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We propose and analyze shear discontinuities as a new type of defect in photonic crystals. This defect can support guided modes with minimum group velocity dispersion (GVD) and maximum bandwidth, provided that the shear shift equals half the lattice constant. A mode gap emerges when the shear shift is different than half the lattice constant. The shear shift can be used to tune the bandwidth, group velocity, and group velocity dispersion (GVD) of the guided mode. The necessary condition for the existence of guided modes along the shear plane is discussed.
©2006 Optical Society of America
1. Introduction
Photonic crystals are made of periodic arrays of dielectric media with high contrast modulation of the dielectric constant [1, 2, 3, 4, 5]. Electromagnetic waves propagate in photonic crystals as Bloch waves. The coupling between these Bloch waves gives rise to the band structure, which may include forbidden gaps where the waves cannot propagate. Certain defects in photonic crystals can lead to localized states. For example, by introducing a line defect, we can induce a guided mode along the defect axis for a band of frequencies inside the band gap [6]. Such a waveguide does not rely on the total internal reflection as regular dielectric waveguides do. Because of that, the evanescent region is virtually zero. Light can be guided without appreciable losses for a wide range of frequencies and transmitted efficiently around sharp corners [7], even if the radius of curvature of the bend is on the order of one wavelength. Conventional photonic crystal waveguides consist of a missing row of rods or holes in a two-dimensional (2-D) array of dielectric rods or air holes. These structures have been studied extensively in both theory [6, 7, 8] and experiments [9, 10]. However, many of the proposed waveguide structures suffer from a large group velocity dispersion (GVD) and exhibit relatively small guiding bandwidth because of the distributed Bragg reflection (DBR) along the guiding direction. Techniques that have been proposed to mitigate guided mode dispersion have been either successful for relatively small bandwidth or involve a combination of slab mode and photonic crystal confinement [11, 12, 13].
In this paper, we propose a new type of defect, consisting of a shear discontinuity in an otherwise periodic photonic crystal lattice, as shown in Fig. 1. Such a defect can confine optical waves to propagate along the shear plane. Such guided waves are sometimes referred to as zero mode [14]. The confined propagation mode is effective over the entire band gap, provided that the shear shift equals half the lattice constant. The guided modes avoid large GVD due to flattening of the dispersion curve. This is because the local period is half the lattice constant and thus breaks the DBR condition. The low GVD makes this structure very promising for high speed transmission, high speed optical signal processing and highly integrated optical circuits. Alternatively, the shear shift can be adjusted as a parameter to tailor a particular dispersion response. If the shear shift is not equal to half the lattice constant, a mode gap [15] emerges inside the band gap. This property can be used to implement a tunable optical filter or optical switch. We also investigate the coupling efficiency between a guided mode external to the photonic crystal and the shear mode, with the shear shift as a parameter. The group velocity can also be tuned by changing the shear shift. This enables us to realize tunable slow light devices. We find that the existence of surface waves for each half of the sheared photonic crystals is a necessary condition for the existence of guided modes. The mode gap introduced by the shear shift can also be used to induce bound states [15]. By changing the shape of circular rods at the interface (height h in Fig. 1(b)), we can further optimize the design of our sheared photonic crystals to achieve minimum GVD or other requirements.
Fig. 1. Two dimensional photonic crystals: (a) square lattice of dielectric rods in air, with lattice constant a and radius r = 0.2a (b) photonic crystal lattice with shear discontinuity (sheared photonic crystals) with shear shift s = a/2 and cylinder section height at the interface h = r.
The remainder of the paper continues with detailed analysis of localized propagation modes guided by shear discontinuities in Sec. 2. In Sec. 3, we investigate the dependence of the guiding bandwidth and group velocity on the shear shift. The bound states (localized resonance) are also discussed in detail. In Sec. 4, we discuss the necessary condition for the existence of guided modes and further optimize sheared photonic crystals by changing the shape of circular rods at the interface. Conclusions and discussion are presented in Sec. 5.
2. Localized propagation modes guided by shear discontinuities
We start with a conventional 2-D photonic crystal consisting of dielectric rods in air on a square array with lattice constant a, as shown in Fig. 1(a). Just as the regular arrangement of atoms in a crystal gives rise to band gaps, here the spatial periodicity of the dielectric index may prevent electromagnetic waves of certain frequencies from propagating inside the photonic crystal. As a numerical example, we assume that the refractive index of the rods is 3.0 and the radius is r = 0.2a. The crystal has a TM (magnetic field in-plane) band gap which extends from frequency ω = 0.323 × 2πc/a to ω = 0.443 × 2πc/a. The gap range corresponds to the canonical free-space wavelength for light between 451nm and 619nm when a = 0.2μm. Subsequent simulations use this value of a and center wavelength λ 0 = 550nm. Here we restrict our analysis to TM modes.
We introduce a shear discontinuity in the middle row, as shown in Fig. 1(b). The circular dielectric rods in the middle row are cut in half (the height h = r). In this section, we will restrict our analysis to shear shift s = a/2, i.e. exactly one half the lattice constant. General shear shifts will be discussed in Section 3. When light of frequency within the band gap enters the photonic crystal along the shear plane, we can expect that the light will be well confined near the shear plane. This is because the upper and lower halves are still perfect photonic crystals and there are no extended modes into which the propagating wave can couple. Unlike other guided mode structures, the transverse confinement is induced by a zero thickness entity: the shear plane. The computational setup is shown in Fig. 2. A slab waveguide with core index 1.5 and core thickness 0.4μm sandwiched by clading of index 1.0 is used to couple the light source into the sheared photonic crystal with lattice constant a = 0.2μm and radius of rods r = 40nm. Using the Finite-Difference Time-Domain (FDTD) method, we simulated a 10fs pulse with center wavelength of 550nm being injected by an external waveguide near the tip of the shear plane. All the FDTD simulations in this paper are 2-D and perfectly matched layer (PML) boundary conditions are used to minimize back reflections. Because the spectrum of the pulse is mostly inside the band gap of the photonic crystal, we can see from Fig. 2 that the entire pulse is well confined to the shear plane.
Fig. 2. A pulse is coupled in by a slab waveguide and propagates inside the sheared photonic crystal. The pulse duration is 10fs and the center wavelength is 550nm. Plane A is at the end of the slab waveguide and Plane B is located 5.5μm away from Plane A. [Media 1]
The dispersion diagrams for the sheared photonic crystals in this paper are calculated by solving Maxwell’s equations in the frequency domain for given dielectric configurations [4]. A supercell of size 15a × a with periodic boundary conditions is used as the computational domain. Because the guided modes are sufficiently localized and the width of the supercell is large enough, the introduction of the supercell has a negligible effect on the results. When the shear shift equals half the lattice constant and the shear discontinuity is formed by half circular dielectric rods, the dispersion relation is illustrated as the solid line in Fig. 3, indicating the existence of two guided modes inside the band gap.
Two important features of the dispersion diagram are worth pointing out. First, no flattening occurs at the edge of Brillouin zone X (k = π/a), unlike in conventional photonic crystal waveguides. The flattening is very undesirable for optical signal transmission, since it makes guided modes suffer from large GVD. The primary physical reason for the flattening in conventional photonic crystal waveguides is the DBR effect or constructive backward coupling [16] since the spatial periodicity of the photonic crystal waveguides is exactly half of the Bloch wavelength at the edge of the first Brillouin zone. In the sheared photonic crystal, the local period along the shear plane is actually a/2 instead of a, although the period of the entire structure still equals a. Therefore, the condition for strong backward DBR coupling is broken. Since the local period is decreased to one half of the lattice period, the actual Brillouin zone of guided modes doubles. The mode outside the first Brillouin zone of the entire sheared photonic crystal folds back to form a second mode inside the band gap. Second, as an additional benefit of the absence of flattening, no mode gap exists [15]; guided modes span the entire band gab. Without the problems of limited bandwidth and a large GVD, the sheared photonic crystals are very promising for applications such as high speed transmission, high speed optical signal processing and highly integrated optical circuits.
Fig. 3. Dispersion relation for the sheared photonic crystals when s = a/2. Solid line: half circular rods at the interface h = r; dash-dot line: entire circular rods at the interface h = r+a/2.
The very low dispersion of sheared photonic crystals also results in uniform coupling efficiency over the entire width of the band gap. In the FDTD simulation, we calculated the power coupled into the sheared photonic crystals at Plane B, as shown in Fig. 2, which is 5.5μm away from the end of slab waveguide, Plane A. Then we calculated the coupling efficiency as the ratio of the coupled-in power and incident power at Plane A. Figure 4 shows the spectrum of coupling efficiency when a 10fs pulse is coupled into the shear photonic crystal in Fig. 2. The profile of normalized power of incident and coupled-in pulse almost overlap with each other and the coupling efficiency is equal to 1 uniformly for the entire pulse spectrum. In order to obtain the coupling efficiency for the full band gap, we use a 3fs pulse with center wavelength 550nm. The pulse spectrum covers the entire width of the band gap. As shown in Fig. 5, the coupling efficiency is uniformly equal to 1 inside the band gap while some oscillations emerge at the edge of the band gap. The rapid fluctuations of coupling efficiency at the edge of the band gap are not real features of the system and likely to be numerical artifacts [17, 18]. These artifacts arise from the small signal-to-noise ratio outside the band gap and from Gibbs effect inside the band gap.
Fig. 4. Incident power spectrum at Plane A (the reflected power due to impedance mismatch between the slab waveguide and the sheared photonic crystal is subtracted) and coupled-in power calculated at Plane B, and coupling efficiency when a 10fs pulse with center wavelength of 550nm is input into the sheared photonic crystal.
Fig. 5. Incident power spectrum at Plane A (the reflected power due to impedance mismatch between the slab waveguide and the sheared photonic crystal is subtracted) and coupled-in power calculated at Plane B, and coupling efficiency when a 3fs pulse with center wavelength of 550nm is input into the sheared photonic crystal.
3. The effect of shear shifts on GVD and group velocity
It can be expected that guiding by shear discontinuities in photonic crystals will depend strongly on the shear shift s between the upper and lower halves of the lattice. The dispersion diagram for shifts other than half lattice constant is shown in Fig. 6. We can see that a mode gap opens up progressively towards the band gap edges as the shear shift decreases from half lattice constant to zero. At this point, the presence of the mode gap is also because of DBR. So the dispersion curves are flattened and the mode gap re-emerges. This can also be explained from the symmetry in the Fourier domain as follows. The band diagram must continue symmetrically beyond X. Therefore, when there is no crossing point at X, as in the case of half the lattice constant shear shift, all bands must have zero group velocity (i.e. zero slope) at X in order to be analytic functions of the wave vector. Note that sheared photonic crystals with shear shift s are actually the same as with shear shift a – s because of the periodicity of the whole structure. So we only consider the case s < 0.5a here. Because no guided modes exist within the frequency range of the mode gap, light of wavelength inside the mode gap can not propagate in sheared photonic crystals. Thus, tunable sheared photonic crystals can be used to implement optical filtering or optical switching. The relationship between the mode gap and shear shift follows Fig. 7. Using FDTD simulations, we obtain the spectra of coupling efficiency for different shear shifts as shown in Fig. 8. For wavelengths inside the mode gap, the coupling efficiency decreases to zero. Wavelengths outside the mode gap but inside the band gap retain very high coupling efficiency. The fluctuations near the band edges and outside the band gap result from numerical artifacts in this calculation. Within the band gap, however, the FDTD calculation is very accurate, as can be seen by the excellent match between the mode gap calculated by FDTD and by the dispersion relation (Fig. 7).
Fig. 6. Dispersion relation for the sheared photonic crystal with different shear shifts. Half circular rods are at the interface h = r.
Fig. 7. Mode gap versus shear shift s of the sheared photonic crystals. Cross symbols indicate the mode gap measured from FDTD simulations.
Fig. 8. Coupling efficiency spectra for different values of shear shift s. 3fs pulses with center wavelength of 550nm are input into the sheared photonic crystals.
These phenomena can also be explained intuitively by the argument of surface waves as follows. The fields in the lower and upper halves of the sheared photonic crystal can be thought of as surface waves localized at the shear surface. Half lattice constant shear shift will phase match the two Bloch surface waves. They can then strongly couple into each other, resulting in a guided mode that is well confined along the shear plane. When the shear shift is between half lattice constant and zero, the two surface waves are partially phase matched, and so mode gaps emerge. When the shear shift is equal to zero (perfect photonic crystal without defect), the two surface waves are totally phase mismatched, and so no wavelengths can propagate within the band gap.
From Fig. 6, the slope of the dispersion curves (i.e., the group velocity) also depend on the shear shift. The flattened dispersion curves at the edge of the Brillouin zone result in small group velocities. Thus, the shear photonic crystal structure with mechanically controlled shear shift can be used for active control of the group velocity. Figure 9 shows the group velocity spectra for different shear shifts. The group velocity can be tuned from zero to approximately its value in the bulk material with the same averaged index as the sheared photonic crystal, as the shear shift increases. The flattened dispersion curves in Fig. 6 are well inside the band gap and isolated from the continuum of modes that lie outside the band gap. This is in contrast with some photonic crystal waveguides which achieve low group velocity near the band edge at the cost of poor field confinement. Our proposed approach is unique in that it utilizes structure to control the group velocity. Tuning via shear does not require special media like cold atomic gases, electronic transitions in crystalline solids or other nonlinear optical effects and electrical heating process [19, 20, 21, 22]. Thus, our approach is applicable at any wavelength range, particularly in the low loss window of optical devices, and independent of operation temperature. Our approach also provides high flexibility because it is decoupled from nonlinear, electro-optic or other effects that are best reserved for other purposes in optical systems.
Fig. 9. Group velocity spectra for different values of shear shift s of the sheared photonic crystals.
Figure 6 shows that not only wider mode gaps exist for smaller shear shifts, but also that part of the guided modes for larger shear shifts always fall into the mode gap for smaller shear shifts. This means that bound states can exist [15] whenever a slice of sheared photonic crystal with larger shear shift is sandwiched by two semi-infinite sheared photonic crystals with smaller shear shifts. We indeed find a bound state at ω = 0.334 × 2π/a by choosing a configuration such that a slice of sheared photonic crystal with half lattice constant shift had length 2a and was sandwiched by two semi-infinite sheared photonic crystals with quarter lattice constant shift. The electric field for this case is shown in Fig. 10. The shear discontinuity is a totally new method to induce bound states instead of changing the width of guiding region in conventional photonic crystal waveguides or metallic waveguides. Actually, in our case, the width of the guiding region does not change at all.
Fig. 10. (a) Geometry of a slice of sheared photonic crystal with shear shift s = a/2 and thickness 2a sandwiched between two semi-infinite sheared photonic crystals of s = a/4. (b) Electric field for the bound state at ω= 0.334×2π/a in the geometry shown in (a).
4. The effect of truncated rod shapes near the interface
The shape of dielectric rods at the interface is also crucial to the presence and properties of the surface waves and thus the guided modes. The shape of dielectric rods in this paper is defined by the height h: the distance from the top of the circular rods to the shear plane as shown in Fig. 1(b). For example, h = r means half circular rods at the interface, while h = r+a/2 means that entire circular rods are at the interface and the distance between the centers of circular rods of lower and upper halves near the interface is a. As the dashed line in Fig. 3, no guided modes exist inside the band gap when whole rods (h = r + a/2) are at the shear interface. The disappearance of guided modes for this case can also be explained by the lack of a surface mode in this geometry [23]. The existence of surface waves for each half of the sheared photonic crystals is a necessary condition for guided modes.
Removing dielectric material resembles adding acceptor atoms in semiconductors [24]. This gives rise to acceptor modes which have their origin at the top of the dielectric (valence) band. So by gradually truncating the rods near the shear plane, we can “pull” the dispersion curves into the band gap, thus creating guided modes that were originally in the dielectric band, as shown in Fig. 11.
From Fig. 3, we see that point C is actually below the center of the band gap when h = r. So half circular rods at the interface may not fully utilize the advantages of half lattice constant shear shift, and may not achieve the minimum GVD. By truncating the rods at the interface by more than half, we can “pull” the dispersion curves further till the center of the band gap. As shown in Fig. 12, when 42% of rods (s = 0.84r) left at the interface, point C moves to the center of the band gap. It is worth noting that when only 20% of rods (s = 0.4r) is left, guided modes may still have relatively small GVD even though the point C moves out of the band gap. This is understood as follows: the first guided mode occupies the entire band gap although the second guided mode is “pulled” out of the band gap. We calculated the average GVD parameter β 2 = d2 k/dω 2 of the guided modes inside the band gap for different heights h, as shown in Fig. 13. We can see that guided modes have minimum GVD when the height h is around 0.8r and relatively smaller GVD when h = 0.4r than when h = 0.5r. The shape of dielectric rods at the interface provides us with another degree of freedom to design and optimize photonic crystal waveguides corresponding to different requirements.
Fig. 11. Dispersion relations for sheared photonic crystals with different values of h when s = a/2. Truncating rods at the interface creates guided modes originated from dielectric band.
5. Conclusions and discussion
In summary, we have described a new type of waveguide that consists of a shear discontinuity in a photonic crystal. The shear shift was shown to be a design parameter that can be varied to achieve dispersion properties. For example, we can achieve guiding over the entire band gap with minimal GVD by selecting shear shift of half the lattice constant. Alternatively, by changing the shear shift we can tune the mode gap and the group velocity. This type of tuning can be implemented mechanically for dynamic reconfiguration by shearing two plates containing the half lattices relative to each other; or the shear shift may be lithographically defined and fixed to satisfy specific device requirement. In both cases, such devices can be useful for telecommunication and other information processing operations involving large bandwidths. In sheared photonic crystals, light can also be transmitted through sharp bends without appreciable losses for a wide range of frequencies [25] as in conventional photonic crystal waveguides [7]. The transmission efficiency is also similar as in the case of conventional photonic crystals [25]. By adjusting the length of the bend section [7, 25], we should be able to achieve zero reflection through sharp bends. In a following paper [26], we also present a periodic modification of the shear photonic crystal that can be used as a coupled-resonator optical waveguide (CROW) [27] with tunable group velocity.
References and links
1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals (Princeton University Press, 1995).
2. K. M. Ho, C. T. Chan, and C. M. Soulouis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152 (1990). [CrossRef] [PubMed]
3. E. Yablonovitch, T. J. Gmiter, and L. M. Leung, “Photonic band structure: the face-certered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991). [CrossRef] [PubMed]
4. R. D. Meade, A. M. Rappe, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B , 48, 8434 (1993). [CrossRef]
5. P. Yeh, Optical waves in layered media (Wiley, New York, 1988).
6. R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low-loss bends and high Q cavlties,” J. Appl. Phys. 75, 4753 (1994). [CrossRef]
7. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. B 58, 4809 (1998). [CrossRef]
8. S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751 (1999). [CrossRef]
9. S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demostration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274 (1998). [CrossRef] [PubMed]
10. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremly large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]
11. W. Tung and S. Fan, “Creating large bandwidth line defects by embedding dielectric waveguides into photonic crystal slabs,” Appl. Phys. Lett. 81, 3915 (2002). [CrossRef]
12. A. Jafarpour, A. Adibi, Y. Xu, and R. K. Lee, “Mode dispersion in biperiodic photonic crystal waveguides,” Phys. Rev. B 68, 233102 (2003). [CrossRef]
13. T. Sondergaard and A. Lavrinenko, “Large-bandwidth planar photonic crystal waveguides,” Opt. Commun. 203, 263 (2002). [CrossRef]
14. M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682 (2003). [CrossRef] [PubMed]
15. A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B 58, 4809 (1998). [CrossRef]
16. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).
17. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837 (1996). [CrossRef]
18. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal optical waveguides,” Phys. Rev. B 64, 033308 (2001). [CrossRef]
19. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) 397, 594 (1999). [CrossRef]
20. M. D. Stenner, D. J. Gauthies, and M. A. Neifeld, “The speed of information in a “fast-light” optical medium,” Nature (London) 425695 (2003). [CrossRef]
21. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200 (2003). [CrossRef] [PubMed]
22. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature (London) 438, 65 (2005). [CrossRef]
23. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528 (1993). [CrossRef] [PubMed]
24. E. Yablonovitch, T. J. Gmiter, R. M. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991). [CrossRef] [PubMed]
25. K. Tian, G. Babarstathis, and J. Hong, “Localized propagation modes guided by shear discontinuities in photonic crystals” in Frontiers in Optics/Laser Science conferences, (Optical Society of America, Tucson, Arizona, Oct. 2005), Paper FWI4.
26. K. Tian, G. Babarstathis, and J. Hong, “Tunable group velocity in a coupled-resonator optical waveguide (CROW) formed by shear discontinuities in a photonic crystal,” under preparation.
27. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711 (1999). [CrossRef]
