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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 8 — Apr. 19, 2004
  • pp: 1605–1610
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All-optical tunability of a nonlinear photonic crystal channel drop filter

Nicolae C. Panoiu, Mayank Bahl, and Richard M. Osgood, Jr  »View Author Affiliations


Optics Express, Vol. 12, Issue 8, pp. 1605-1610 (2004)
http://dx.doi.org/10.1364/OPEX.12.001605


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Abstract

We report a numerical analysis of an optically tunable channel drop filter that consists of a resonant cavity side-coupled to a waveguide embedded in a two-dimensional nonlinear photonic crystal. We first introduce a numerical method that allows us to calculate the photonic band structure of a nonlinear photonic crystal, as well as the frequency and field profile of cavity and waveguide modes. Then, we use this numerical method to study the dependence of the resonant frequency of a cavity side-coupled to a waveguide, on the optical power in the waveguide.

© 2004 Optical Society of America

1. Introduction

Since their discovery more than a decade ago [1

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

, 2

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

], photonic crystals (PCs) have attracted a rapidly growing research interest. These optical structures have a spatially periodic dielectric constant (index of refraction), which induces striking changes of the photon density of states. In particular, the ability of PCs to induce a band gap in the frequencies of the electromagnetic modes that can propagate in such structures has leaded to advances in the basic understanding of light-matter interactions, as well as to optoelectronic devices with new and improved functionality. Furthermore, inserting structural defects in PCs not only fundamentally changes their physical properties but also leads to an unprecedented flexibility in the ability to control and manipulate light. For instance, by introducing line (1D) or point (0D) defects in the periodic structure of a PC, one can guide light around sharp corners with negligible losses [3

3. 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. Lett. 77, 3787–3790 (1996). [CrossRef] [PubMed]

, 4

4. S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998). [CrossRef] [PubMed]

] and create ultra-small laser nanocavities [5

5. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two–dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]

] or efficient optical isolators [6

6. M. J. Steel, M. Levy, and R. M. Osgood, “High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects,” IEEE Photon. Tecnol. Lett. 12, 1171–1173 (2000). [CrossRef]

].

To employ PCs in advanced technologies, e.g. for ultra-fast optical functions and all-optical switching, it is crucially important to have dynamic tunability of their properties. In recent years, several promising schemes for achieving this feature have been proposed. For example, tunability of the photonic band-gap (PBG) has been obtained by modulating the PC’s refractive index through the electro-optic effect [7

7. D. Scrymgeour, N. Malkova, S. Kim, and V. Gopalan, “Electro-optic control of the superprism effect in photonic crystals,” Appl. Phys. Lett. 82, 3176–3178 (2003). [CrossRef]

] or through temperature-induced changes in the PC’s refractive index [8

8. K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, and M. Ozaki, “Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal,” Appl. Phys. Lett. 75, 932–934 (1999). [CrossRef]

]. One promising approach, which is suitable for ultra-fast devices, is to employ intrinsic optical nonlinearities in the PC material. Thus, experimental studies have recently demonstrated both the ultrafast response of semiconductor-based PC waveguides as well as enhanced nonlinearity effects [9

9. A. D. Bristow, J.-P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851–853 (2003). [CrossRef]

, 10

10. L. Brzozowski, V. Sukhovatkin, E. H. Sargent, A. J. SpringThorpe, and M. Extavour, “Intensity-dependent reflectance and transmittance of semiconductor periodic structures,” IEEE J. Quantum Electron. 39, 924–930 (2003). [CrossRef]

], which are specific to PC structures. Furthermore, it recently has been predicted that such nonlinear optical PC devices can be used to achieve unique functionalities, namely switching [11

11. M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002). [CrossRef]

13

13. M. Bahl, N. C. Panoiu, and R. M. Osgood, “Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,” Phys. Rev. E 67, 056604(1–9) (2003). [CrossRef]

], all-optical control of their enhanced dispersive properties [14

14. N. C. Panoiu, M. Bahl, and R. M. Osgood, “Optically tunable superprism effect in nonlinear photonic crystals,” Opt. Lett. 28, 2503–2505 (2003). [CrossRef] [PubMed]

, 15

15. N. C. Panoiu, M. Bahl, and R. M. Osgood, “Ultrafast optical tuning of superprism effect in nonlinear photonic crystals,” J. Opt. Soc. Am. B, (submitted).

], or optical limiting and higher harmonics generation [13

13. M. Bahl, N. C. Panoiu, and R. M. Osgood, “Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,” Phys. Rev. E 67, 056604(1–9) (2003). [CrossRef]

].

In this paper, we introduce a numerical method that allows us to calculate the field profile and dispersion curves of waveguide modes, as well as the modes and resonant frequencies corresponding to resonant cavities; in both cases these structural defects are embedded in nonlinear PCs. Furthermore, as an example, we apply this numerical method to investigate the optical response of a channel drop filter consisting of a resonant cavity side-coupled to a waveguide PC whose optical properties can be tuned by means of a pump beam.

2. Numerical method

In this section, we describe the numerical method we used to find the photonic band structure (PBS) of nonlinear PCs as well as the frequencies and the corresponding modes associated to structural defects in nonlinear PCs, namely a plane wave expansion (PWE) method [16

16. K. M. Ho, K. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]

], modified to self-consistently include the changes in the refractive index caused by the nonlinear effects [17

17. V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E 63, 027602(1–4) (2001). [CrossRef]

]. We first present the method that is applied to a PC without defects and then discuss the modifications that are required when describing structural defects, namely the supercell approach. A generic geometry of a PC with 1D and 0D defects is shown in Fig. 1.

Fig. 1. Schematics of a 1D and 0D defects in a 2D PC with square symmetry (a is the lattice pitch). The two rectangles represent the supercells used to find the defect modes and the corresponding mode frequencies. In the lower panels, the electric field (TM polarization) of the 0D cavity mode (right) and 1D waveguide mode (left); the mode frequency is ωr .

To begin with, we consider the propagation of a monochromatic electromagnetic field in a medium with a periodic distribution of the dielectric constant, ε(r)=ε(r+R), with R a lattice vector. Then, the electromagnetic field distribution is governed by the following wave equation,

×[ε1(r)×H(r)]=(ω2c2)H(r),
(1)

where H(r) and ω are the magnetic field and wave frequency, respectively. Using Bloch’s theorem, we express the field as H(r)=Hk (r)e ik·r, where k is the Bloch wavevector and Hk (r) is a periodic field. By Fourier expanding Hk (r) and ε -1(r), HK (r)=∑G∑λ=1,2 h λ Ge iG ·rê λ G and ε -1(r)=∑GεG1 e iG·r, and inserting them in Eq. (1), one yields the eigenvalue matrix equation:

Gλ=1,2𝓜Gλ,GλhG'λ=(ω2c2)hGλ.
(2)

Here, the matrix elements M Gλ,Gλ′=εGG1 [(k+G)×ê λG′, with G a reciprocal lattice vector, hGλ and εG1 are Fourier coefficients, and êGλ are unit vectors orthogonal on k+G. In a 2D geometry, in the particular cases that correspond to TE or TM polarizations, in Eq. (2) there is only one summation, as for the TE (TM) polarization there is only one unit vector êG, oriented perpendicular (parallel) to the plane of the PC.

The eigenvalue matrix equation (2) is solved for k in the first Brillouin zone; the eigenvalues {ωn } give the PBS whereas the eigenvectors {hGλ }, expressed in the k-space, determine the propagating eigenmodes. The field distribution Hk (r) in the position space is then obtained by Fourier transforming these coefficients. Finally, the electric field is calculated by using the relation iωε0ε (r)E(r)=-∇×H(r).

To determine the altered PBS of the PC, in the presence of an illumination pump beam with the frequency ωp and power per unit length in the transverse direction, per unit cell, P, one proceeds as follows. First, one determines the PBS using the dielectric constant ε 0(r)≡ε (r), with P=0, i.e., the PBS of the linear PC. Then, by using an interpolation procedure, one numerically computes the wavevector kp0 that corresponds to the frequency ωp , the field distribution of the corresponding eigenmode, and its group velocity, vg0 =∇k ω(k=kp0 ). The group velocity can be calculated by using the Hellmann-Feynman theorem [18

18. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, Heidelberg, 2001).

], h ·∇k ·h=[2ω(k)/c 2]∇k ω(k), or by direct numerical differentiation of the frequency manifold ω(k).

Once the group velocity is calculated, the electric field in the pumping mode E0(r)Ekp0(r) is rescaled such that the equation

P=ε0ε(r)|Ek(r)|2vG(k)a2,
(3)

is satisfied. Here, 〈·〉 means spatial averaging over the unit cell. Equation 3 can be readily derived from the relationship between the average of the Poynting vector and the average of the mode energy density, 〈Sk (r)〉=vg (k)〈𝓤 k(r)〉. After E 0(r) is calculated, one determines the change in the refractive index n of the Kerr material, δn 0(r)=ε0cnn2|E 0(r)|2/2, and then the new value of the dielectric constant, ε1(r)=[ε(r)δn0(r)]2. These steps are repeated until the variation of ε between two iterations is below some threshold, δε≡∑i|ε j+1(ri )-ε j (ri)|/N<10-8. Here, the sum is taken over the grid points of the mesh that covers the unit cell. Then, the final value of ε is used to calculate the PBS of the pumped nonlinear crystal.

When structural defects are introduced into the PC, this method is modified as follows. First, consider that a 1D defect is introduced into the PC, as shown in Fig. 1, with the optical power propagating in the waveguide mode being P. To compute the waveguide mode, one considers that the unit cell of the PC is a supercell whose transverse dimension is large enough to ensure that the whole waveguide mode is contained within. Then, the iterative procedure described before is used, with the only modification that in this case the Eq. (3) becomes

P=ε0ε(r)Ek(r)2vg(k)Nla2,
(4)

where N l is the number of unit cells in the transverse direction and the average is taken over the supercell. A 0D defect is treated similarly. Generally, both for a pure periodic PC and for a PC with structural defects the algorithm converged in less than ten iterations; however, if the pump frequency was close to k=0, which corresponds to a region of low group velocity, the number of iterations required to reach convergence increased by about a factor of two.

3. Tunable channel drop filter

In what follows, we illustrate how the numerical method introduced in Section 2 can be used to analyze a tunable channel drop filter consisting of a resonant cavity side-coupled to a waveguide in a nonlinear PC. The linear version of this device has been recently demonstrated, both as a single- [19

19. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000). [CrossRef] [PubMed]

] as well as multi-frequency [20

20. B.-S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science 300, 1537 (2003). [CrossRef] [PubMed]

] operating device. We consider a 2D PC composed of a square lattice of circular rods made from nonlinear instantaneous Kerr dielectric with refractive index n=3.4, e.g., corresponding to Si or GaAs semiconductors, and Kerr coefficient n 2=3·10-16m2/W. For simplicity, the refractive index of the background medium is set to n=1. The structural parameter of the PC is r/a=0.18, where r and a are the rod radius and the lattice pitch, respectively. Note that the results presented here can be extended to PC slab waveguides, but the numerical analysis becomes much more computationally demanding. The waveguide PC is obtained by removing one row of rods, whereas the resonant cavity by removing one rod (see Fig. 1). The waveguide mode dispersion, calculated for the linear PC, as well as the corresponding resonant frequency of the cavity are shown in Fig. 2. Also shown in Fig. 2 is the result of the finite-difference time-domain (FDTD) simulation of the resonant CW excitation of the cavity side-coupled to the waveguide.

The waveguide mode dispersion curve in Fig. 2 was determined by using the supercell approach discussed in Section 2, the size of the supercell being Nl ×1, with Nl =9. In the PWE, a number of 1024 plane waves was used. In the case of the cavity mode, an Np ×Np , with Np =5, supercell was used, the number of plane waves in the PWE being 4096. We analyzed three cases, in which the resonant cavity was placed one, two, and three rows apart from the waveguide PC. The values of the quality factors Q that correspond to these cases are illustrated in Fig. 3, and were computed by using FDTD calculations. This figure shows that as the cavity is placed closer to the waveguide PC, Q decreases, due to increased coupling losses. Also, the resonant frequency ωr increases, which is explained by a slight decrease in the effective refractive index of the cavity mode. Finally, as the distance between the cavity and the waveguide PC changes from two to three rows there is hardly any change in Q, a behavior that is explained by the fact that the overlap between the cavity and waveguide modes remains almost unchanged.

Fig. 2. Left, mode dispersion of the waveguide PC (red line), at P=0; the blue line corresponds to the resonant frequency of the cavity, also at P=0. Right, FDTD simulation of propagation of a low power CW, at ωr0 , in the waveguide PC side-coupled to the cavity.
Fig. 3. Spectral densities corresponding to resonant cavities placed one row (Q 1), two rows (Q 2), and three rows (Q 3) from the waveguide PC.

Next, we consider addition of a pump beam, and, in particular, that a mode with optical power P and frequency ωp propagates in the waveguide PC. As a result, the refractive index in the region covered by the waveguide mode changes due to the induced Kerr effect. This change leads to a variation of the coupling between the cavity and the waveguide, as well as the resonant frequency ωr . To describe quantitatively these effects, we used the numerical method previously introduced and proceeded as follows. We first numerically determined the waveguide mode corresponding to a given power P and then, from the field distribution, we computed the change in the refractive index in the region of the cavity. Then, by using this distribution of the refractive index in the spatial domain occupied by the cavity, we computed the corresponding resonant frequency ωr . Note that in these calculations we considered that the pump frequency ωp is far from the resonant frequency ωr , and therefore the waveguide mode is not affected by the presence of the cavity (very close to the resonance this assumption no longer holds). Moreover, as the grids of the supercells used to compute the waveguide and cavity modes do not coincide, we used a bilinear interpolation procedure to determine the latter from the former one. These calculations were repeated for different pump powers P, for several pump frequencies ωp . The results we obtained are summarized in Fig. 4. The waveguide mode dispersion, calculated at pump powers P=400W/µm and P=900W/µmand pump frequencies (expressed in normalized units of c/2π a) ωp =0.33 and ωp =0.35, respectively, are shown in Fig. 4(a). Note that, although in the latter case the power P is more than twice as much as the power in the former case, the mode frequency shift is almost the same. This behavior is explained by the reduced group velocity at ωp =0.33 (0.295c at ωp =0.33 as compared to 0.426c at ωp =0.35). Thus, as Eq. (4) suggests, for a certain power P the electric field in the waveguide mode is inverse proportional to the group velocity, so that waveguide modes that correspond to lower group velocities induce stronger nonlinear effects. The dependence of the resonant frequency ωr on the power P, calculated for the case in which the cavity is placed one and two rows apart from the waveguide, is illustrated in Fig. 4(b). Regarding this figure, we mention that at ωp =0.33 we investigated a smaller range of powers P, because in this case at large powers the numerical algorithm converges at a slower rate; again, this is due to the reduced group velocity. Notice that at low powers, if the cavity is placed closer to the waveguide PC, ωr shifts to higher values, which is in agreement with the behavior seen in Fig. 3. In addition, if the cavity is closer to the waveguide PC, its resonant frequency ωr depends stronger on the power P. Furthermore, Fig. 4(b) shows that at a given power the shift in ωr is much larger if the cavity is closer to the waveguide, a consequence of an increased variation of the refractive index in the region covered by the cavity mode.

Fig. 4. Mode dispersion curves of the waveguide PC, determined at different pump power P and pump frequency ωp (a); in (b), resonant frequency ωr vs. power P, determined for ωp =0.33 (—) and ωp =0.35 (-·-·-). The cavity is placed one row (red curves) and two rows (blue curves) apart from the waveguide PC.

4. Discussion and conclusions

In conclusion, we have introduced a numerical method that can be used to compute the frequencies and the corresponding modes associated with 1D and 0D structural defects in nonlinear PCs. The method was applied to characterize the optical response of a device consisting of a resonant cavity side-coupled to a waveguide, both embedded in a nonlinear PC. Furthermore, note that the optical power required to induce a specific variation in the operating frequency of the device can be reduced if materials with large Kerr nonlinearity, e.g. InSb or ZnSe-based semiconductors or polymers are used. Finally, instead of the waveguide PC discussed here, one can use waveguide PCs whose mode dispersion is specially engineered, so as to have an extremely low group velocity. For instance, using coupled-resonator optical waveguides [21

21. S. Mookherjea, “Coupled resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 448–456 (2002). [CrossRef]

], one can achieve group velocities less than 0.01c, and thus a large decrease in the operating power.

Acknowledgments

This work was supported by the NIST Advanced Technology Program Cooperative Agreement, Grant No 70NANB8H4018 and also in part by the DoD STTR, Grant No FA9550-04-C-0022.

References and links

1.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

2.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

3.

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. Lett. 77, 3787–3790 (1996). [CrossRef] [PubMed]

4.

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998). [CrossRef] [PubMed]

5.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two–dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]

6.

M. J. Steel, M. Levy, and R. M. Osgood, “High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects,” IEEE Photon. Tecnol. Lett. 12, 1171–1173 (2000). [CrossRef]

7.

D. Scrymgeour, N. Malkova, S. Kim, and V. Gopalan, “Electro-optic control of the superprism effect in photonic crystals,” Appl. Phys. Lett. 82, 3176–3178 (2003). [CrossRef]

8.

K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, and M. Ozaki, “Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal,” Appl. Phys. Lett. 75, 932–934 (1999). [CrossRef]

9.

A. D. Bristow, J.-P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851–853 (2003). [CrossRef]

10.

L. Brzozowski, V. Sukhovatkin, E. H. Sargent, A. J. SpringThorpe, and M. Extavour, “Intensity-dependent reflectance and transmittance of semiconductor periodic structures,” IEEE J. Quantum Electron. 39, 924–930 (2003). [CrossRef]

11.

M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002). [CrossRef]

12.

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003). [CrossRef]

13.

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,” Phys. Rev. E 67, 056604(1–9) (2003). [CrossRef]

14.

N. C. Panoiu, M. Bahl, and R. M. Osgood, “Optically tunable superprism effect in nonlinear photonic crystals,” Opt. Lett. 28, 2503–2505 (2003). [CrossRef] [PubMed]

15.

N. C. Panoiu, M. Bahl, and R. M. Osgood, “Ultrafast optical tuning of superprism effect in nonlinear photonic crystals,” J. Opt. Soc. Am. B, (submitted).

16.

K. M. Ho, K. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]

17.

V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E 63, 027602(1–4) (2001). [CrossRef]

18.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, Heidelberg, 2001).

19.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000). [CrossRef] [PubMed]

20.

B.-S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science 300, 1537 (2003). [CrossRef] [PubMed]

21.

S. Mookherjea, “Coupled resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 448–456 (2002). [CrossRef]

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.4360) Nonlinear optics : Nonlinear optics, devices
(230.1150) Optical devices : All-optical devices
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Focus Issue: Photonic crystals and holey fibers

History
Original Manuscript: February 27, 2004
Revised Manuscript: March 18, 2004
Manuscript Accepted: March 21, 2004
Published: April 19, 2004

Citation
Nicolae C. Panoiu, Mayank Bahl, and Richard M. Osgood, "All-optical tunability of a nonlinear photonic crystal channel drop filter," Opt. Express 12, 1605-1610 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1605


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References

  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]
  3. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996). [CrossRef] [PubMed]
  4. S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998). [CrossRef] [PubMed]
  5. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two–dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]
  6. M. J. Steel, M. Levy, R. M. Osgood, “High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects,” IEEE Photon. Tecnol. Lett. 12, 1171–1173 (2000). [CrossRef]
  7. D. Scrymgeour, N. Malkova, S. Kim, V. Gopalan, “Electro-optic control of the superprism effect in photonic crystals,” Appl. Phys. Lett. 82, 3176–3178 (2003). [CrossRef]
  8. K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, M. Ozaki, “Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal,” Appl. Phys. Lett. 75, 932–934 (1999). [CrossRef]
  9. A. D. Bristow, J.-P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851–853 (2003). [CrossRef]
  10. L. Brzozowski, V. Sukhovatkin, E. H. Sargent, A. J. SpringThorpe, M. Extavour, “Intensity-dependent reflectance and transmittance of semiconductor periodic structures,” IEEE J. Quantum Electron. 39, 924–930 (2003). [CrossRef]
  11. M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002). [CrossRef]
  12. M. F. Yanik, S. Fan, M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003). [CrossRef]
  13. M. Bahl, N. C. Panoiu, R. M. Osgood, “Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,” Phys. Rev. E 67, 056604(1–9) (2003). [CrossRef]
  14. N. C. Panoiu, M. Bahl, R. M. Osgood, “Optically tunable superprism effect in nonlinear photonic crystals,” Opt. Lett. 28, 2503–2505 (2003). [CrossRef] [PubMed]
  15. N. C. Panoiu, M. Bahl, R. M. Osgood, “Ultrafast optical tuning of superprism effect in nonlinear photonic crystals,” J. Opt. Soc. Am. B, (submitted).
  16. K. M. Ho, K. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]
  17. V. Lousse, J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E 63, 027602(1–4) (2001). [CrossRef]
  18. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, Heidelberg, 2001).
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