## Self-imaging in periodic dielectric waveguides

Optics Express, Vol. 17, Issue 1, pp. 365-378 (2009)

http://dx.doi.org/10.1364/OE.17.000365

Acrobat PDF (1405 KB)

### Abstract

Self-imaging phenomena in periodic dielectric waveguides has been predicted and investigated based on multimode interference effect by using the plane wave expansion method and the finite-difference time-domain method. Asymmetric and symmetric interferences were discussed and respective imaging positions were calculated. As examples of application, a demultiplexer and a filter with ultracompact and simple structures were designed and demonstrated theoretically for optical communication wavelengths.

© 2009 Optical Society of America

## 1. Introduction

*et al*. [1

1. R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. **27**, 337–339 (1975). [CrossRef]

2. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. **13**, 615–627 (1995). [CrossRef]

3. J. B. Xiao, X. Liu, and X. H. Sun, “Design of an ultracompact MMI wavelength demultiplexer in slot waveguide structures,” Opt. Express **15**, 8300–8308 (2007). [CrossRef] [PubMed]

4. J. K. Hong and S. S. Lee, “1 × 2 Wavelength multiplexer with high transmittances using extraneous self-imaging phenomenon,” J. Lightwave Technol. **25**, 1264–1268 (2007). [CrossRef]

5. David. J. Y. Feng and T. S. Lay, “Compact multimode interference couplers with arbitrary power splitting ratio,” Opt. Express **16**, 7175–7180 (2008). [CrossRef] [PubMed]

6. X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, “Compact variable optical attenuator based on multimode interference coupler,” IEEE Photon. Technol. Lett. **17**, 2361–2363 (2005). [CrossRef]

7. S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multimode interference photonic switches,” J. Lightwave Technol. **20**, 675–681 (2002). [CrossRef]

8. F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. **18**, 421–423 (2006). [CrossRef]

4. J. K. Hong and S. S. Lee, “1 × 2 Wavelength multiplexer with high transmittances using extraneous self-imaging phenomenon,” J. Lightwave Technol. **25**, 1264–1268 (2007). [CrossRef]

6. X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, “Compact variable optical attenuator based on multimode interference coupler,” IEEE Photon. Technol. Lett. **17**, 2361–2363 (2005). [CrossRef]

8. F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. **18**, 421–423 (2006). [CrossRef]

9. H. J. Kim, I. Park, B. H. O, S. G. Park, E. H. Lee, and S. G. Lee, “Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing,” Opt. Express **12**, 5625–5633 (2004). [CrossRef] [PubMed]

9. H. J. Kim, I. Park, B. H. O, S. G. Park, E. H. Lee, and S. G. Lee, “Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing,” Opt. Express **12**, 5625–5633 (2004). [CrossRef] [PubMed]

10. Y. Zhang, Z. Li, and B. Li, “Multimode interference effect and self-imaging principle in two-dimensional silicon photonic crystal waveguides for terahertz waves,” Opt. Express **14**, 2679–2689 (2006). [CrossRef] [PubMed]

12. L. W. Chung and S. L. Lee, “Multimode-interference-based broad-band demultiplexers with internal photonic crystals,” Opt. Express **14**, 4923–4927 (2006). [CrossRef] [PubMed]

13. Z. Li, Y. Zhang, and B. Li, “Terahertz photonic crystal switch in silicon based on self-imaging principle,” Opt. Express **14**, 3887–3892 (2006) [CrossRef] [PubMed]

9. H. J. Kim, I. Park, B. H. O, S. G. Park, E. H. Lee, and S. G. Lee, “Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing,” Opt. Express **12**, 5625–5633 (2004). [CrossRef] [PubMed]

12. L. W. Chung and S. L. Lee, “Multimode-interference-based broad-band demultiplexers with internal photonic crystals,” Opt. Express **14**, 4923–4927 (2006). [CrossRef] [PubMed]

14. S. Fan, N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J.D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B **12**, 1267–1272 (1995). [CrossRef]

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

16. P. G. Luan and K. D Chang, “Periodic dielectric waveguide beam splitter based on co-directional coupling,” Opt. Express **15**, 4536–4545 (2007). [CrossRef] [PubMed]

17. D. S. Gao, R. Hao, and Z. P. Zhou, “Mach-Zehnder interferometer based on coupled dielectric pillars,” Chin. Phys. Lett. **24**, 3172–3174 (2007). [CrossRef]

18. W. Huang, Y. Zhang, and B. Li, “Ultracompact wavelength and polarization splitters in periodic dielectric waveguides,” Opt. Express **16**, 1600–1609 (2008). [CrossRef] [PubMed]

19. Y. Zhang, W. Huang, and B. Li, “Fabry-Pérot microcavities with controllable resonant wavelengths in periodic dielectric waveguides,” Appl. Phys. Lett. **93**, 0311101–3 (2008). [CrossRef]

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

21. A. Lavrinenko, P. I. Borel, L. H. Frandsen, M. Thorhauge, A. HarpϘth, M. Kristensen, T. Niemi, and H. M. H. Chong, “Comprehensive FDTD modelling of photonic crystal waveguide components,” Opt. Express **12**, 234–248 (2004). [CrossRef] [PubMed]

## 2. Guided modes in PDWs

*r*is the radius of the PD rods and

*a*is the center-to-center distance between two adjacent PD rods. The refractive index of the PD rods is

*n*= 3.45 and the radius

*r*= 0.46

*a*. Its band structure for TM mode (E-polarization) was calculated by the plane wave expansion method and was depicted in Fig. 1(c). The inset denotes the supercell with a size of

*a*×9

*a*, which was used for calculation. The shaded region is for extended modes, which is not suitable for light guiding. The orange is light line. The solid curve below the light line is guided mode. It can be seen that, in the single row PDW, there is only one guided mode (single mode) at a frequency range of 0.132(

*a*/

*λ*) to 0.156(

*a*/

*λ*). Figure 1(b) shows four parallel PD rows in air [22

22. D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, “Nanopillars photonic crystal waveguides,” Opt. Express **12**, 617–622 (2004). [CrossRef] [PubMed]

23. D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, “Numerical characterization of nanopillar photonic crystal waveguides and directional couplers,” Opt. Quant. Electron. **37**, 331–341 (2005). [CrossRef]

*a*. Fig. 1(d) shows its band structure for TM mode, which was calculated by using the supercell with a size of

*a*×13

*a*(inset). From Fig. 1(d), we can see that, there are three and four guided modes at frequencies of 0.132(

*a*/

*λ*) and 0.156(

*a*/

*λ*), respectively. Therefore, the four parallel PD rows are multimode PDW.

*n*= 3.45) from 0.46

*a*to 0.45

*a*and did the calculations. Figure 2(a) shows the single row PDW with

*r*= 0.45

*a*and Fig. 2(c) shows the calculated band structure for TM mode. Figure 2(b) shows the multimode PDW with five parallel rows of PD rods (

*r*= 0.45

*a*) in air. The row-to-row space between two adjacent rows was set to be

*d*= 1.5

*a*. The calculated band structure was plotted in Fig. 2(d). For the multimode PDW formed by the row-to-row space of

*d*= 1.5

*a*, there are five guided modes (0th to 4th) at a frequency range of 0.119(

*a*/

*λ*) to 0.140(

*a*/

*λ*). The insets of Figs. 2(c) and 2(d) represent supercells with size of

*a*×9

*a*and

*a*×16

*a*, respectively, for calculations.

## 3. Analysis of self-imaging phenomena

*a*and 3

*a*, respectively. For this structure, when an input optical field Ψ(0,

*y*) is introduced into the multimode region through the input waveguide, a mirrored image at

*x*=

*L*and a direct image at

_{m}*x*=

*L*will be reproduced, as shown in Fig. 3(b).

_{d}*x*,

*y*) in the multimode region can be found in Ref. [9

**12**, 5625–5633 (2004). [CrossRef] [PubMed]

*x*=

*L*and the direct image at

_{m}*x*=

*L*can be expressed as

_{d}*β*is the propagation constant. Therefore, the positions of the mirrored image and the direct image can be obtained by the Eqs. (1) and (2) if appropriate positive integers for each

_{n}*k*are decided.

_{n}*a*/

*λ*) and 0.156(

*a*/

*λ*) in the model-I of Fig. 3(a). From the distributions, asymmetric multimode interference effect and self-imaging phenomena are clearly observed in the multimode region. Especially in the Poynting vector distribution [Fig. 4(b)], it can be seen that, for the operating frequency of 0.132(

*a*/

*λ*), two mirrored images were reproduced at positions A

_{1}and A

_{2}, and two direct images were reproduced at positions B

_{1}and B

_{2}. From Fig. 4(d), we can see that, for the operating frequency of 0.156(

*a*/

*λ*), there are four mirrored images at positions A

_{1}, A

_{2}, A

_{3}and A

_{4}, and two direct images at positions B

_{1}and B

_{2}. The mirrored image reproduced at the position A

_{4}is the clearest one. By comparing Figs. 4(b) and 4(d), at the same position

*x*= 50

*a*along the propagation direction, the clearest direct image at B

_{2}for the frequency of 0.132(

*a*/

*λ*) and the clearest mirrored image at A

_{4}for the frequency of 0.156(

*a*/

*λ*) were reproduced, simultaneously. For potential application, if we choose

*x*= 50

*a*as the length of the multimode region, the structure can be used as a wavelength demultiplexer for the frequencies of 0.132(

*a*/

*λ*) and 0.156(

*a*/

*λ*).

*a*/

*λ*) and 0.156(

*a*/

*λ*) were excited by the input optical fields, therefore, they all contributed to the self-imaging. In calculation, the values of the wave vectors for each modes at frequencies of 0.132(

*a*/

*λ*) and 0.156(

*a*/

*λ*) were taken out from the guided mode curves in Fig. 1(d). The values of the propagation constant

*β*were derived from the values of the wave vectors, accordingly. The average values of

_{n}*L*and

_{d}*L*were calculated by utilizing the self-imaging conditions derived from propagation analysis for the guided modes. In general, there are no exact solutions for Eqs. (1) and (2), so an approximate calculation was considered as follows: first, we found the nearest positive integers for each

_{m}*k*, then calculated the

_{n}*L*and

_{d}*L*for each

_{m}*n*by substituting the values of

*k*and

_{n}*β*into the Eqs. (1) and (2), accordingly. Last, the average values of

_{n}*L*and

_{d}*L*were obtained. The parameters and the calculated results are summarized in Tables 1 and 2. It is easy to see that, for the model-I, a direct image for 0.132(

_{m}*a*/

*λ*) and a mirrored image for 0.156(

*a*/

*λ*) were reproduced at the same position of

*x*= 50

*a*along the propagation direction. Along

*y*axis direction, the direct image is at

*y*= 1.5

*a*and the mirrored image is at

*y*= -1.5

*a*. This is the reason, if the length of the multimode region for the model-I is set to be 50

*a*, a 1 -to-2 wavelength demultiplexer can be achieved.

*a*, and the width is 6

*a*. From the band structure showed in Fig. 2(d), there are five modes (two odd modes and three even modes) at each of frequencies of 0.119(

*a*/

*λ*) and 0.140(

*a*/

*λ*). Therefore, for the model-II, when an input optical field Ψ(0,0) is introduced into the multimode region through the input waveguide, symmetric interference phenomena will be occurred. As a result, three kinds of images will be reproduced, i.e. single image (mirrored image or direct image), two-fold images and three-fold images. For simplicity, only imaging positions of single and two-fold images were depicted in Fig. 5(b) schematically. The single image (mirrored or direct image) was assumed to be reproduced at

*x*= L

_{s}and the two-fold images were reproduced at

*x*=

*L*.

_{f}10. Y. Zhang, Z. Li, and B. Li, “Multimode interference effect and self-imaging principle in two-dimensional silicon photonic crystal waveguides for terahertz waves,” Opt. Express **14**, 2679–2689 (2006). [CrossRef] [PubMed]

*x*=

*L*can be expressed as

_{s}*a*/

*λ*) and 0.140(

*a*/

*λ*) were launched into the input waveguide individually, the simulated steady-state electric field distributions and Poynting vector (x-component) distributions in the model-II are shown in Fig. 6. The symmetric interference effect and self-imaging phenomena are obviously observed in the multimode region. In the Poynting vector distribution [Fig. 6(b)], for the operating frequency of 0.119(

*a*/

*λ*), there are six two-fold images reproduced at the positions A

_{1}, A

_{2}, A

_{3}, A

_{4}, A

_{5}and A

_{6}, four three-fold images reproduced at the positions B

_{1}, B

_{2}, B

_{3}and B

_{4}, and two single images reproduced at the positions C

_{1}and C

_{2}. From the Poynting vector distribution [Fig. 6(d)] we can see that, for the operating frequency of 0.140(

*a*/

*λ*), four two-fold images reproduced at the positions A

_{1}, A

_{2}, A

_{3}, and A

_{4}, three three-fold images reproduced at the positions B

_{1}, B

_{2}, and B

_{3}, and only one single image reproduced at the position C

_{1}. From Figs. 6(b) and 6(d), we further found that, the positions of all three-fold images are between the positions of the two-fold images.

*a*/

*λ*), the first single image (mirrored image) was reproduced at

*x*=

*L*

_{s1}= 23

*a*(C

_{1}) and the second single image (direct image) was reproduced at

*x*=

*L*

_{s2}= 46

*a*(C

_{2}). From the positions of the two-fold images A

_{2}(

*x*=

*L*

_{f1}= 11

*a*) and A

_{5}(

*x*=

*L*

_{f2}= 34

*a*), we got that the interval between the two-fold images is 23

*a*. From Figs. 6(c) and 6(d), it can be seen that, for the operating frequency of 0.140(

*a*/

*λ*), a single image (mirrored image) was reproduced at

*x*=

*L*

_{s1}= 35

*a*(C

_{1}) and a twofold image was reproduced at

*x*=

*L*

_{f1}= 17

*a*(A

_{2}). We predict that more self-images can be observed if the length of the multimode region is sufficient long. As approximate descriptions, the following two formulas were used to express the inherent relation of

*L*

_{s1}and

*L*

*[10*

_{fk}10. Y. Zhang, Z. Li, and B. Li, “Multimode interference effect and self-imaging principle in two-dimensional silicon photonic crystal waveguides for terahertz waves,” Opt. Express **14**, 2679–2689 (2006). [CrossRef] [PubMed]

*L*

_{s1}is the imaging position of the first single image, and

*L*is the position of the twofold images.

_{fk}*L*

_{s1}can be calculated by the self-imaging conditions. In calculation, the values of the wave vectors for each modes at frequencies of 0.119(

*a*/

*λ*) and 0.140(

*a*/

*λ*) were taken out from the guided mode curves of Fig. 2(d). The values of the propagation constants

*β*were derived from the values of the wave vectors, accordingly. First, we tried to find the nearest positive integers for each

_{n}*k*, then calculated

_{n}*L*

_{s1}for each

*n*by substituting the values of

*k*and

_{n}*β*into the Eq. (3), accordingly. Last, the average values of

_{n}*L*

_{s1}were obtained. Tables 3 and 4 list the parameters and calculated results of

*L*

_{s1}for the frequencies of 0.119(

*a*/

*λ*) and 0.140(

*a*/

*λ*), respectively. By substituting the average value

*L*

_{s1}= 22.9049

*a*(Table 3) into the Eqs. (4) and (5) for the frequency of 0.119(

*a*/

*λ*), we calculated that

*L*

_{f1}= 11.4525

*a*and

*L*

_{f2}= 34.3574

*a*.

*L*

_{f1}= 17.6452

*a*was also obtained by substituting the average value

*L*

_{s1}= 35.2904

*a*(Table 4) into the Eq. (4) for the frequency of 0.140(

*a*/

*λ*). Simulated results agree well with the theoretical results of the imaging positions. Above analysis shows that a two-fold image for 0.119(

*a*/

*λ*) and a single image for 0.140(

*a*/

*λ*) were reproduced at the same position of

*x*= 35

*a*along the propagation direction in the model-II. Therefore, if the length of the multimode region for the model-II is set to be 35

*a*, a filter can be achieved.

## 4. Applications and discussions

*L*= 49

*a*(close to 50

*a*), which is to reproduce a direct image for 0.132(

*a*/

*λ*) and a mirrored image for 0.156(

*a*/

*λ*). To avoid coupling effect between the two outputs, a bending angle of 90° was designed. It should be emphasized that the center-to-center distance between the two adjacent dielectric rods is still

*a*, the radii of the curvature for each bend is

*R*= 0.5

*a*/(sin5°) = 5.74

*a*, which supports low bending loss [15

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

_{1}= 1550 nm [0.132(

*a*/

*λ*)] and

*λ*

_{2}= 1310 nm [0.156(

*a*/

*λ*)] applications, we choose

*a*= 204 nm. So the calculated total length of the multimode region is about 10 μm and the width is about 0.6 μm.

*λ*

_{1}= 1550 nm, the normalized output powers in the outputs 1 and 2 are

*P*

_{11}= 83.6% and

*P*

_{21}= 1.6%, respectively. There is about 15% energy loss (flow into the air) in the propagation. At

*λ*

_{2}= 1310 nm, the normalized output powers in the outputs 1 and 2 are

*P*

_{12}= 1.9% and

*P*

_{22}= 92.5% (5.6% energy loss), respectively. Calculated crosstalks are 10log(

*P*

_{21}/

*P*

_{11}) = -17.2 dB for 1550 nm and 10log(

*P*

_{12}/

*P*

_{22}) = -16.9 dB for 1310 nm. If we change the length of the multimode region a little from 49

*a*to 50

*a*, corresponding normalized optical power spectrums will be a little left shift (dashed lines). It can be seen that the peak power at the output 2 decreased a little while the normalized power at the output 1 decreased to below 80% at 1550 nm. That is the reason why we choose

*L*= 49

*a*as the length of the multimode region for the demultiplexer.

*a*/

*λ*) and a single image for 0.140(

*a*/

*λ*) will be reproduced at

*x*= 35

*a*. To filter the wavelength of

*λ*

_{1}= 1550 nm, the length of the filter was set to be 35

*a*,

*a*is specified as

*a*= 184 nm. So the length of the multimode region is 6.4 μm and the width is 1.1 μm.

*P*

_{1}= 5.3% (most energy was flowed into the air and/or reflected back to the input waveguide) and

*P*

_{2}= 94.0% (only few energy was flowed into the air) for 0.119(

*a*/

*λ*) (

*λ*

_{1}= 1550 nm) and 0.140(

*a*/

*λ*) (

*λ*

_{2}= 1310 nm), respectively. The calculated extinction ratio is 12.5 dB.

4. J. K. Hong and S. S. Lee, “1 × 2 Wavelength multiplexer with high transmittances using extraneous self-imaging phenomenon,” J. Lightwave Technol. **25**, 1264–1268 (2007). [CrossRef]

3. J. B. Xiao, X. Liu, and X. H. Sun, “Design of an ultracompact MMI wavelength demultiplexer in slot waveguide structures,” Opt. Express **15**, 8300–8308 (2007). [CrossRef] [PubMed]

**12**, 5625–5633 (2004). [CrossRef] [PubMed]

**12**, 5625–5633 (2004). [CrossRef] [PubMed]

*a*× 4.3

*a*) for the PCW filter is relatively large [10

**14**, 2679–2689 (2006). [CrossRef] [PubMed]

3. J. B. Xiao, X. Liu, and X. H. Sun, “Design of an ultracompact MMI wavelength demultiplexer in slot waveguide structures,” Opt. Express **15**, 8300–8308 (2007). [CrossRef] [PubMed]

**12**, 5625–5633 (2004). [CrossRef] [PubMed]

**14**, 2679–2689 (2006). [CrossRef] [PubMed]

18. W. Huang, Y. Zhang, and B. Li, “Ultracompact wavelength and polarization splitters in periodic dielectric waveguides,” Opt. Express **16**, 1600–1609 (2008). [CrossRef] [PubMed]

24. J. Smajic, C. Hafner, and D. Erni, “On the design of photonic crystal multiplexers,” Opt. Express **11**, 566–571 (2003). [CrossRef] [PubMed]

25. M. Y. Tekeste and J. M. Yarrison-Rice, “High efficiency photonic crystal based wavelength demultiplexer,” Opt. Express **14**, 7931–7942 (2006). [CrossRef] [PubMed]

26. R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. **15**, 401–403 (2003). [CrossRef]

## 5. Conclusion

## Acknowledgments

## References and links

1. | R. Ulrich and G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. |

2. | L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. |

3. | J. B. Xiao, X. Liu, and X. H. Sun, “Design of an ultracompact MMI wavelength demultiplexer in slot waveguide structures,” Opt. Express |

4. | J. K. Hong and S. S. Lee, “1 × 2 Wavelength multiplexer with high transmittances using extraneous self-imaging phenomenon,” J. Lightwave Technol. |

5. | David. J. Y. Feng and T. S. Lay, “Compact multimode interference couplers with arbitrary power splitting ratio,” Opt. Express |

6. | X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, “Compact variable optical attenuator based on multimode interference coupler,” IEEE Photon. Technol. Lett. |

7. | S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multimode interference photonic switches,” J. Lightwave Technol. |

8. | F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. |

9. | H. J. Kim, I. Park, B. H. O, S. G. Park, E. H. Lee, and S. G. Lee, “Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing,” Opt. Express |

10. | Y. Zhang, Z. Li, and B. Li, “Multimode interference effect and self-imaging principle in two-dimensional silicon photonic crystal waveguides for terahertz waves,” Opt. Express |

11. | D. Modotto, M. Conforti, A. Locatelli, and C. D. Angelis, “Imaging properties of multimode photonic crystal waveguides and waveguide arrays,” J. Lightwave Technol. |

12. | L. W. Chung and S. L. Lee, “Multimode-interference-based broad-band demultiplexers with internal photonic crystals,” Opt. Express |

13. | Z. Li, Y. Zhang, and B. Li, “Terahertz photonic crystal switch in silicon based on self-imaging principle,” Opt. Express |

14. | S. Fan, N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J.D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B |

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

16. | P. G. Luan and K. D Chang, “Periodic dielectric waveguide beam splitter based on co-directional coupling,” Opt. Express |

17. | D. S. Gao, R. Hao, and Z. P. Zhou, “Mach-Zehnder interferometer based on coupled dielectric pillars,” Chin. Phys. Lett. |

18. | W. Huang, Y. Zhang, and B. Li, “Ultracompact wavelength and polarization splitters in periodic dielectric waveguides,” Opt. Express |

19. | Y. Zhang, W. Huang, and B. Li, “Fabry-Pérot microcavities with controllable resonant wavelengths in periodic dielectric waveguides,” Appl. Phys. Lett. |

20. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

21. | A. Lavrinenko, P. I. Borel, L. H. Frandsen, M. Thorhauge, A. HarpϘth, M. Kristensen, T. Niemi, and H. M. H. Chong, “Comprehensive FDTD modelling of photonic crystal waveguide components,” Opt. Express |

22. | D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, “Nanopillars photonic crystal waveguides,” Opt. Express |

23. | D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, “Numerical characterization of nanopillar photonic crystal waveguides and directional couplers,” Opt. Quant. Electron. |

24. | J. Smajic, C. Hafner, and D. Erni, “On the design of photonic crystal multiplexers,” Opt. Express |

25. | M. Y. Tekeste and J. M. Yarrison-Rice, “High efficiency photonic crystal based wavelength demultiplexer,” Opt. Express |

26. | R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: December 1, 2008

Revised Manuscript: December 23, 2008

Manuscript Accepted: December 24, 2008

Published: January 2, 2009

**Citation**

Shunquan Zeng, Yao Zhang, and Baojun Li, "Self-imaging in periodic dielectric waveguides," Opt. Express **17**, 365-378 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-1-365

Sort: Year | Journal | Reset

### References

- R. Ulrich and G. Ankele, "Self-imaging in homogeneous planar optical waveguides," Appl. Phys. Lett. 27, 337-339 (1975). [CrossRef]
- L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995). [CrossRef]
- J. B. Xiao, X. Liu, and X. H. Sun, "Design of an ultracompact MMI wavelength demultiplexer in slot waveguide structures," Opt. Express 15, 8300-8308 (2007). [CrossRef] [PubMed]
- J. K. Hong and S. S. Lee, "1 × 2 Wavelength multiplexer with high transmittances using extraneous self-imaging phenomenon," J. Lightwave Technol. 25, 1264-1268 (2007). [CrossRef]
- D. J. Y. Feng, and T. S. Lay, "Compact multimode interference couplers with arbitrary power splitting ratio," Opt. Express 16, 7175-7180 (2008). [CrossRef] [PubMed]
- X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005). [CrossRef]
- S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, "Multimode interference photonic switches," J. Lightwave Technol. 20, 675-681 (2002). [CrossRef]
- F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, "Optical switch based on multimode interference coupler," IEEE Photon. Technol. Lett. 18, 421-423 (2006). [CrossRef]
- H. J. Kim, I. Park, B. H. O, S. G. Park, E. H. Lee, and S. G. Lee, "Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing," Opt. Express 12, 5625-5633 (2004). [CrossRef] [PubMed]
- Y. Zhang, Z. Li, and B. Li, "Multimode interference effect and self-imaging principle in two-dimensional silicon photonic crystal waveguides for terahertz waves," Opt. Express 14, 2679-2689 (2006). [CrossRef] [PubMed]
- D. Modotto, M. Conforti, A. Locatelli, and C. D. Angelis, "Imaging properties of multimode photonic crystal waveguides and waveguide arrays," J. Lightwave Technol. 25, 402-409 (2007). [CrossRef]
- L. W. Chung and S. L. Lee, "Multimode-interference-based broad-band demultiplexers with internal photonic crystals," Opt. Express 14, 4923-4927 (2006). [CrossRef] [PubMed]
- Z. Li, Y. Zhang, and B. Li, "Terahertz photonic crystal switch in silicon based on self-imaging principle," Opt. Express 14, 3887-3892 (2006) [CrossRef] [PubMed]
- S. Fan, N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J.D. Joannopoulos, "Guided and defect modes in periodic dielectric waveguides," J. Opt. Soc. Am. B 12, 1267-1272 (1995). [CrossRef]
- P. G. Luan and K. D. Chang, "Transmission characteristics of finite periodic dielectric waveguides," Opt. Express 14, 3263-3272 (2006). [CrossRef] [PubMed]
- P. G. Luan and K. D Chang, "Periodic dielectric waveguide beam splitter based on co-directional coupling," Opt. Express 15, 4536-4545 (2007). [CrossRef] [PubMed]
- D. S. Gao, R. Hao, and Z. P. Zhou, "Mach-Zehnder interferometer based on coupled dielectric pillars," Chin. Phys. Lett. 24, 3172-3174 (2007). [CrossRef]
- W. Huang, Y. Zhang, and B. Li, "Ultracompact wavelength and polarization splitters in periodic dielectric waveguides," Opt. Express 16, 1600-1609 (2008). [CrossRef] [PubMed]
- Y. Zhang, W. Huang, and B. Li, "Fabry-Pérot microcavities with controllable resonant wavelengths in periodic dielectric waveguides," Appl. Phys. Lett. 93, 0311101-3 (2008). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001). [CrossRef] [PubMed]
- A. Lavrinenko, P. I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. M. H. Chong, "Comprehensive FDTD modelling of photonic crystal waveguide components," Opt. Express 12, 234-248 (2004). [CrossRef] [PubMed]
- D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Nanopillars photonic crystal waveguides," Opt. Express 12, 617-622 (2004). [CrossRef] [PubMed]
- D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Numerical characterization of nanopillar photonic crystal waveguides and directional couplers," Opt. Quant. Electron. 37, 331-341 (2005). [CrossRef]
- J. Smajic, C. Hafner, and D. Erni, "On the design of photonic crystal multiplexers," Opt. Express 11, 566-571 (2003). [CrossRef] [PubMed]
- M. Y. Tekeste and J. M. Yarrison-Rice, "High efficiency photonic crystal based wavelength demultiplexer," Opt. Express 14, 7931-7942 (2006). [CrossRef] [PubMed]
- R. Costa, A. Melloni, and M. Martinelli, "Bandpass resonant filters in photonic-crystal waveguides," IEEE Photon. Technol. Lett. 15, 401-403 (2003). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1. |
Fig. 2. |
Fig. 3. |

Fig. 4. |
Fig. 5. |
Fig. 6. |

Fig. 7. |
Fig. 8. |
Fig. 9. |

Fig. 10. |
Fig. 11. |
Fig. 12. |

OSA is a member of CrossRef.