## Photonic band gap filter for wavelength division multiplexer

Optics Express, Vol. 11, Issue 3, pp. 230-239 (2003)

http://dx.doi.org/10.1364/OE.11.000230

Acrobat PDF (162 KB)

### Abstract

An optical multiplexer-demultiplexer based on an index-confined photonic band gap waveguide is proposed. The dropping of electromagnetic waves having a given frequency or a certain frequency band is obtained via a phase-shifted grating obtained by breaking the uniform period sequence to include a defect layer. The selective filtering properties of the proposed structure are simulated by means of a computer code relying on a bidirectional beam propagation method based on the method of lines.

© 2002 Optical Society of America

## 1. Introduction

*et al.*[3

3. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. **80**, 960–963 (1998). [CrossRef]

4. C. Jin, S. Han, X. Meng, B. Cheng, and D. Zhang, “Demultiplexer using directly resonant tunneling between point defects and waveguides in a photonic crystal,” J. Appl. Phys. **91**, 4771–4773 (2002). [CrossRef]

5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. **74**, 1370–1372 (1999). [CrossRef]

6. 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]

7. B. E. Nelson, M. Gerken, D. A. B. Miller, R. Piestun, C.-C. Lin, and J. S. Harris, “Use of a dielectric stack as a one-dimensional photonic crystal for wavelength demultiplexing by beam shifting,” Opt. Lett. **25**, 1502–1504 (2000). [CrossRef]

8. M. Koshiba, “Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,” J. Lightwave Technol. **19**, 1970–1975 (2001). [CrossRef]

9. A. Sharkawy, S. Shi, and D. W. Prather, “Multichannel wavelength division multiplexing with photonic crystals,” Appl. Opt. **40**, 2247–2252 (2001). [CrossRef]

10. E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, “Investigation of localized coupled-cavity modes in two-dimensional photonic band gap structures,” IEEE J. Quantum Electron. **38**, 837–843 (2002). [CrossRef]

11. S.Y. Lin and J.G. Fleming, “A three-dimensional Optical Photonic Crystal,” J.Lightwave Technol. , **17**, 1944–1947 (1999). [CrossRef]

13. G.I. Stegeman and D.G. Hall, “Modulated index structure,” J.Opt. Soc. Am. A , **7**, 1387–1398 (1996). [CrossRef]

14. C.R. Giles, “Lightwave applications of fiber Bragg gratings,” J.Lightwave Technol. , **15**, 1391–1404, (1997). [CrossRef]

15. R.W. Ziolkowski and T. Liang, “Design and characterization of a grating-assisted coupler enhanced by a photonic-band-gap structure for effective wavelength-division demultiplexing,” Optics Letters , **22**, 1033–1035, (1997). [CrossRef] [PubMed]

16. C.F. Lam, R.B. Vrijen, P.P.L. Chang-Chien, D.F. Sievenpiper, and E. Yablonovitch, “A tunable wavelength demultiplexer using logarithmic filter chains,” J. Lightwave Technol. **16**, 9, 1657–1662 (1998). [CrossRef]

17. R. Zengerle and O. Leminger, “Phase shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. , **13**, 2354–2358 (1995). [CrossRef]

18. L. Wei and J.W.Y. Lit, “Phase-shifted Bragg Grating Filters with symmetrical structures,” J. Lightwave Technol. , **15**,1405–1410, (1997). [CrossRef]

_{1}, L

_{2}, .....L

_{M}, and (M-1) intermediate regions, providing the phase shifts ϕ

_{1}, ϕ

_{2},...... ϕ

_{M}, all stacked in series. These filters exhibit nearly rectangular pass band characteristics,

*i.e.*, a drastic slope to the stop band. The finite width of the stop band in the center of which the transmission takes place and the bandwidth needed for sensitivity penaltyfree transmission are serious drawbacks in terms of channel number and channel selection for the use of these phase-shifted filters in WDM systems. To overcome these limits, more complicated structures have to be considered. For adjusting the pass band central wavelength in each filter to the corresponding channel waveguide, grating regions with different periods have to be included. To increase the channel number, the ratio between the stop band width and the transmission bandwidth ought to be high: therefore gratings characterized by a maximum coupling factor κ are necessary, the width of the stop band being proportional to κ. The structure proposed in this paper, having a single defect layer, appears to be even simpler and shorter than the compound phase-shifted Bragg grating reported in [17

17. R. Zengerle and O. Leminger, “Phase shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. , **13**, 2354–2358 (1995). [CrossRef]

20. S. Helfert and R. Pregla, “Efficient analysis of periodic structures,” J. Lightwave Technol. , **16**, 1694–1702 (1998). [CrossRef]

## 2. Theoretical remarks

**φ**, the components of which are the scalar potentials φ

_{i}in the original domain:

**Q**is the matrix of the discretized difference operator and u=k

_{0}z is the normalized longitudinal axis, k

_{0}being the vacuum wavenumber. Because the matrix

**Q**is not diagonal, the potentials φ

_{i}are each other coupled and the wave Eq. (1) cannot be solved in this form. By diagonalizing

**Q**, both the vector of the eigenvalues λ and the matrix of the eigenvectors are evaluated:

**F**e

^{-λu}and

**B**e

^{λu}are the contributions of the forward and the backward fields in the transformed domain, respectively. The field

**φ**(u) in the original domain can be obtained by the inverse domain transformation. In order to obtain the reflection operator, Eq. (2) is analytically solved for the forward and backward travelling waves in both the left (L) and the right (R) regions of each discontinuity. After having calculated the forward coefficient F

_{L}of Eq. (3) in the left region, the unknown left backward and right forward coefficients B

_{L}, F

_{R}are evaluated by placing the right backward coefficient B

_{R}=0 and by matching the electromagnetic field transverse components at the discontinuity interfaces.

*i.e.*, a periodically strong etched waveguide grating, originating from the European COST 268 action, has given excellent results compared with those obtained by two home-made simulators based on the analytical Transfer Matrix Method (TMM) and the numerical 2D Finite Difference Time Domain (FDTD) method [19]. On the other hand, the CPU computing times (of about 5 min on a personal computer Pentium III 800 MHz) of the MoL-BBPM are only a little longer than those occurring for the TMM, while are much more shorter than those evaluated by the FDTD (of about 180 min).

## 3. Numerical results

*i.e,*. a single mode optical slab waveguide equipped with an appropriate periodically corrugated overlay (see Fig. 1 without the defect waveguide). The GaAs/Al

_{2}O

_{3}waveguide is characterized by the following parameters: core refractive index n

_{g}=3.4, substrate refractive index n

_{s}=1.6 at the operating wavelength λ=1300 nm; core thickness d

_{g}=240 nm. The grating is constituted by a finite number of N alternating layers of air (n

_{a}=1) and GaAs having the same width (w

_{1}=w

_{2}=116 nm), spatial period Λ=w

_{1}+w

_{2}=232 nm and tooth depth h=60 nm.

_{Λ}, the tooth depth h and by the duty cycle,

*i.e.*the ratio of tooth width to groove width w

_{2}/w

_{1}. In particular we have found that, for N=63 alternating layers and number of spatial periods N

_{Λ}=32, the structure acts as a stop band reflection filter, for the TE polarization, centered at the Bragg wavelength λ

_{B}=1305 nm, with a 3 dB bandwidth equal to 50 nm (see blue line in Fig. 2). For the chosen geometrical and physical parameter values, the Bragg wavelength of the waveguide grating having rectangular grooves can be evaluated according to the approximated formula obtained by neglecting the scattering into radiation modes:

_{t}and n

_{g}are the refractive effective indices of the two three-layered waveguides corresponding to the tooth and to the groove of the Bragg grating unit cell, respectively.

_{t}=2.937 for the waveguide having thickness d

_{g}and n

_{g}=2.738 for the waveguide having thickness (d

_{g}-h), thus Eq. (4) gives λ

_{B}=1316 nm. However, the “exact” Bragg wavelength value directly obtained by the MoL-BBPM is λ

_{B}=1305 nm.

*i.e.*, with a segment of GaAs/Al

_{2}O

_{3}waveguide, having core thickness d

_{g}. Really the introduction of a single defect in any part of the grating does not give rise to significant changes in the stop band while the increase of the defect number, introduced especially in the central part of the grating, gives the maximum efficiency in terms of maximum reflectance in the stop band and channel bandwidth reduction. In fact, the transmittance and reflectance spectra of the structure do not change significantly when the defects are all confined in the neighbourhood of either the input or the output ports, because in this case the defect inclusion induces only the optical shift of the propagating signal in the input or the output port, respectively. On the other hand, the signal optical shift in both the longitudinal propagation directions, induced on the propagating signal along the defect region, assumes an increasing important role as the waveguide inclusion is shifted from the extreme to the central position of the Bragg grating. Anyway, a further increase of the defect number, to parity of device length, reducing drastically the total spatial period number, deteriorates the filter spectral response. For this reason, the spatial period number has been fixed to 32 and a unique defected region L long is introduced in the middle of the IC-1D-PBG waveguide.

_{2}O

_{3}waveguide having length L and core depth d

_{g}.

_{c}=1293 nm, 3 dB bandwidth B

_{λ}=12 nm, maximum value of transmission coefficient T=0.95 and minimum reflection coefficient R=0.05. The stop bands centered at λ=1275 nm and λ=1318 nm exhibit a maximum R value of 0.70 and 0.90, respectively. By increasing L to 3.6 µm the spectra do not change significantly, the only difference being the value of the reflectance in the first stop band which reaches the maximum of 0.90.

_{c}increases by increasing the defect waveguide L length. This parameter becomes fundamental to design a demultiplexer: the empirical relationship between the channel number, the defect waveguide L length and the Bragg wavelength λ

_{B}derived by simulations is N

_{c}≅L/2λ

_{B}. In the case of L=35 µm, we calculate N

_{c}=12 channels which increase to 22 for L=67 µm. Anyway, in the following investigations we will count the number of channels by considering transmission channels only those having, on their left and right, stop bands characterized by reflection coefficient values greater than 0.75. This number will be named useful channel number N

_{c,u}. Figure 3 shows the useful channel number against the defect waveguide L length. Following the definition, we count N

_{c,u}=8 channels for L=35 µm: this number increases to 18 for L=67 µm.

_{λ}=2 nm, while the channel spacing is 6 nm. These values drastically reduce when an index confined photonic band gap demultiplexer characterized by a defect waveguide length of L=67 µm is considered: in this case a 3 dB bandwidth of only 0.8 nm and a channel spacing of 2.5 nm are obtained. The spectral behaviour of the examined structures can be understood by considering a Fabry-Perot cavity of length L having distributed reflectors at each end consisting of short strong Bragg gratings. This is a special case of filter with one-phase shift section [17

17. R. Zengerle and O. Leminger, “Phase shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. , **13**, 2354–2358 (1995). [CrossRef]

18. L. Wei and J.W.Y. Lit, “Phase-shifted Bragg Grating Filters with symmetrical structures,” J. Lightwave Technol. , **15**,1405–1410, (1997). [CrossRef]

_{c}, obtained by the MoL-BBPM simulations, accounts for these terms.

_{λ})

^{2}, resulting for our structure Δ/B

_{λ}≅3, we have obtained CT≅-54 dB.

_{y}component calculated by the MoL-BBPM, for the wavelength values of maximum transmission (λ=1299 nm, R=0.10, T=0.90) and of 3 dB bandwidth (λ=1300 nm, R=0.45, T=0.55), respectively. The pink lines draft the geometrical profile of the IC-1D-PBG filter with the inclusion of the defect waveguide as in Fig. 1. The different behaviour of the EM field propagation for the two wavelength values is outlined in Figs. 7 and 8, that sketch the patterns of the electric field E

_{y}component of the signal launched in the input port (blue line), the reflected field in the input port (red line) and the transmitted field in the output port (green line). We can see that for the wavelength of maximum transmission (λ=1299 nm) the reflected field E

_{y}component (R=0.10) is negligible with respect to the transmitted one, whereas for the 3 dB bandwidth wavelength (λ=1300 nm) the two reflected and transmitted components assumes almost the same values (R=0.45, T=0.55).

_{2}substrate (n

_{s}=1.45), Si

_{3}N

_{4}core (n

_{c}=2), w

_{1}=w

_{2}=215 nm, N=39, d

_{g}=500 nm, h=250 nm, have evidenced excellent properties to be exploited for the design of WDM filters useful for applications in the third window of the optic fiber telecommunications at the central wavelength λ=1485 nm.

## 4. Conclusion

## References and Links

1. | J.D. Joannopoulos, R.D. Meade, and J.N. Winn, |

2. | A. D’Orazio, M. De Sario, V. Petruzzelli, and F. Prudenzano: “Numerical modeling of photonic band gap waveguiding structures”, Recent Research Developments in Optics, S.G. Pandalai Editor, 2002. |

3. | S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. |

4. | C. Jin, S. Han, X. Meng, B. Cheng, and D. Zhang, “Demultiplexer using directly resonant tunneling between point defects and waveguides in a photonic crystal,” J. Appl. Phys. |

5. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. |

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

7. | B. E. Nelson, M. Gerken, D. A. B. Miller, R. Piestun, C.-C. Lin, and J. S. Harris, “Use of a dielectric stack as a one-dimensional photonic crystal for wavelength demultiplexing by beam shifting,” Opt. Lett. |

8. | M. Koshiba, “Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,” J. Lightwave Technol. |

9. | A. Sharkawy, S. Shi, and D. W. Prather, “Multichannel wavelength division multiplexing with photonic crystals,” Appl. Opt. |

10. | E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, “Investigation of localized coupled-cavity modes in two-dimensional photonic band gap structures,” IEEE J. Quantum Electron. |

11. | S.Y. Lin and J.G. Fleming, “A three-dimensional Optical Photonic Crystal,” J.Lightwave Technol. , |

12. | R. Kashyap, |

13. | G.I. Stegeman and D.G. Hall, “Modulated index structure,” J.Opt. Soc. Am. A , |

14. | C.R. Giles, “Lightwave applications of fiber Bragg gratings,” J.Lightwave Technol. , |

15. | R.W. Ziolkowski and T. Liang, “Design and characterization of a grating-assisted coupler enhanced by a photonic-band-gap structure for effective wavelength-division demultiplexing,” Optics Letters , |

16. | C.F. Lam, R.B. Vrijen, P.P.L. Chang-Chien, D.F. Sievenpiper, and E. Yablonovitch, “A tunable wavelength demultiplexer using logarithmic filter chains,” J. Lightwave Technol. |

17. | R. Zengerle and O. Leminger, “Phase shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. , |

18. | L. Wei and J.W.Y. Lit, “Phase-shifted Bragg Grating Filters with symmetrical structures,” J. Lightwave Technol. , |

19. | A. D’Orazio, M. De Sario, V. Petruzzelli, and F. Prudenzano: “Bidirectional Beam Propagation Method based on the Method of Lines for the Analysis of Photonic Band Gap Structures,” accepted for publication in Optical and Quantum Electronics, 2002. |

20. | S. Helfert and R. Pregla, “Efficient analysis of periodic structures,” J. Lightwave Technol. , |

21. | G. Murtuza and J. M. Senior, “Analytical tools for the assessment of optical crosstalk in WDM systems,” IEE Colloquium on, Digest |

**OCIS Codes**

(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers

(060.4230) Fiber optics and optical communications : Multiplexing

(230.7400) Optical devices : Waveguides, slab

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 19, 2002

Revised Manuscript: January 22, 2003

Published: February 10, 2003

**Citation**

A. D'Orazio, M. De Sario, Vincenzo Petruzzelli, and F. Prudenzano, "Photonic band gap filter for wavelength division multiplexer," Opt. Express **11**, 230-239 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-3-230

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### References

- J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals. Molding the Flow of Light (Princeton University Press, 1995).
- A.D??Orazio, M.De Sario, V.Petruzzelli, F.Prudenzano: ??Numerical modeling of photonic band gap waveguiding structures,?? Recent Research Developments in Optics, S.G.Pandalai Editor, 2002.
- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, ??Channel drop tunneling through localized states,?? Phys. Rev. Lett. 80, 960??963 (1998). [CrossRef]
- C. Jin, S. Han, X. Meng, B. Cheng, and D.Zhang, ??Demultiplexer using directly resonant tunneling between point defects and waveguides in a photonic crystal,?? J. Appl. Phys. 91, 4771??4773 (2002 [CrossRef]
- H.Kosaka,T.Kawashima, A.Tomita, M.Notomi, T.Tamamura, T.Sato and S.Kawakami, ??Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,?? Appl. Phys. Lett. 74, 1370-1372 (1999). [CrossRef]
- 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]
- B. E. Nelson, M. Gerken, D. A. B. Miller, R. Piestun, C.-C. Lin, and J. S. Harris, ??Use of a dielectric stack as a one-dimensional photonic crystal for wavelength demultiplexing by beam shifting,?? Opt. Lett. 25, 1502??1504 (2000). [CrossRef]
- M. Koshiba, ??Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,?? J. Lightwave Technol. 19, 1970??1975 (2001). [CrossRef]
- A. Sharkawy, S. Shi, and D. W. Prather, ??Multichannel wavelength division multiplexing with photonic crystals,?? Appl. Opt. 40, 2247??2252 (2001). [CrossRef]
- E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, ??Investigation of localized coupled-cavity modes in twodimensional photonic band gap structures,?? IEEE J. Quantum Electron. 38, 837??843 (2002). [CrossRef]
- S.Y. Lin, J.G. Fleming, ??A three-dimensional Optical Photonic Crystal,?? J. Lightwave Technol. 17, 1944-1947 (1999). [CrossRef]
- R.Kashyap, Fiber Bragg Gratings, (Academic press, San Diego, 1999).
- G.I. Stegeman, D.G. Hall, ??Modulated index structure,?? J. Opt. Soc. Am. A 7, 1387-1398 (1996). [CrossRef]
- C.R. Giles, ??Lightwave applications of fiber Bragg gratings,?? J. Lightwave Technol. 15, 1391-1404 (1997). [CrossRef]
- R.W.Ziolkowski, T.Liang, ??Design and characterization of a grating-assisted coupler enhanced by a photonic-band-gap structure for effective wavelength-division demultiplexing,?? Opt. Lett. 22, 1033-1035 (1997). [CrossRef] [PubMed]
- C.F. Lam, R.B. Vrijen, P.P.L. Chang-Chien, D.F. Sievenpiper, E. Yablonovitch, ??A tunable wavelength demultiplexer using logarithmic filter chains,?? J. Lightwave Technol. 16, 1657-1662 (1998). [CrossRef]
- R.Zengerle, O.Leminger, ??Phase shifted Bragg-grating filters with improved transmission characteristics,?? J. Lightwave Technol. 13, 2354-2358 (1995). [CrossRef]
- L.Wei, J.W.Y.Lit, ??Phase-shifted Bragg Grating Filters with symmetrical structures,?? J. Lightwave Technol. 15, 1405-1410 (1997). [CrossRef]
- A. D'Orazio, M. De Sario, V. Petruzzelli, F. Prudenzano: Bidirectional Beam Propagation Method based on the Method of Lines for the Analysis of Photonic Band Gap Structures, accepted for publication in Optical and Quantum Electronics, 2002.
- S.Helfert, R.Pregla, ??Efficient analysis of periodic structures,?? J. Lightwave Technol. 16, 1694-1702 (1998). [CrossRef]
- G. Murtuza, J. M. Senior, ??Analytical tools for the assessment of optical crosstalk in WDM systems,?? IEE Colloquium on, Digest 1997/036, 16/1-16/4, (1997).

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