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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 8736–8745
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Degenerate band edge resonances in coupled periodic silicon optical waveguides

Justin R. Burr, Nadav Gutman, C. Martijn de Sterke, Ilya Vitebskiy, and Ronald M. Reano  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 8736-8745 (2013)
http://dx.doi.org/10.1364/OE.21.008736


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Abstract

Using full three-dimensional analysis we show that coupled periodic optical waveguides can exhibit a giant slow light resonance associated with a degenerate photonic band edge. We consider the silicon-on-insulator material system for implementation in silicon photonics at optical telecommunications wavelengths. The coupling of the resonance mode with the input light can be controlled continuously by varying the input power ratio and the phase difference between the two input arms. Near unity transmission efficiency through the degenerate band edge structure can be achieved, enabling exploitation of the advantages of the giant slow wave resonance.

© 2013 OSA

1. Introduction

The possibility of a giant transmission resonance in a structure depends on the presence of a DBE in the Bloch dispersion relation. An essential element is that the structure has two modes which are coupled. In periodic layered structures, a DBE can exist because of the coupling between two polarization modes when each unit cell contains layers with sufficiently strong and misaligned birefringence [1

1. A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3), 036619 (2005). [CrossRef] [PubMed]

,5

5. A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photonics Rev. 5(2), 201–213 (2011). [CrossRef]

]. A more practical alternative is provided by using waveguides. The two modes that are essential for a DBE can be created by tuning the waveguide width. DBEs were shown to exist in different waveguiding structures such as optical fiber with multiple gratings [2

2. A. A. Sukhorukov, C. J. Handmer, C. M. de Sterke, and M. J. Steel, “Slow light with flat or offset band edges in few-mode fiber with two gratings,” Opt. Express 15(26), 17954–17959 (2007). [CrossRef] [PubMed]

,6

6. N. Gutman, C. M. de Sterke, A. Sukhorukov, and L. Botten, “Slow and frozen light in optical waveguides with multiple gratings Degenerate band edges and stationary inflection points,” Phys. Rev. A 85(3), 033804 (2012). [CrossRef]

,7

7. N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett. 36(16), 3257–3259 (2011). [CrossRef] [PubMed]

] and coupled periodic waveguides [4

4. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B 25(12), C65–C74 (2008). [CrossRef]

,8

8. N. Gutman, W. H. Dupree, Y. Sun, A. A. Sukhorukov, and C. M. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express 20(4), 3519–3528 (2012). [CrossRef] [PubMed]

].

The paper is organized as follows. In section 2, the design of the coupled periodic dielectric waveguides exhibiting a DBE is presented. The transmission and resonance properties of finite length DBE structures are described in section 3. The quality factor scaling of the DBE resonance is also shown as a function of the number of periods and the gap width between waveguides. Section 4 discusses the location of the resonances and Section 5 provides a conclusion.

2. Design

The structure under consideration is shown in top view in Fig. 1(a) and in cross-sectional view in Fig. 1(b). Two dielectric strip waveguides which have two periodic arrays of cylindrical holes are brought into close proximity. The core material of the waveguides is silicon. The cladding and the material in the cylindrical holes is silicon dioxide (SiO2). The cross-sectional width and height of an individual waveguide are 450 nm and 250 nm respectively. These values are chosen so that the feed strip waveguides into the periodic structure are single mode for TE optical polarization (dominant electric field component in the x direction). We denote the gap separation between the waveguides with parameter xo and the longitudinal offset between waveguides with parameter zo. The longitudinal period between the cylindrical holes is denoted with parameter a. The coupled structure is a four-port network. The ports are labeled as indicated in Fig. 1(a).

Dispersion relationships were calculated using the freely available MIT Photonic Bands (MPB) software package that implements the plane-wave expansion method in three-dimensions [19

19. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]

]. We use a refractive index of 3.48 for the silicon and 1.45 for the silicon dioxide. The way to create a DBE in two coupled nanowires is explained in Fig. 2
Fig. 2 (a)-(f) Dispersion diagrams. (a) Single mode waveguide; (b) Two coupled single mode waveguides; (c) Single periodic dielectric waveguide; (d) Two coupled periodic dielectric waveguides; (e) Two coupled periodic waveguides with a 0.5a lateral shift; (f) Lateral shift of 0.37a to realize a DBE; (g) Frequency difference versus wavenumber difference on loglog scale for RBE; (h) Frequency difference versus wavenumber difference on loglog scale for DBE (red) and the lower spectral branch (blue).
.

Figure 2(a) shows the band structure, in the first Brillouin zone, of a reference single mode feed waveguide with no holes. Figure 2(b) shows the splitting of the single mode by the coupling between two waveguides in close proximity. The introduction of holes in the strip waveguides causes a band gap to open as shown in Figs. 2(c) and 2(d). A non-zero longitudinal offset between the coupled periodic dielectric waveguides leads to a split band edge, shown in the top band in Fig. 2(c), and a DBE, shown in Fig. 2(f) (red). At the DBE longitudinal offset value, ∂ω/∂k = ∂2ω/∂k2 = 0. In order to scale the DBE wavelength to be near 1500 nm, the period is chosen to be a = 325 nm. The gap between the waveguides is set to x0 = 50 nm and the cylindrical holes in the periodic section are set to a radius of 100 nm. The lateral offset needed for a DBE is zo = 0.37a = 120 nm.

A plot of frequency difference versus wavenumber difference from the degenerate band edge, plotted on a loglog scale in Fig. 2(h), shows a slope of 4, verifying the quartic nature of the band gap. The spectral region where the lowest order of the band edge has a quartic dependence is approximately Δωa/(2πc) < 10−4 and if fitted to
(ωωD)=(kkD)4/ξ4
(1)
yields ξ4 = −0.319/(a3c) for degenerate band edge frequency ωDa/(2πc) = 0.216. For comparison, frequency difference versus wavenumber difference for a periodic single nanowire is plotted in Fig. 2(g). On the loglog scale, the slope is 2 (quadratic), as expected.

3. Transmission and resonance properties near the degenerate band edge

Figure 3
Fig. 3 Outgoing power from all ports for excitation at port 1 only (N = 65). (a) Outgoing power from ports 1 and 3. (b) Outgoing power from ports 2 and 4. The band diagram in the inset indicates the three lowest order modes. The second mode (indicated in red) exhibits a DBE. The yellow region indicates the frequency range of the plotted s-parameters.
shows the outgoing power from all ports for N = 65 and excitation at port 1 only (ie. β = 0). The reflected power from port 1 is denoted |S11|2 and the outgoing power from port 3 is denoted |S31|2. We observe three resonance peaks corresponding to the three lowest order resonances of the finite periodic structure. The narrowest resonance, near 1.508 µm, is the fundamental resonance closest to the band edge. On the fundamental resonance, the reflected power from port 1 is relatively low. Off resonance, the reflected power is relatively large. Figure 3(b) shows that the outgoing power from ports 2 and 4 are relatively low for all wavelengths. At the fundamental resonance, there is nearly equal outgoing power from both ports 2 and 4. Figure 4
Fig. 4 Outgoing power from all ports for excitation at port 3 only (N = 65). (a) Outgoing power from ports 1 and 3. (b) Outgoing power from ports 2 and 4.
shows the outgoing power from all ports for the case of excitation at port 3 only. In this case, the reflected power from the excitation port, |S33|2, shows a smaller value of extinction ratio for the fundamental resonance and larger values of extinction ratio for the second and third resonances, when compared to |S11|2. The extinction ratio differences between Figs. 3 and 4 are due to the asymmetry of the physical structure. In each excitation case, the evanescent and propagating modes are coupled with different intensities which affect the total transmission and extinction ratio. Furthermore, larger transmission peaks are observed for the outgoing power through port 2, for the second and third resonances, while the fundamental resonance is slightly smaller. Finally, the |S13|2 is observed to be equal to |S31|2, as expected, since the 4-port network is reciprocal.

To scan the possible transmissions and reflections for arbitrary input into both ports 1 and 3, Fig. 5
Fig. 5 Outgoing power from all ports at the fundamental resonance (λ = 1508.3 nm), for excitation at both ports 1 and 3, parameterized by |α|2 and ϕ (N = 65). (a)-(c) Reflected power from ports 1 and 3, respectively. (b)-(d) Transmitted power from ports 2 and 4, respectively.
shows the outgoing power from all ports given a parametric sweep of |α|2 (input power into port 1) and ϕ (the phase difference between the excitations at port 1 and port 3) for a 65 period DBE at the fundamental resonance. The corresponding input power into port 3 is |β|2 = 1 - |α|2. We observe two regions on the surface plot. The first region corresponds to maximum outgoing power (transmission) at ports 2 and 4. The second region is characterized by maximum outgoing power (reflection) at the excitation ports 1 and 3. The maximum transmission state occurs when |α|2 = 0.5588, |β|2 = 0.4413, and ϕ = 89.46° (condition 1). Likewise, the minimum transmission state (maximum reflection) is observed for |α|2 = 0.4413, |β|2 = 0.5588, and ϕ = 269.1° (condition 2). The observation that conditions 1 and 2 are complementary can be understood from the analysis in [7

7. N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett. 36(16), 3257–3259 (2011). [CrossRef] [PubMed]

].

We also plot the sum of the outgoing power from ports 1 and 3 (Fig. 6(a)
Fig. 6 Outgoing power versus wavelength for excitation at both ports 1 and 3 (N = 65). Red lines correspond to the excitation condition for maximum transmission from ports 2 and 4 (condition 1). Blue lines correspond to the excitation condition for maximum reflection from ports 1 and 3 (condition 2). (a) Sum of the reflected power from ports 1 and 3. (b) Sum of the transmitted powers from ports 2 and 4.
) and ports 2 and 4 (Fig. 6(b)) as a function of wavelength when the excitation corresponds to either maximum combined transmission out of ports 2 and 4 (labeled as condition 1) or minimum combined transmission out ports 2 and 4 (labeled as condition 2). The excitation is at both ports 1 and 3. Red lines correspond to condition 1 and blue lines correspond to condition 2. At condition 1, the peak transmission at the fundamental resonance approaches unity.

4. Spectral location of resonances

In close vicinity of a DBE, inside the transmission band, there are four Bloch eigenmodes – a pair of propagating modes with equal and opposite real wave numbers, and a pair of evanescent modes with complex conjugate wave numbers [1

1. A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3), 036619 (2005). [CrossRef] [PubMed]

,3

3. A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A 76(5), 053839 (2007). [CrossRef]

,5

5. A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photonics Rev. 5(2), 201–213 (2011). [CrossRef]

8

8. N. Gutman, W. H. Dupree, Y. Sun, A. A. Sukhorukov, and C. M. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express 20(4), 3519–3528 (2012). [CrossRef] [PubMed]

]. The frequency dependence of the four Bloch wave numbers can be approximated by the following expression
kj=kD+ξ4(ωωD)4=kD+Δkeiπ2j,j=0,1,2,3,
(4)
where Δk=|ξ4(ωωD)4|. The wave numbers of the propagating and evanescent modes are, respectively

k0,2=kD±Δkandk1,3=kD±iΔk.
(5)

At the resonance frequency, the forward and the backward propagating modes constructively interfere in the middle of the waveguide. For this to happen a round trip of a propagating wave inside the cavity must be equal to a multiple of 2π. The phase gathered by a propagating wave inside the structure is equal to kL, where L is the length of the cavity. Further, at the interfaces, when the propagating mode is reflected it gathers a phase φ. At the resonance, the total phase gathered in a round trip must be equal to
2πm=2φ+2kL,
(6)
where m is an integer. According to equation Eq. (4), k = kD + Δk, where kD is the edge of the Brillioun zone and is equal to π/a. The length is taken to be an integer, N, periods multiplied by a single period length, a. Hence, 2kDaN is always equal to a multiple of 2π, so
2πm=2φ+2[ξ4(ωωD)]1/4aN.
(7)
The resonance frequency versus the number of periods follows
ω=ωD(πmφ)4/ξ4(aN)4.
(8)
We fit Eq. (8) to the resonance frequencies calculated via 3D FDTD (Fig. 8(b)) and find φ = 0.18(2π) for the DBE. For comparison, we also plot (Fig. 8(c)) the resonance frequencies versus the number of periods for the RBE depicted in Fig. 2(c).

The quality factor scaling shown in Fig. 7(a), together with the decomposition of the field in Fig. 8(a), demonstrate that a slow light giant DBE resonance can be created in waveguides, which can be effectively integrated. Thus slow light resonances can potentially have very large quality factors that scale as N5, which can be used instead of reflector based cavities, for such application as delay lines, sensors, light amplification [22

22. J. K. Yang, H. Noh, M. J. Rooks, G. S. Solomon, F. Vollmer, and H. Cao, “Lasing in localized modes of a slow light photonic crystal waveguide,” Appl. Phys. Lett. 98(24), 241107 (2011). [CrossRef]

], and nonlinear enhancement [23

23. N. Gutman, A. A. Sukhorukov, F. Eilenberger, and C. M. de Sterke, “Bistability suppression and low threshold switching using frozen light at a degenerate band edge waveguide,” Opt. Express 20(24), 27363–27368 (2012). [CrossRef] [PubMed]

] where large distributed electric fields are important. Further benefit in using waveguides for giant DBE resonances is the ability to create higher order DBEs, such as sextic [ω - ωD ~(k - kD)6], for which the quality factor will scale as N7 [7

7. N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett. 36(16), 3257–3259 (2011). [CrossRef] [PubMed]

], with the possibility of achieving higher quality factors in comparatively shorter waveguides.

5. Conclusion

Acknowledgments

This research was supported by the Air Force Office of Scientific Research through LRIR 09RY04COR (Dr. Arje Nachman), the AOARD Window-On-Science (Dr. Misoon Mah), the Air Force Research Lab contract FA8650-09-C-1619, and the Australian Research Council.

References and links

1.

A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3), 036619 (2005). [CrossRef] [PubMed]

2.

A. A. Sukhorukov, C. J. Handmer, C. M. de Sterke, and M. J. Steel, “Slow light with flat or offset band edges in few-mode fiber with two gratings,” Opt. Express 15(26), 17954–17959 (2007). [CrossRef] [PubMed]

3.

A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A 76(5), 053839 (2007). [CrossRef]

4.

A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B 25(12), C65–C74 (2008). [CrossRef]

5.

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photonics Rev. 5(2), 201–213 (2011). [CrossRef]

6.

N. Gutman, C. M. de Sterke, A. Sukhorukov, and L. Botten, “Slow and frozen light in optical waveguides with multiple gratings Degenerate band edges and stationary inflection points,” Phys. Rev. A 85(3), 033804 (2012). [CrossRef]

7.

N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett. 36(16), 3257–3259 (2011). [CrossRef] [PubMed]

8.

N. Gutman, W. H. Dupree, Y. Sun, A. A. Sukhorukov, and C. M. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express 20(4), 3519–3528 (2012). [CrossRef] [PubMed]

9.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994). [CrossRef]

10.

S. Kim, B.-H. Ahn, J.-Y. Kim, K.-Y. Jeong, K. S. Kim, and Y.-H. Lee, “Nanobeam photonic bandedge lasers,” Opt. Express 19(24), 24055–24060 (2011). [CrossRef] [PubMed]

11.

B. Wang, M. A. Dündar, R. Nötzel, F. Karouta, S. He, and R. W. van der Heijden, “Photonic crystal slot nanobeam slow light waveguides for refractive index sensing,” Appl. Phys. Lett. 97(15), 151105 (2010). [CrossRef]

12.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(2), R1078–R1081 (1996). [CrossRef] [PubMed]

13.

M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13(18), 7145–7159 (2005). [CrossRef] [PubMed]

14.

A. E. Willner, B. Zhang, L. Zhang, L. Yan, and I. Fazal, “Optical signal processing using tunable delay elements based on slow light,” IEEE J. Sel. Top. Quantum Electron. 14(3), 691–705 (2008). [CrossRef]

15.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73(10), 1368–1371 (1994). [CrossRef] [PubMed]

16.

A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J. 4(5), 1306–1315 (2012). [CrossRef]

17.

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006). [CrossRef]

18.

P. Sun and R. M. Reano, “Cantilever couplers for intra-chip coupling to silicon photonic integrated circuits,” Opt. Express 17(6), 4565–4574 (2009). [CrossRef] [PubMed]

19.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]

20.

D. Pozar, Microwave Engineering (John Wiley & Sons, 2005).

21.

A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Dispersion extraction with near-field measurements in periodic waveguides,” Opt. Express 17(5), 3716–3721 (2009). [CrossRef] [PubMed]

22.

J. K. Yang, H. Noh, M. J. Rooks, G. S. Solomon, F. Vollmer, and H. Cao, “Lasing in localized modes of a slow light photonic crystal waveguide,” Appl. Phys. Lett. 98(24), 241107 (2011). [CrossRef]

23.

N. Gutman, A. A. Sukhorukov, F. Eilenberger, and C. M. de Sterke, “Bistability suppression and low threshold switching using frozen light at a degenerate band edge waveguide,” Opt. Express 20(24), 27363–27368 (2012). [CrossRef] [PubMed]

OCIS Codes
(230.7370) Optical devices : Waveguides
(250.5300) Optoelectronics : Photonic integrated circuits
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: January 11, 2013
Revised Manuscript: March 19, 2013
Manuscript Accepted: March 22, 2013
Published: April 2, 2013

Citation
Justin R. Burr, Nadav Gutman, C. Martijn de Sterke, Ilya Vitebskiy, and Ronald M. Reano, "Degenerate band edge resonances in coupled periodic silicon optical waveguides," Opt. Express 21, 8736-8745 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8736


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References

  1. A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.72(3), 036619 (2005). [CrossRef] [PubMed]
  2. A. A. Sukhorukov, C. J. Handmer, C. M. de Sterke, and M. J. Steel, “Slow light with flat or offset band edges in few-mode fiber with two gratings,” Opt. Express15(26), 17954–17959 (2007). [CrossRef] [PubMed]
  3. A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A76(5), 053839 (2007). [CrossRef]
  4. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B25(12), C65–C74 (2008). [CrossRef]
  5. A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photonics Rev.5(2), 201–213 (2011). [CrossRef]
  6. N. Gutman, C. M. de Sterke, A. Sukhorukov, and L. Botten, “Slow and frozen light in optical waveguides with multiple gratings Degenerate band edges and stationary inflection points,” Phys. Rev. A85(3), 033804 (2012). [CrossRef]
  7. N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett.36(16), 3257–3259 (2011). [CrossRef] [PubMed]
  8. N. Gutman, W. H. Dupree, Y. Sun, A. A. Sukhorukov, and C. M. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express20(4), 3519–3528 (2012). [CrossRef] [PubMed]
  9. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys.75(4), 1896–1899 (1994). [CrossRef]
  10. S. Kim, B.-H. Ahn, J.-Y. Kim, K.-Y. Jeong, K. S. Kim, and Y.-H. Lee, “Nanobeam photonic bandedge lasers,” Opt. Express19(24), 24055–24060 (2011). [CrossRef] [PubMed]
  11. B. Wang, M. A. Dündar, R. Nötzel, F. Karouta, S. He, and R. W. van der Heijden, “Photonic crystal slot nanobeam slow light waveguides for refractive index sensing,” Appl. Phys. Lett.97(15), 151105 (2010). [CrossRef]
  12. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(2), R1078–R1081 (1996). [CrossRef] [PubMed]
  13. M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express13(18), 7145–7159 (2005). [CrossRef] [PubMed]
  14. A. E. Willner, B. Zhang, L. Zhang, L. Yan, and I. Fazal, “Optical signal processing using tunable delay elements based on slow light,” IEEE J. Sel. Top. Quantum Electron.14(3), 691–705 (2008). [CrossRef]
  15. M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett.73(10), 1368–1371 (1994). [CrossRef] [PubMed]
  16. A. Brimont, A. M. Gutierrez, M. Aamer, D. J. Thomson, F. Y. Gardes, J.-M. Fedeli, G. T. Reed, J. Martí, and P. Sanchis, “Slow-light-enhanced silicon optical modulators under low-drive-voltage operation,” IEEE Photon. J.4(5), 1306–1315 (2012). [CrossRef]
  17. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006). [CrossRef]
  18. P. Sun and R. M. Reano, “Cantilever couplers for intra-chip coupling to silicon photonic integrated circuits,” Opt. Express17(6), 4565–4574 (2009). [CrossRef] [PubMed]
  19. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8(3), 173–190 (2001). [CrossRef] [PubMed]
  20. D. Pozar, Microwave Engineering (John Wiley & Sons, 2005).
  21. A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Dispersion extraction with near-field measurements in periodic waveguides,” Opt. Express17(5), 3716–3721 (2009). [CrossRef] [PubMed]
  22. J. K. Yang, H. Noh, M. J. Rooks, G. S. Solomon, F. Vollmer, and H. Cao, “Lasing in localized modes of a slow light photonic crystal waveguide,” Appl. Phys. Lett.98(24), 241107 (2011). [CrossRef]
  23. N. Gutman, A. A. Sukhorukov, F. Eilenberger, and C. M. de Sterke, “Bistability suppression and low threshold switching using frozen light at a degenerate band edge waveguide,” Opt. Express20(24), 27363–27368 (2012). [CrossRef] [PubMed]

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