## All-optical tunability of a nonlinear photonic crystal channel drop filter

Optics Express, Vol. 12, Issue 8, pp. 1605-1610 (2004)

http://dx.doi.org/10.1364/OPEX.12.001605

Acrobat PDF (705 KB)

### 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

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]

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]

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]

*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

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]

*ε*(

**r**)=

*ε*(

**r**+

**R**), with

**R**a lattice vector. Then, the electromagnetic field distribution is governed by the following wave equation,

**H**(

**r**) and

*ω*are the magnetic field and wave frequency, respectively. Using Bloch’s theorem, we express the field as

**H**(

**r**)=

**H**

_{k}(

**r**)

*e*

^{ik·r}, where

**k**is the Bloch wavevector and

**H**

_{k}(

**r**) is a periodic field. By Fourier expanding

**H**

_{k}(

**r**) and

*ε*

^{-1}(

**r**),

**H**

_{K}(

**r**)=∑G∑λ=1,2

*h*

^{λ}G

*e*

^{iG}·

^{r}ê

^{λ}G and

*ε*

^{-1}(

**r**)=∑G

*e*

^{iG·r}, and inserting them in Eq. (1), one yields the eigenvalue matrix equation:

*M*

_{Gλ,G′}λ′=

**k**+

**G**)×ê

^{λ}′

_{G}′, with

**G**a reciprocal lattice vector,

**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 ê

_{G}, oriented perpendicular (parallel) to the plane of the PC.

**k**in the first Brillouin zone; the eigenvalues {

*ω*

_{n}} give the PBS whereas the eigenvectors {

**k**-space, determine the propagating eigenmodes. The field distribution

**H**

_{k}(

**r**) in the position space is then obtained by Fourier transforming these coefficients. Finally, the electric field is calculated by using the relation

*iωε*0

*ε*(

**r**)

**E**(

**r**)=-∇×

**H**(

**r**).

*ω*

_{p}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

*ω*

_{p}, the field distribution of the corresponding eigenmode, and its group velocity,

_{k}

*ω*(

**k**=

**h**

^{†}·∇

_{k}

*ℳ*·

**h**=[2

*ω*(

**k**)/

*c*

^{2}]∇

_{k}

*ω*(

**k**), or by direct numerical differentiation of the frequency manifold

*ω*(

**k**).

**S**

_{k}(

**r**)〉=

*v*

_{g}(

**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**)=

*ε*0

*cnn*2|

**E**

^{0}(

**r**)|

^{2}/2, and then the new value of the dielectric constant,

*ε*between two iterations is below some threshold,

*δε*≡∑

*i*|

*ε*

^{j+1}(

**r**

_{i})-

*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.

*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

*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

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]

*n*=3.4, e.g., corresponding to Si or GaAs semiconductors, and Kerr coefficient

*n*

_{2}=3·10

^{-16}m

^{2}/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.

*N*

_{l}×1, with

*N*

_{l}=9. In the PWE, a number of 1024 plane waves was used. In the case of the cavity mode, an

*N*

_{p}×

*N*

_{p}, with

*N*

_{p}=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

*ω*

_{r}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.

*P*and frequency

*ω*

_{p}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

*ω*

_{r}. 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

*ω*

_{r}. Note that in these calculations we considered that the pump frequency

*ω*

_{p}is far from the resonant frequency

*ω*

_{r}, 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

*ω*

_{p}. The results we obtained are summarized in Fig. 4. The waveguide mode dispersion, calculated at pump powers

*P*=400W/

*µ*m and

*P*=900W/µmand pump frequencies (expressed in normalized units of

*c*/2

*π a*)

*ω*

_{p}=0.33 and

*ω*

_{p}=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

*ω*

_{p}=0.33 (0.295

*c*at

*ω*

_{p}=0.33 as compared to 0.426

*c*at

*ω*

_{p}=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

*ω*

_{r}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

*ω*

_{p}=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,

*ω*

_{r}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

*ω*

_{r}depends stronger on the power

*P*. Furthermore, Fig. 4(b) shows that at a given power the shift in

*ω*

_{r}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.

## 4. Discussion and conclusions

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

*c*, and thus a large decrease in the operating power.

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

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. |

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 |

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 |

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. |

7. | D. Scrymgeour, N. Malkova, S. Kim, and V. Gopalan, “Electro-optic control of the superprism effect in photonic crystals,” Appl. Phys. Lett. |

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. |

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. |

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. |

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 |

12. | M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. |

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 |

14. | N. C. Panoiu, M. Bahl, and R. M. Osgood, “Optically tunable superprism effect in nonlinear photonic crystals,” Opt. Lett. |

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. |

17. | V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E |

18. | K. Sakoda, |

19. | S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature |

20. | B.-S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science |

21. | S. Mookherjea, “Coupled resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. |

**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

- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- N. C. Panoiu, M. Bahl, R. M. Osgood, “Optically tunable superprism effect in nonlinear photonic crystals,” Opt. Lett. 28, 2503–2505 (2003). [CrossRef] [PubMed]
- N. C. Panoiu, M. Bahl, R. M. Osgood, “Ultrafast optical tuning of superprism effect in nonlinear photonic crystals,” J. Opt. Soc. Am. B, (submitted).
- 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]
- 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]
- K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, Heidelberg, 2001).
- S. Noda, A. Chutinan, M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000). [CrossRef] [PubMed]
- B.-S. Song, S. Noda, T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science 300, 1537 (2003). [CrossRef] [PubMed]
- S. Mookherjea, “Coupled resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 448–456 (2002). [CrossRef]

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