## Surface modes in two-dimensional photonic crystal slabs with a flat dielectric margin

Optics Express, Vol. 14, Issue 16, pp. 7368-7377 (2006)

http://dx.doi.org/10.1364/OE.14.007368

Acrobat PDF (1542 KB)

### Abstract

We report numerical simulations of surface modes in two-dimensional high-index-contrast photonic crystal slabs with a flat dielectric margin of width on the order of the photonic crystal periodicity. Our calculations using plane wave expansion method reveal multiple surface guided modes within the photonic band gap, with some high-order modes exhibit relatively flat dispersion curves. We calculate the finite-length surface waveguide modes transmission and field patterns using two-dimensional finite-difference time-domain method. We verify the surface mode dispersion curves by using spatial Fourier transform of the mode field patterns. Our study on surface modes under small ambient refractive index changes (5 × 10^{-3}) shows that lower order modes exhibit larger wavelength shifts on the order of 1 nm. We also design a 4-port 3-channel bidirectional coupler using a conventional dielectric waveguide side coupled to the multimode surface waveguide.

© 2006 Optical Society of America

## 1. Introduction

13. K. K. Tsia and A. W. Poon, “Dispersion-guided resonances in two-dimensional photonic-crystal-embedded microcavities,” Opt. Express **12**, 5711–5722 (2004). [CrossRef] [PubMed]

14. K. K. Tsia and A. W. Poon, “Dispersion-guided and bandgap-guided resonances in semiconductor waveguide-coupled hexagonal photonic-crystal-embedded microcavities,” presented at 2005 Conference on Lasers and Electro Optics/ Quantum Electronics and Laser Science Conference (CLEO/QELS), Baltimore, USA, 22–27 May 2005.

## 2. Plane wave expansion calculations

*r*. The surface waveguide axis is in the

*z*-direction (ΓK symmetry direction). The waveguide transverse dimension is in the

*x*-direction (ΓM symmetry direction). The dielectric margin of width

*m*spanning between the flat sidewall and the edge of the first row of holes constitutes an asymmetric waveguide. We focus on the E-modes (electric field parallel to plane of the periodicity) in such PC slab surface waveguides.

*r*= 0.3

*a*and refractive index

*n*= 3.5 (e.g. silicon refractive index), the PC slab displays a band gap between ~0.205

*c*/

*a*and ~0.275

*c*/

*a*, where

*c*is the speed of light in vacuum. We consider surface waveguides for a range of margin widths from

*m*= 0.1

*a*to

*m*= 2.0

*a*. We define a super-cell as one unit cell in the

*z*-direction (ΓK) and 11 unit cells in the

*x*-direction (ΓM), as shown in Fig. 1(b). We discretize each unit cell into 32 computation steps. From the calculated dispersion diagrams, we extract several pertinent parameters about the surface waveguide modes including the group refractive index

*n*

_{g}, the guided mode bandwidth, and the effective span in projected wavevector

*k*that displays a relatively large

*n*

_{g}.

*z*- direction) for surface waveguides of

*m*= 0.1

*a*, 0.7

*a*, and 1.2

*a*. The band gap can be clearly discerned. The thin dark lines are the PC slab modes in the 1

^{st}and 2

^{nd}PC bands. The dashed line is the light line. We highlight the surface modes and label them as A0, A1, etc. For

*m*= 0.1

*a*[Fig. 2(a)], we observe only a single surface mode A0. For

*m*= 0.7

*a*[Fig. 2(b)], two surface modes A1 and A2 exist within the band gap, whereas mode A0 redshifts to below the lower PC zone edge frequency. For

*m*= 1.2

*a*[(Fig. 2(c)], three surface modes A2, A3, and A4 exist within the band gap, whereas modes A0 and A1 redshift to below the lower PC zone edge frequency. It is noteworthy that mode A3 displays a flat band (~0.004

*c*/

*a*bandwidth) over a relatively wide

*k*span (~0.3 (2

*π*/

*a*) to ~0.5 (2

*π*/

*a*)).

*m*’s. Modes A0 and A1 exhibit familiar field patterns as previously shown in corrugated surface waveguides [11

11. J.-K. Yang, S.-H. Kim, G.-H. Kim, H.-G. Park, Y.-H. Lee, and S.-B. Kim, “Slab-edge modes in two-dimensional photonic crystals,” Appl. Phys. Lett. **84**, 3016–3018 (2004). [CrossRef]

*x*-direction.

*m*spanning from 0.1

*a*to 2.0

*a*. We identify a total of six different guided modes. As

*m*increases, the surface modes dispersion curves and thereby the zone edge frequencies redshift, and the number of surface modes within the band gap increases. The surface waveguide exhibits single guided mode (fundamental mode) in the PC band gap only for

*m*below ~0.3

*a*.

*k*span (Δ

*k*) within which the modes exhibit a relatively large n

_{g}(arbitrarily set at exceeding

*n*

_{g}= 50) as a function of the zone-edge frequency. We find that Δ

*k*overall narrows as the mode redshifts with the margin width. For example, mode A2 displays a Δ

*k*that is narrowed from ~0.06 (2

*π*/a) to ~0.01 (2

*π*/a) with

*m*widens from 0.6

*a*to 1.4

*a*. Mode A2 exhibits ~3 to ~5 times wider Δ

*k*than mode A1 at the same zone-edge frequencies. Whereas mode A4 only displays marginally wider Δ

*k*than mode A1. It is noteworthy that mode A3 Δ

*k*exceeds ~0.1 (2

*π*/

*a*) for a relatively wide span of

*m*between 1.1

*a*and 1.7

*a*[see Fig. 2(h)]. We remark that the lowest order mode A0 also exhibits wide Δ

*k*up to 0.1 (2

*π*/

*a*) in

*m*= 0.1

*a*waveguide, which is however prohibitively narrow and not favorable for device fabrication. Increasing the margin width to 0.2

*a*renders A0 mode Δ

*k*narrows to 0.01 (2

*π*/

*a*). Figures 3(b)–3(d) show the PWE-calculated dispersion diagrams for

*m*= 1.1

*a*,

*m*= 1.7

*a*, and

*m*= 1.9

*a*, with mode A3 Δ

*k*of ~0.128 (2

*π*/

*a*), ~0.112 (2

*π*/

*a*), and 0.054 (2

*π*/

*a*), respectively.

## 3. Finite-difference time-domain calculations

*L*= 16

*a*and

*L*= 32

*a*by means of 2-D finite-difference time-domain (FDTD) simulations. We employ 36 computation points per period in our FDTD calculations. We launch at longitudinal position z = 0 a single-pulse source (pulse width

*c*

*Δt*~ 4.3

*a*) with center frequency ~0.2322

*c*/

*a*, and detect the transmission field at the waveguide forward end and the reflection field at the waveguide backward end. We apply perfectly matched layers (PMLs) of reflectivity 10

^{-8}and thickness of 0.5 μm to all four boundaries of the computation domain [Fig. 1(a)]. The finite simulation domain effectively imposes mode-mismatch reflections at the two waveguide ends.

*m*= 0.8

*a*with

*L*= 16

*a*and

*L*= 32

*a*. The two waveguides show largely comparable spectra with sharper spectral features in the case of

*L*= 32

*a*. In both cases we observe two transmission bands. We contrast the transmission spectra with the PWE-calculated dispersion diagram, as shown in Fig. 4(a). We find that the two transmission bands lie in the band gap. The minor one corresponds to mode A1, with peak transmissions below ~0.3 and spans from ~0.2

*c*/

*a*to ~0.209

*c*/

*a*. The transmission bandwidth is bound by mode A1 zone-edge frequency and the maximum frequency of the mode A1 dispersion curve. The major one corresponds to mode A2, with peak transmissions of ~0.75 and spans from ~0.233

*c*/

*a*to ~0.26

*c*/

*a*. The transmission bandwidth is bound by mode A2 zone-edge frequency and the intersection between the light line and the mode A2 dispersion curve. In between the two transmission bands is a forbidden band with a bandwidth of ~0.024

*c*/

*a*.

*L*= 32

*a*spectrum than in the

*L*= 16

*a*spectrum. We attribute the spectral oscillations to the Fabry-Perot (FP) reflections between the waveguide end-faces (mode-mismatch reflections), similar to the experimentally observed FP resonance spectra in line defect waveguides [15

15. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. **87**, 253902 (2001). [CrossRef] [PubMed]

**H**-field intensity patterns (

*L*= 32

*a*,

*m*= 0.8

*a*) at (a) mode A1 transmission peak at 0.20016

*c*/

*a*(near the zone-edge frequency) and (b) mode A2 transmission peak at 0.23314

*c*/

*a*(near the zone-edge frequency). Figures 5(c) and 5(d) show the zoom-in view of the steady-state

**H**-field amplitude patterns.

**k**-space information of the surface modes. Figure 5(e) shows the Fourier transform of mode A1 field pattern [Fig. 5(c)]. We identify three peaks in the

**k**-space, with major

*k*

*component at ~0.5098 (2*

_{z}*π*/

*a*), and minor

*k*components at ~0.246 (2

_{z}*π*/

*a*) and ~0.754 (2

*π*/

*a*). We fold

*k*components exceeding 0.5 (2

_{z}*π*/

*a*) back into the 1

^{st}Brillouin zone (BZ). The three

*k*components are then reduced to two projected wavevectors (marked with *’s) at ~0.4902 (2

_{z}*π*/

*a*) and ~0.246 (2

*π*/

*a*), as shown in Fig. 4(a).

**k**-space at

*k*~ 0.50098 (2

_{z}*π*/

*a*), corresponding to

*k*~ 0.49902 (2

_{z}*π*/

*a*) in the 1

^{st}BZ. Other retrieved wavevector values in Fig. 4(a) are obtained in the same fashion from the field patterns at various frequencies using

*L*= 32

*a*. The Fourier-transform retrieved

**k**-vectors show excellent agreement with the PWE-calculated surface mode dispersion curves.

*m*= 1.5

*a*waveguide. Figure 6(a) shows the PWE-calculated projected dispersion diagram. The Fourier-transform retrieved

**k**-vectors are shown as symbols. Figures 6(b) and 6(c) show the FDTD-simulated

**H**-field patterns for modes A3 at 0.2185

*c*/

*a*and A4 at 0.2391

*c*/

*a*. Figure 6(d) shows the

**k**-space distribution of the simulated mode A3 field pattern [Fig. 6(c)]. We observe five peaks in the

**k**-space distribution, corresponding to three wavevectors (marked with *’s) on mode A3 dispersion curve in Fig. 6(a). We see that the retrieved

**k**-vectors confirm the flat band characteristics of mode A3.

**k**-space peaks are folded to two wavevectors on mode A4 dispersion curve in Fig. 6(a). The retrieved

**k**-vectors to the left of the light line are due to the

**k**-vectors in the 2

^{nd}BZ folded back to the 1

^{st}BZ.

## 4. Ambient refractive index sensing

16. D. Erickson, T. Rockwood, T. Emery, A. Scherer, and D. Psaltis, “Nanofluidic tuning of photonic crystal circuits,” Opt. Lett. **31**, 59–61 (2006). [CrossRef] [PubMed]

*n*

_{c}= 1.33, with

*Δn*= 0.005. We assume surface waveguide of periodicity

_{c}*a*= 0.36 μm and

*L*= 32

*a*. We use the Fabry-Perot spectra to identify the wavelength shifts.

18. E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, “Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity,” Opt. Lett. **29**, 1093–1095 (2004). [CrossRef] [PubMed]

*n*induced fractional frequency shifts |Δ(

_{c}*c/a*)| / (

*c/a*) as a function of center frequency (at selected transmission peaks) for modes A1, A2, and A3. The fractional frequency shift increases with the center frequency (with narrowing margin widths). Mode A1 exhibits largest fractional frequency shifts (~1.0×10

^{-3}) to ambient refractive index variations among the three modes. Figures 7(c)–7(e) show the time-averaged

**E**-field intensity distribution for modes A1, A2 and A3. The

**E**-field intensity filling factor in the ambient region (integrated

**E**-field intensity normalized by the computation area as shown) for mode A1 is ~58.1%, and for modes A2 and A3 are ~51.1% and ~50.6%, respectively.

## 5. Asymmetric 4-port 3-channel bi-directional coupler design

*a*= 0.33 μm, margin of 0.23 μm (

*m*~ 0.7

*a*), and a strip waveguide of

*w*= 0.25 μm. The lateral interaction length is 15.18 μm (

*L*= 46

*a*) and the coupling air gap distance is 0.2 μm.

*k*

_{1}*ω*

_{1}) ≈ (0.4406 (2

*π*/

*a*), 0.2138

*c*/

*d*) and (

*k*,

_{2}*ω*

_{2}) ≈ (0.3775 (2

*π*/

*a*), 0.2516

*c/a*), suggesting phase matching between the surface modes and the strip waveguide mode. The transmission dips wavelengths correspond well to the two intersections normalized frequencies.

**H**-field intensity patterns at wavelengths 1.3114 μm (forward coupling), 1.49 μm (throughput transmission), and 1.5494 μm (backward coupling). It is noteworthy that the waveguide-coupled surface mode field intensities suggest field enhancement.

*λ*= 1.3114 μm and

*λ*= 1.5494 μm. For the forward coupling mode, we only discern one major k-space component in the 2

^{nd}BZ with

*k*~ 0.624 (2

_{z}*π*/

*a*), representing co-propagating phase-matched strip waveguide mode and mode A

_{2}. We remark that the

*k*value corresponds to ~0.376 (2

_{z}*π/a*) in the 1

^{st}BZ. For the backward coupling mode, we discern one major k-space component in the 1

^{st}BZ with

*k*~ 0.445 (2

_{z}*π/a*), representing forward propagating strip waveguide mode, and another one in the 2

^{nd}BZ in the negative domain with

*k*~ -0.553 (2

_{z}*π/a*), representing backward propagating mode A1. We note that the kz value in the 2

^{nd}BZ corresponds to ~0.447 (2

*π/a*) in the 1

^{st}BZ.

^{st}and part of 2

^{nd}BZ’s, as shown in Fig. 8(h). According to the Fourier-transformed k-space, modes A1 and A2 dispersion lines are in the 2

^{nd}BZ, whereas the waveguide dispersion line extends from the 1

^{st}to the 2

^{nd}BZ. We see that the forward propagating waveguide mode at

*ω*

_{2}phase matches with mode A

_{2}(

*k*=

_{WG2}*k*), suggesting directional coupling. In contrast, the forward propagating waveguide mode at

_{A2}*ω*

_{1}only phase matches with mode A1 with an additional momentum of 2π/a (

*k*= 2

_{WG1}*π/a*+

*k*), suggesting Bragg-reflection assisted counter-directional coupling.

_{A1}## 6. Conclusions

*a*to 2.0

*a*. Our numerical analysis based on plane wave expansion method suggests that some high-order modes demonstrate group refractive indices exceeding 50 with a relatively wide k-span exceeding 0.1 (2

*π/a*). We used two-dimensional FDTD to evaluate the finite-length surface guided mode transmission and reflection and their mode-field patterns. We apply spatial Fourier analysis on the simulated mode-field patterns and verify the flat bands in surface wave dispersion curves. For ambient index variation of

*Δn*/

_{c}*n*= 0.005/1.33, wavelength redshifts of |

_{c}*Δ(c/a)*|

*/ (c/a)*~ 1×10

^{-3}can be attained.

## Acknowledgment

## References and Links

1. | P. Yeh, “Guided waves in layered media,” in |

2. | R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B |

3. | F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B |

4. | S. Enoch, E. Popov, and N. Bonod, “Analysis of the physical origin of surface modes on finite-size photonic crystals,” Phys. Rev. B |

5. | W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. |

6. | E. Moreno, F. J. Garcìa-Vidal, and L. Martìn-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B |

7. | M. Laroche, R. Carminati, and J.-J. Greffet, “Resonant optical transmission through a photonic crystal in the forbidden gap,” Phys. Rev. B |

8. | W. T. Lau and S. Fan, “Creating large bandwidth line defects by embedding dielectric waveguides into photonic crystal slabs,” Appl. Phys. Lett. |

9. | Yu. A. Vlasov, N. Moll, and S. J. McNab, “Mode mixing in asymmetric double-trench photonic crystal waveguides,” J. Appl. Phys. |

10. | Y. A. Vlasov, N. Moll, and S. J. McNab, “Observation of surface states in a truncated photonic crystal slab,” Opt. Lett. |

11. | J.-K. Yang, S.-H. Kim, G.-H. Kim, H.-G. Park, Y.-H. Lee, and S.-B. Kim, “Slab-edge modes in two-dimensional photonic crystals,” Appl. Phys. Lett. |

12. | S. Xiao and M. Qiu, “Surface-mode microcavity,” Appl. Phys. Lett. |

13. | K. K. Tsia and A. W. Poon, “Dispersion-guided resonances in two-dimensional photonic-crystal-embedded microcavities,” Opt. Express |

14. | K. K. Tsia and A. W. Poon, “Dispersion-guided and bandgap-guided resonances in semiconductor waveguide-coupled hexagonal photonic-crystal-embedded microcavities,” presented at 2005 Conference on Lasers and Electro Optics/ Quantum Electronics and Laser Science Conference (CLEO/QELS), Baltimore, USA, 22–27 May 2005. |

15. | M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. |

16. | D. Erickson, T. Rockwood, T. Emery, A. Scherer, and D. Psaltis, “Nanofluidic tuning of photonic crystal circuits,” Opt. Lett. |

17. | H. Ouyang, C. C. Striemer, and P. M. Fauchet, “Quantitative analysis of sensitivity of porous silicon optical biosensors,” Appl. Phys. Lett. |

18. | E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, “Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity,” Opt. Lett. |

**OCIS Codes**

(230.3120) Optical devices : Integrated optics devices

(230.3990) Optical devices : Micro-optical devices

(240.6690) Optics at surfaces : Surface waves

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 14, 2006

Revised Manuscript: July 25, 2006

Manuscript Accepted: July 25, 2006

Published: August 7, 2006

**Citation**

Hui Chen, Kevin K. Tsia, and Andrew W. Poon, "Surface modes in two-dimensional photonic crystal slabs with a flat dielectric margin," Opt. Express **14**, 7368-7377 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7368

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### References

- P. Yeh, "Guided waves in layered media," in Optical waves in layered media, (John Wiley & Sons, New York, 1988).
- R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961 - 10964 (1991). [CrossRef]
- F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 59, 15112 - 15120 (1999). [CrossRef]
- S. Enoch, E. Popov, and N. Bonod, "Analysis of the physical origin of surface modes on finite-size photonic crystals," Phys. Rev. B 72, 155101 (2005). [CrossRef]
- W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Observation of surface photons on periodic dielectric arrays," Opt. Lett. 7, 528-531 (1993). [CrossRef]
- E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004). [CrossRef]
- M. Laroche, R. Carminati, and J.-J. Greffet, "Resonant optical transmission through a photonic crystal in the forbidden gap," Phys. Rev. B 71, 155113 (2005). [CrossRef]
- W. T. Lau and S. Fan, "Creating large bandwidth line defects by embedding dielectric waveguides into photonic crystal slabs," Appl. Phys. Lett. 81, 3915 - 3917 (2002). [CrossRef]
- Yu. A. Vlasov, N. Moll, and S. J. McNab, "Mode mixing in asymmetric double-trench photonic crystal waveguides," J. Appl. Phys. 95, 4538 - 4544 (2004). [CrossRef]
- Y. A. Vlasov, N. Moll, and S. J. McNab, "Observation of surface states in a truncated photonic crystal slab," Opt. Lett. 29, 2175 - 2177 (2004). [CrossRef] [PubMed]
- J.-K. Yang, S.-H. Kim, G.-H. Kim, H.-G. Park, Y.-H. Lee, and S.-B. Kim, "Slab-edge modes in two-dimensional photonic crystals," Appl. Phys. Lett. 84, 3016-3018 (2004). [CrossRef]
- S. Xiao and M. Qiu, "Surface-mode microcavity," Appl. Phys. Lett. 87, 111102 (2005). [CrossRef]
- K. K. Tsia and A. W. Poon, "Dispersion-guided resonances in two-dimensional photonic-crystal-embedded microcavities," Opt. Express 12, 5711-5722 (2004). [CrossRef] [PubMed]
- K. K. Tsia and A. W. Poon, "Dispersion-guided and bandgap-guided resonances in semiconductor waveguide-coupled hexagonal photonic-crystal-embedded microcavities," presented at 2005 Conference on Lasers and Electro Optics/ Quantum Electronics and Laser Science Conference (CLEO/QELS), Baltimore, USA, 22-27 May 2005.
- M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]
- D. Erickson, T. Rockwood, T. Emery, A. Scherer, and D. Psaltis, "Nanofluidic tuning of photonic crystal circuits," Opt. Lett. 31, 59-61 (2006). [CrossRef] [PubMed]
- H. Ouyang, C. C. Striemer, and P. M. Fauchet, "Quantitative analysis of sensitivity of porous silicon optical biosensors," Appl. Phys. Lett. 88, 163108 (2006). [CrossRef]
- E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, "Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity," Opt. Lett. 29, 1093-1095 (2004). [CrossRef] [PubMed]

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