## Periodic dielectric waveguide beam splitter based on co-directional coupling

Optics Express, Vol. 15, Issue 8, pp. 4536-4545 (2007)

http://dx.doi.org/10.1364/OE.15.004536

Acrobat PDF (484 KB)

### Abstract

A compact beam splitter consisting of three branches of periodic dielectric waveguides (PDW) is designed and analyzed theoretically. Both the symmetrical and asymmetrical configurations of the beam splitter are studied. The band structure for the guided modes is calculated by using finite-difference time-domain (FDTD) method with Bloch-type boundary conditions applying in an appropriate supercell. The field patterns for the whole structure and the transmissions for the output ports are calculated using the multiple scattering method. By utilizing the co-directional coupling mechanism, the light injected into the input branch can be efficiently transferred into the two output branches if the phase matching conditions are satisfied. The coupling length is short and the broad-band requirement can be achieved. Bending loss is small and high transmission (above 95 %) can be preserved for arbitrarily bent PDW if the bend radius of each bend exceeds five wavelengths. This feature indicates that the periodic dielectric waveguide beam splitter (PDWBS) is a high efficiency device for power redistribution while avoiding the lattice orientation restriction of the photonic crystal waveguides (PCW).

© 2007 Optical Society of America

## 1. Introduction

3. A. Mekis, J. C. Chen, I. Kurland, S.-H. 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.-H. Fan, S. G. Johnson, J. D. Joannopoulos, G. Manoatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B **18**, 162–165 (2001). [CrossRef]

5. S. Boscolo, M. Midrio, and T. F. Krauss, “Y junctions in photonic crystal channel waveguides: high transmission and impedance matching,” Opt. Lett. **27**, 1001–1003 (2002). [CrossRef]

6. C. C. Chen, H. T. Chien, and P. G. Luan, “Photonic crystal beam splitter,” Appl. Opt. **43**, 6187, (2004). [CrossRef] [PubMed]

7. S.-Y. Shi, A. Sharkawy, G.-H. Chen, D. M. Pustai, and D. W. Prather “Dispersion-based beam splitter in photonic crystal,” Opt. Lett. **29**, 617–619 (2004). [CrossRef] [PubMed]

8. X.-F. Yu and S.-H. Fan, “Bends and splitters for self-collimated beams in photonic crystal,” Appl. Phys. Lett. **83**, 3251–3253 (2003). [CrossRef]

9. S. G. Lee, S. S. Oh, J. E. Kim, H. Y. Parka, and C. S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. **87**, 181106 (2005). [CrossRef]

10. P. Pottier, S. Mastroiacovo, and R. M. D. L. Rue, “Power and polarization beam-splitters, mirrors, and integrated interferometers based on air-hole photonic crystals and lateral large index-contrast waveguides,” Opt. Express **14**, 5617–5633 (2006). [CrossRef] [PubMed]

13. P. G. Luan and K. D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Express **14**, 3263–3272 (2006). [CrossRef] [PubMed]

13. P. G. Luan and K. D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Express **14**, 3263–3272 (2006). [CrossRef] [PubMed]

## 2. Review of the previous work on PDW

13. P. G. Luan and K. D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Express **14**, 3263–3272 (2006). [CrossRef] [PubMed]

*ε*= 11.56 (GaAs), and the radius of the cylinders is

*r*= 0.2

*a*,

*a*is the lattice constant. Only TM modes (with

**E**field parallel to the cylinders) are considered. The first band is below the light line, which indicates that the first-band modes are indeed guided modes. Transmission for finite S-shaped bent PDW are obtained using multiple scattering method [15

15. Y. Y. Chen and Z. Ye, “Acoustic Attenuation by Two-Dimensional Arrays of Rigid Cylinders,” Phys. Rev. Lett. **87**, 1843011–4 (2001). [CrossRef]

*a*/λ < 0.25) high transmission (> 90%) can be achieved at any bend angle (from 0

*°*to 90

*°*) for any bend radius larger than 11.5

*a*(about three wavelengths). This feature indicates that PDW has the advantage that it can be bent to any arbitrary shape while still preserves its high transmission property, avoiding the lattice orientation restriction of PCW devices. In this work, we adopt this advantage and use the directional coupling effect to design a PDWBS.

## 3. Dispersion relations of the modes in the coupling section

*ε*= 11.56 and the radius

*r*= 0.2

*a*of the cylinders are chosen as the same as that used in [13

**14**, 3263–3272 (2006). [CrossRef] [PubMed]

*L*= 1.3

_{y}*a*, the corresponding band structures (dispersion relations) are shown in Fig. 1(a2), (b2). The number of low-frequency guiding bands is equal to the number of rows. In addition, for a definite frequency, different modes correspond to different

*k*vectors and different field patterns (See the insets of Fig. 1(a2), (b2)).

*k*, say,

*k*

_{1}and

*k*

_{2}. Assuming that

*k*

_{1}>

*k*

_{2}, then

*k*

_{1}represents the first band mode and

*k*

_{2}the second band mode. The energy of the

*k*

_{1}mode is concentrated in the central row, whereas the energy of the

*k*

_{2}mode is mostly distributed in the two side-rows. Along the propagation direction, the two modes interfere with each other via establishing the phase difference, thus the field pattern changes its manifestation from one to another in a propagation distance (coupling length)

*k*

_{1}mode is almost the same as the original single-row mode, the length of the coupling section of the PDWBS for splitting one beam to two can be approximately set as the coupling length

*L*. This is the central idea for designing the PDWBS based on the directional-coupling mechanism. Here we have utilized only the simplest concept of mode interference. More thorough treatment of coupled waves can be found in [14].

_{c}*L*and recalculate the dispersion relations of the modes, and then derive the corresponding coupling lengths according to Eq.(1). For

_{y}*L*= 1.0

_{y}*a*, 1.2

*a*, and 1.4

*a*, the results are shown in Fig. 2(a) and (b). As one can see, when we reduce the width of the coupling section (the three rows region), the coupling length is also reduced. For a given width

*L*the coupling length for structure A is a monotonically decreasing function of frequency. Besides, there is a cut-off frequency. For example, when

_{y}*L*= 1.2

_{y}*a*, the cut-off frequency is

*a*/λ = 0.23. This frequency is the top frequency of the first band. In fact, it is impossible to couple two modes, one in the 1st-band and one in the 2nd-band, above this frequency. This explains why there is the cut-off frequency. On the other hand, the coupling length for structure B is almost a constant in a wide range of frequency from 0.2 to 0.24, insensitive to the frequency variation. This feature indicates that structure B is the better choice to be used as the coupling section of a wideband PDWBS. In the following simulations we choose structure B of length 8

*a*as the coupling section of the PDWBS, and the width

*L*is assumed to be 1.2

_{y}*a*.

## 4. System description and numerical results

*L*= 8

_{c}*a*and width

*L*= 1.2

_{y}*a*. An S-shaped arm (excluding the output straight PDW) contains 61 rods, forming the two oppositely curved bends and one straight PDW in between. Hereafter we denote the two arms as Arm

*and Arm*

_{a}*. For any one of the two arms, say, Arm*

_{b}*, the two bends (arcs) inside are identical, and can be characterized by the bending angle*

_{a}*θ*. The bending angles

_{a}*θ*for the two arms take arbitrary values from 0° to 90°, and the bend radius

_{a,b}*R*is chosen to be 19.1

*a*. In a bend, one lattice spacing a contributes 3° to the bending angle. Thus for a 90° bent S-shaped arm the straight PDW between the two bends disappears. In order to calculate the transmissions

*T*, we insert three planes of width 6

_{a,b}*a*in the input and output PDWs at the positions indicated by Fig. 3(b),(c) and evaluate the energy fluxes

*P*, and

_{i}*P*. The transmissions through the two output PDWs that connected with Arm

_{a,b}*are then defined as*

_{a,b}*T*=

_{a,b}*P*/

_{a,b}*P*.

_{i}*θ*on the transmission spectra

_{a,b}*T*(ω). We first study the symmetric case of

_{a,b}*θ*=

_{a}*θ*, here we define

_{b}*θ*=

_{sp}*θ*as the splitting angle. It is obvious that in this case we must have

_{a}*T*=

_{a}*T*. In order to reduce the interference effect between the two arms, we only consider the situations that

_{b}*θ*> 9°. The transmission spectrum

_{sp}*T*(ω) for different values of

_{a,b}*θ*is shown in Fig. 4. The simulation result reveals that on average for one arm the transmission is higher than 0.425, that is, the averaged total transmission is higher than 0.85, if 9° <

_{sp}*θ*< 90° are considered. The total transmission is high enough so this symmetric structure can indeed be used as a beam splitter. Furthermore, high splitting ability (> 0.46) can be achieved around

_{sp}*a*/λ = 0.219. It seems that power transfer can be accomplished more efficiently in the central part of the frequency range we considered.

*θ*and study the influence of the bending angle

_{a}*θ*on the transmission spectra

_{b}*T*(ω). Two simplest situations are considered. In the first situation we set

_{a,b}*θ*= 0° and vary

_{a}*θ*from 9° to 90°, and the calculated

_{b}*T*(ω) and

_{a}*T*(ω) are shown in Fig. 5(a1) and (b1), respectively. Similarly, in the second situation we set

_{b}*θ*= 90°, and

_{a}*T*(ω) are shown in Fig. 5(a2) and (b2). From the results of Fig. 5(a1) and (b1), we find that

_{a,b}*T*is in general larger than

_{a}*T*, and on average we have

_{b}*T*:

_{a}*T*≈ 0.63 : 0.37, implying that the internal reflection of wave in the straight arm is much smaller than in the bent arm. Moreover, the typical value of total transmission is larger than 95%, exceeding that of the symmetric structure (see Fig.4). For case of

_{b}*θ*= 90°, the results are very different, as shown in Fig. 5(a2) and (b2). In this situation, light powers are more equally distributed between the two arms. The average splitting ratio

_{a}*T*:

_{a}*T*is 0.44 : 0.56. It indicates that the asymmetric configuration of three-branch structure can indeed be used as a practical beam splitter.

_{b}*θ*and

_{a}*θ*on the transmissions

_{b}*T*. We choose

_{a,b}*a*/

*λ*= 0.219 and plot the calculated

*T*in Fig. 6. The horizontal and vertical axes are

_{b}*θ*and

_{a}*θ*, respectively. By symmetry consideration, the result for

_{b}*T*can be easily obtained by the reflection mapping with respect to the 45°-oriented line (the line described by the equation

_{a}*θ*=

_{a}*θ*). According to the numerical results, the ratio

_{a}*θ*:

_{a}*θ*on average is 0.48 : 0.52. In addition, for the major portion of the transmission map shown in Fig. 6,

_{b}*T*and

_{a}*T*are almost equal. This indicates that asymmetric PDWBS can work well in most situations.

_{b}*a*/λ = 0.219 in Fig.7 (a) and (b). In Fig. 7(a), the splitting angle for the symmetric PDWBS is taken to be

*θ*= 60°. In Fig. 7(b), we set

_{sp}*θ*= 75° and

_{a}*θ*= 39° for the asymmetric PDWBS. The field patterns confirm again the beam splitting abilities of the two kinds of structures. We believe both the symmetric and asymmetric structures of PDWBS can be used as practical devices in the optical integrating circuits.

_{b}16. Y. Zhang and B.-J. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express **14**, 5723–5732 (2006). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

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

2. | K. Sakoda, |

3. | A. Mekis, J. C. Chen, I. Kurland, S.-H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. |

4. | S.-H. Fan, S. G. Johnson, J. D. Joannopoulos, G. Manoatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B |

5. | S. Boscolo, M. Midrio, and T. F. Krauss, “Y junctions in photonic crystal channel waveguides: high transmission and impedance matching,” Opt. Lett. |

6. | C. C. Chen, H. T. Chien, and P. G. Luan, “Photonic crystal beam splitter,” Appl. Opt. |

7. | S.-Y. Shi, A. Sharkawy, G.-H. Chen, D. M. Pustai, and D. W. Prather “Dispersion-based beam splitter in photonic crystal,” Opt. Lett. |

8. | X.-F. Yu and S.-H. Fan, “Bends and splitters for self-collimated beams in photonic crystal,” Appl. Phys. Lett. |

9. | S. G. Lee, S. S. Oh, J. E. Kim, H. Y. Parka, and C. S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. |

10. | P. Pottier, S. Mastroiacovo, and R. M. D. L. Rue, “Power and polarization beam-splitters, mirrors, and integrated interferometers based on air-hole photonic crystals and lateral large index-contrast waveguides,” Opt. Express |

11. | T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Multimode Interference-Based Photonic Crystal Waveguide Power Splitter,” J. Lightwave Technol. |

12. | N. Yamamoto, T. Ogawa, and K. Komori, “Photonic crystal directional coupler switch with small switching length and bandwidth,” Opt. Express |

13. | P. G. Luan and K. D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Express |

14. | A. Yariv and P. Yeh, |

15. | Y. Y. Chen and Z. Ye, “Acoustic Attenuation by Two-Dimensional Arrays of Rigid Cylinders,” Phys. Rev. Lett. |

16. | Y. Zhang and B.-J. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: February 8, 2007

Revised Manuscript: March 22, 2007

Manuscript Accepted: March 22, 2007

Published: April 3, 2007

**Citation**

Pi-Gang Luan and Kar-Der Chang, "Periodic dielectric waveguide beam splitter based on co-directional coupling," Opt. Express **15**, 4536-4545 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4536

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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).
- K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).
- A. Mekis, J. C. Chen, I. Kurland, S.-H. 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]
- S.-H. Fan, S. G. Johnson, J. D. Joannopoulos, G. Manoatou, and H. A. Haus, "Waveguide branches in photonic crystals," J. Opt. Soc. Am. B 18, 162-165 (2001). [CrossRef]
- S. Boscolo, M. Midrio, and T. F. Krauss, "Y junctions in photonic crystal channel waveguides: high transmission and impedance matching," Opt. Lett. 27, 1001-1003 (2002). [CrossRef]
- C. C. Chen, H. T. Chien, and P. G. Luan, "Photonic crystal beam splitter," Appl. Opt. 43, 6187, (2004). [CrossRef] [PubMed]
- S.-Y. Shi, A. Sharkawy, G.-H. Chen, D.M. Pustai, and D.W. Prather "Dispersion-based beam splitter in photonic crystal," Opt. Lett. 29, 617-619 (2004). [CrossRef] [PubMed]
- X.-F. Yu, and S.-H. Fan, "Bends and splitters for self-collimated beams in photonic crystal," Appl. Phys. Lett. 83, 3251-3253 (2003). [CrossRef]
- S. G. Lee, S. S. Oh, J. E. Kim, H. Y. Parka, and C. S. Kee, "Line-defect-induced bending and splitting of selfcollimated beams in two-dimensional photonic crystals," Appl. Phys. Lett. 87, 181106 (2005). [CrossRef]
- P. Pottier, S. Mastroiacovo, and R. M. D. L. Rue, "Power and polarization beam-splitters, mirrors, and integrated interferometers based on air-hole photonic crystals and lateral large index-contrast waveguides," Opt. Express 14, 5617-5633 (2006). [CrossRef] [PubMed]
- T. Liu, A. R. Zakharian,M. Fallahi, J. V. Moloney, andM.Mansuripur, "Multimode Interference-Based Photonic Crystal Waveguide Power Splitter," J. Lightwave Technol. 22, 2842-2846 (2004). [CrossRef]
- N. Yamamoto, T. Ogawa, and K. Komori, "Photonic crystal directional coupler switch with small switching length and bandwidth," Opt. Express 14, 1223-1229 (2006). [CrossRef] [PubMed]
- P. G. Luan, and K. D. Chang, "Transmission characteristics of finite periodic dielectric waveguides," Opt. Express 14, 3263-3272 (2006). [CrossRef] [PubMed]
- A. Yariv, P. Yeh, Optical Waves in Crystals (John Wiley & Sons, New York, 1984).
- Y. Y. Chen and Z. Ye, "Acoustic Attenuation by Two-Dimensional Arrays of Rigid Cylinders," Phys. Rev. Lett. 87, 1843011-4 (2001). [CrossRef]
- Y. Zhang and B.-J. Li, "Photonic crystal-based bending waveguides for optical interconnections," Opt. Express 14, 5723-5732 (2006). [CrossRef] [PubMed]

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