.0 0.8 0.6 0.4 0.2 0 0.26
0.27
0.28
0.29
0.3
0.28
0.29
0.3
0.28
0.29
0.3
ωa 2πc
ωa 2πc
Transmittance
(b) Waveguide B
1.0 0.8 0.6 0.4 0.2 0 0.26
0.27
ωa 2πc
(c) Waveguide C Fig. 3. Transmission spectra of a waveguide with two bent guides at its ends. (a) Waveguide A has 60-degree cornered waveguides. (b) Waveguide B has circular ends with a curvature of 2.0°/a. (c) Waveguide C is a sinusoidal guide that connects smoothly to the straight waveguide. The length of the straight waveguide between the bending points is 100a. The corresponding spectra are shown to the right of each geometrical sketch.
respectively. At the input port of each waveguide, oscillating dipoles were placed, and the component of the Poynting vector parallel to the waveguide was observed at the output port. In order to prevent reflecting at the waveguide terminal, the light output from the waveguide terminals were radiated to GaAs slab, and finally absorbed by PML (Perfectly Matched Layer). Next, the transmission spectra were normalized to that of a straight waveguide of length 10a. The results are shown in Fig. 2. These spectra show that incident light can be confined by and propagate along a waveguide bent into a circular arc. If the curvature of the waveguide is less than 0.5°/a, the transmittance is nearly equal to that of a straight waveguide. For larger curvatures, we see that, although the normalized lower cut-off frequency remains 0.273, the useful transmission band starts at somewhat higher frequencies. This effect results from the positional dependence of photonic bandgap as the crystalline structure of the waveguide is distorted. At this moment, the free-curve waveguides cannot have smaller curvature radius than the ridge waveguides. To increase the curvature limit, we should eliminate the positional dependence of photonic band gap by some geometrical modulations such as radii of air holes. The 2D-FDTD method cannot simulate the radiation from the surface of the slab, thus, the cut-off for higher frequency side cannot appear in the transmission spectra. The higher cut-off frequency can be found from the dispersion diagram, which is discussed in the next subsection. In order to confirm the reduction in the reflectance at the bending points, we calculated the transmittance of a straight waveguide (of
length 100a) with bent waveguides connected to both of its ends. The reflection can be observed as a fluctuation in the transmittance due to Fabry–Perot interferences between two bending points. For this purpose, we prepared three configurations of bent waveguides as illustrated in Fig. 3: A – a sharply cornered waveguide, B – a circular arc waveguide, and C – a sinusoidally curved waveguide. The waveguide C is a free-curve waveguide that can match the waveguide curvature with the straight waveguides at the connecting point. The angle between the input and output ports are 60° in all three cases, and the maximum curvature of waveguide C is equal to the (constant) curvature of waveguide B, namely 1°/a. The graphs in Fig. 3a–c plot the transmittance of waveguide A, B, and C, respectively. The sharp periodic peaks in Fig. 3a are due to Fabry–Perot interferences between the two sharp corners. On the other hand, in Fig. 3b, only a small fluctuation remains due to a weak reflection from the connecting point between the circular and straight waveguides. Finally, for waveguide C, fluctuations cannot be observed in the spectrum in Fig. 3b because it was designed to have matching waveguide curvatures at the connecting points. The result of waveguide C also shows that the shape of our suggested waveguide is not limited to the circular curve. 4.2. Dispersion relation The plane-wave expansion method and 2D-FDTD method with Bloch boundary conditions are standard techniques for calculating dispersion relations for a photonic band. The PhC free-curve wave-
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0.33
0.31
0.29 0.28 0.27
0.20
0.30 0.40 Normalized wavenumber [2π/a]
Fig. 5. Dispersion relation for propagation along a straight waveguide (squares), along a free-curve waveguide with curvature 1.0°/a (open circles and solid line), and along a free-curve waveguide with curvature 2.0°/a (filled circles). The dashed curve is the light line.
Power spectrum [a.u.]
Power spectrum [a.u.] 0
0
60
120
Length [a]
Power spectrum [a.u.]
magnetic field magnetic field [a.u.] magnetic field [a.u.]
c
0.30
Fig. 6. SEM images of PhC circular waveguides with curvatures of (a) 1.0°/a and (b) 2.0°/a.
a
b
Straight waveguide Free curve waveguide (1.0degree/a) Free curve waveguide (2.0degree/a) Light line
0.32
ωa 2πc
guide, however, has no translational symmetry, and thus Bloch boundaries cannot be defined. Instead the following method [22] was adopted, as illustrated in Fig. 4. First, the field distribution of light propagating with a frequency x is calculated. Second, the magnetic field perpendicular to the 2D-PhC slab is determined along the center line of the waveguide. It is Fourier transformed to find the peaks. We obtained four peaks, derived from two peaks folded into the first Brillouin zone. These two peaks correspond to backward and forward traveling waves having the same propagation mode. We thereby find a relation between frequency and wavenumber for the confined light, and so obtain the dispersion relation. To calculate the field distributions, as shown in Fig. 4a, we set the length and angle between the input and output ports to be 120a and 120°, respectively (the curvature becomes 1.0°/a). The normalized wavenumber resolution is therefore 1/120. The resulting dispersion relation is graphed in Fig. 5. In the same way, the dispersion relation of the waveguide with the curvature 2.0°/a was also calculated. The length and the angle between input and output ports were set to 60a and 120°, respectively. The result is also graphed in Fig. 5. The dispersion relation is in excellent agreement with that for a straight waveguide calculated by the 2D-FDTD method. The reason why the wavenumber resolution for the waveguide with the curva-
Wavelength [nm] 1360 1340
1380 150
a
1320 in
100
out
50 0
0.285
0.29
0.295
Wavelength [nm] 1360 1340
1380 300
ωa/2πc
b
1320 in
200 out
100 0
0.285
0.29
0.295
Wavelength [nm] 1360 1340
1380 300
ωa/2πc
c
1320 in
200
out
100 0
0.285
ωa/2πc
0.29
0.295
Fig. 7. Transmission spectra of a waveguide having (a) cornered waveguides, (b) circular waveguides with a curvature of 1.0°/a, and (c) circular waveguides with a curvature of 2.0°/a.
-1
k-1
-k
0
k
-k+1
1
Normalized wavenumber ka/2π Fig. 4. Procedure to obtain the dispersion relation for propagation in a free-curve waveguide: (a) The field distribution is calculated by the 2D-FDTD method. (b) Its value is extracted along the center line of the waveguide. (c) That value is Fourier transformed to obtain the peak location.
ture of 2.0°/a is low is because the length of the calculated waveguide is half of that of 1.0°/a, and the normalized wavenumber resolution is 1/60. The dispersion relation of our waveguides, therefore, does not depend on the curvature, and this result shows
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Next, we imaged the waveguides from above to look for light leakage at the bending points, as shown in Fig. 8. The incident light for the wavelength of 1335.5 nm propagates rightward. One sees bright spots at the sharp corners indicated by the arrows in Fig. 8a, demonstrating that light is radiated at those bends. On the other hand, in the case of the free-curve bends in Fig. 8b and c, no bright spots are observed. The radiation was not observed over the entire transmission band in free-curve waveguide, indicating a considerable reduction in lost light from free-curve waveguides. Fig. 8. The top view of (a) sharply bent waveguide, (b) circular bent waveguide (curvature of 1.0°/a) and (c) circular bent waveguide (curvature of 2.0°/a). Each image is the top view taken with an infrared vidicon. In the case of (a), bright spots (radiation) can be observed (shown by the arrows). On the other hand in the case of free-curve bent waveguide ((b) and (c)), radiations were not observed.
that the dispersion relation of a PhC waveguide is conserved even if the guide is bent along a free-curve. By finding the frequency of the point of intersection of the dispersion curve with the light line (dashed in Fig. 5), a normalized cut-off wavelength of 0.294 is found, in agreement with that of a straight waveguide. 5. Fabrication and testing of free-curve waveguides To investigate the effect of free-curve waveguides on the radiation loss, waveguides were fabricated on GaAs epitaxial wafers, with a 1-lm AlGaAs sacrificial layer on the substrate and 190 nm of GaAs on top of that. We lithographically fabricated PhC waveguides using an electron beam (EB) followed by dry etching of the GaAs top layer using an inductively coupled plasma. Then the sacrificial layer was removed using hydrofluoric acid. Scanning electron microscope (SEM) images of the resulting waveguides are shown in Fig. 6. The lattice constant a was set to 390 nm, and radii of air holes are set to 0.29a. The both ends of all waveguides have planar solid immersion lenses (SIL) [23], which can decrease the insertion loss and Fabry–Perot resonance in the waveguide. The curvatures of the waveguides in panels (a) and (b) are 1.0°/a and 2.0°/a, respectively. Next we measured the transmission spectra of these fabricated waveguides. The output waveguide needs to be parallel to the input waveguides due to the alignment of the measuring system. Therefore, the devices under test had two oppositely bent waveguides. A straight waveguide with a length of 100a was inserted between these bent waveguides, as shown in the insets in Fig. 7. Fig. 7a is a spectrum of a PhC waveguide with sharp corners, while Fig. 7b and c are for free-curve waveguides with curvatures of 1.0°/ a and 2.0°/a, respectively. The results confirm that light is confined and propagates through the waveguides. However, we see that the transmittance at long wavelengths decreases as the curvature of a waveguide increases. The flat transmittance at longer wavelength side in Fig. 7b and c often appear in the spectra of the waveguides which contain the SIL, and that is due to the fabrication error of the shapes of SILs. The spectrum of the waveguide with sharp bends in Fig. 7a oscillates with a period of about 4 nm at short wavelengths and with a period of about 2 nm at long wavelengths. These periods agree with the peak interval for Fabry–Perot resonances across a length of 100a. We conclude that the light reflects at the corners. On the other hand, the spectra of the free-curve waveguides in Fig. 7b and c exhibit minimal oscillations.
6. Conclusion A conventional PhC waveguide with sharp corners exhibits problems such as radiation and reflection at the bending points. As a solution to those problems, we developed a method to fabricate a PhC waveguide that is bent along a free-curve. Their transmittances and dispersion relation were calculated, showing that a free-curve waveguide can confine and propagate light and that the reflection at the bends is diminished. It was found that a free-curve waveguide has nearly the same dispersion relation as a straight waveguide. Free-curve waveguides were fabricated and it was experimentally demonstrated that radiation and reflection at the bending points were negligible. Such free-curve waveguides afford flexibility for the design of nanowire circuits and could help with the development of optical integrated PhC circuits. References [1] A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, Phys. Rev. Lett. 77 (1996) 3787. [2] P.L. Gourley, J.R. Wendt, G.A. Vawter, T.M. Brennan, B.E. Hammons, Appl. Phys. Lett. 64 (1994) 687. [3] T. Baba, N. Fukaya, J. Yonekura, Electron. Lett. 35 (1999) 654. [4] S.-H. Jeong, N. Yamamoto, J. Sugisaka, M. Okano, K. Komori, J. Opt. Soc. Am. 24 (2007) 1951. [5] N. Yamamoto, T. Ogawa, K. Komori, Opt. Express 14 (2006) 1223. [6] Alongkarn Chutinan, Susumu Noda, Jpn. J. Appl. Phys. 39 (2000) L595. [7] M.H. Shih, W.J. Kim, W. Kuang, J.R. Cao, S.-J. Choi, J.D. OfBrien, P.D. Dapkus, Appl. Phys. Lett. 86 (2005) 191104. [8] A. Talneau, M. Kafesaki, C.M. Soukoulis, M. Agio, Appl. Phys. Lett. 80 (2002) 547. [9] J. Smajic, C. Hafner, D. Erni, Opt. Express 11 (2003) 1378. [10] A. Chutinan, M. Okano, S. Noda, Appl. Phys. Lett. 80 (2002) 1698. [11] Y. Zhang, B. Li, Opt. Lett. 32 (2007) 787. [12] P.I. Borel, A. Harpoth, L.H. Frandsen, M. Kristensen, Opt. Express 12 (2004) 1996. [13] L.H. Frandsen, A. Harpoth, P.I. Borel, M. Kristensen, J.S. Jensen, O. Sigmund, Opt. Express 12 (2004) 5916. [14] J.S. Jensen, O. Sigmund, Appl. Phys. Lett. 84 (2004) 2022. [15] Alongkarn Chutinan, Sajeev John, Ovidiu Toader, Phys. Rev. Lett. 90 (2003) 123901. [16] J. Cai, G.P. Nordin, S. Kim, J. Jiang Appl. Opt. 43 (2004) 4244. [17] S.H. Tao, M.B. Yu, J.F. Song, Q. Fang, R. Yang, G.Q. Lo, D.L. Kwong, Appl. Phys. Lett. 92 (2008) 031113. [18] J. Arentoft, T. Sondergaard, M. Kristensen, A. Boltasseva, M. Thorhauge, L. Frandsen, Electron. Lett. 38 (2002) 274. [19] S.I. Bozhevolnyi, V.S. Volkov, T. Sondergaard, A. Boltasseva, P.I. Borel, M. Kristensen, Phys. Rev. B 66 (2002) 235204. [20] N. Horiuchi, Y. Segawa, T. Nozokido, K. Mizuno, H. Miyazaki, Opt. Lett. 30 (2005) 973. [21] J. Zarbakhsh, F. Hagmann, S.F. Mingaleev, K. Busch, K. Hingerl, Appl. Phys. Lett. 84 (2004) 4687. [22] B. Dastmalchi, A. Mohtashami, K. Hingerl, J. Zarbakhsh, Opt. Lett. 32 (2007) 2915. [23] N. Ikeda, H. Kawashima1, Y. Sugimoto, T. Hasama1, K. Asakawa, H. Ishikawa, in: 19th International Conference on Indium Phosphide and Related Materials, 2007, p. 484.