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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 13 — Jun. 21, 2010
  • pp: 14152–14158
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Field penetrations in photonic crystal Fano reflectors

Deyin Zhao, Zhenqiang Ma, and Weidong Zhou  »View Author Affiliations


Optics Express, Vol. 18, Issue 13, pp. 14152-14158 (2010)
http://dx.doi.org/10.1364/OE.18.014152


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Abstract

We report here the field and modal characteristics in photonic crystal (PC) Fano reflectors. Due to the tight field confinement and the compact reflector size, the cavity modes are highly localized and confined inside the single layer Fano reflectors, with the energy penetration depth of only 100nm for a 340 nm thick Fano reflector with a design wavelength of 1550 nm. On the other hand, the phase penetration depths, associated with the phase discontinuity and dispersion properties of the reflectors, vary from 2000 nm to 4000 nm, over the spectral range of 1500 nm to 1580 nm. This unique feature offers us another design freedom of the dispersion engineering for the cavity resonant mode tuning. Additionally, the field distributions are also investigated and compared for the Fabry-Perot cavities formed with PC Fano reflectors, as well conventional DBR reflectors and 1D sub-wavelength grating reflectors. All these characteristics associated with the PC Fano reflectors enable a new type of resonant cavity design for a large range of photonic applications.

© 2010 OSA

1. Introduction

Under surface-normal incidence, DBR, 1D SWG, and 2D PCS reflectors can all exhibit similar reflection properties with extremely high reflection and broad reflection spectral band. However, reflection mechanisms are different. For 1D SWG and 2D PC mirrors, the incident wave couples to the in-plane guided-mode based on phase matching conditions. The wave then reradiates at one edge with a zero phase difference and at another edge with a π phase difference. Consequently, these constructive and destructive interferences result in high reflection and low transmission, respectively [16

16. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008). [CrossRef] [PubMed]

]. While for DBR, the high reflectivity arises from the multiple reflections with constructive interference among these reflected waves. Due to the large index difference, Bragg mirrors possess a broad reflection spectral band [18

18. L. Coldren, and S. Corzine, Diode lasers and photonic integrated circuits, (Wiley New York, 1995).

]. For 1D SWG and 2D PCS mirrors, the broad reflection spectral band most likely originates from the cooperating of the several adjacent guided mode resonances [16

16. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008). [CrossRef] [PubMed]

].

2. Dielectric mirrors configuration and corresponding FP cavities

Here, we consider three types of dielectric mirrors and their corresponding FP cavities, as shown in Fig. 1
Fig. 1 Sketches of different dielectric reflectors and the corresponding Fabry-Perot cavities: (a) cross section (xz plane) of Cavity I consisting of top and bottom reflectors based on 1D single Si-layer sub-wavelength grating (SWG) with the same pattern parameters; (b) overview of Cavity II consisting of top and bottom mirrors based on 2D PCS patterns with a square air holes lattice; (c) cross section (xz plane) of Cavity III consisting of top and bottom mirrors based on 4-pairs of Si/SiO2 (0.11/0.27μm) DBR stacked layers.
. They all consist of two kinds of materials, Si and SiO2. Here Si and SiO2 are assumed to be lossless and dispersion-free, with the refractive indexes are of 3.48 and 1.48, respectively, over the spectral range of interest around 1550 nm. In Fig. 1(a), the top and the bottom mirrors of the FP cavity (denoted as “Cavity I”) are 1D Si SWG structures with the same lattice parameters, with the Si layer thickness h of 0.46μm, the grating period Λ of 0.7μm, and the air slit width w of 0.25Λ. The bottom SWG mirror is formed on a silicon-on-insulator (SOI) substrate, with the buffered oxide (BOX) layer thickness of 0.83μm. These two 1D SWG reflectors exhibit a high reflection over 1.1-2.06μm wavelength band for TM polarization only (H field is parallel to the air slit, y direction) [16

16. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008). [CrossRef] [PubMed]

].

Shown in Fig. 1(b) is the second FP cavity (“Cavity II”) formed with top and bottom 2D PCS Fano reflectors, where the Si slab layers with a thickness h = 0.34um are patterned with 2D square lattice air hole arrays. The PC lattice constant Λ is equal to 0.98μm. Again, the bottom reflector is processed on a SOI wafer, with the BOX layer thickness of 1μm. To enable the reflection bands of the top and bottom mirrors with large spectral overlap, their radius of air hole are set to rt = 0.26Λ and r b = 0.28Λ, respectively. The resulting overlapping reflection range of these two 2D PC mirrors is over 1.49-1.58μm wavelength band for both TE and TM polarizations (Broader and more flat band could be found through carefully optimizing the structure parameters). For comparison, we also consider the third FP cavity (“Cavity III”), based on two classical high index contract Si/SiO2 DBRs, as shown in Fig. 1(c). The DBRs consist of 4-pairs of Si/SiO2 stacked layers, where the thickness of Si and SiO2 are chosen to be 0.11 and 0.27μm, respectively. Such a DBR with a very large index difference possesses a wide reflection band over 1.22-2.07μm. Note all the parameters chosen here are optimized for broadband reflectors with peak reflection around 1550nm.

3. Phase penetration depth and energy penetration depth

The energy storage is always associated to the parameter of energy penetration depth, Le. It is the length that the field intensity decays to 1/e of its maximum from the edge of cavity into the mirrors. However, this method is not suitable to calculate Le of the PC mirrors we discuss here, because the guided modes are excited inside the mirrors. But Le can be estimated from the mirror transmission or reflection based on the following equation,
T=1R=exp(hLe),
(2)
where T is the transmission and h is the mirror thickness [18

18. L. Coldren, and S. Corzine, Diode lasers and photonic integrated circuits, (Wiley New York, 1995).

,21

21. C. Sauvan, J. Hugonin, and P. Lalanne, “Difference between penetration and damping lengths in photonic crystal mirrors,” Appl. Phys. Lett. 95(21), 211101 (2009). [CrossRef]

]. While, for DBRs, Le can be obtained [18

18. L. Coldren, and S. Corzine, Diode lasers and photonic integrated circuits, (Wiley New York, 1995).

]:
Le=meff2(λ4n1+λ4n2),
(3)
where meff = tanh(2mr)/(2r) is the effective period number seen by the incident light. r = (n1-n2)/(n1 + n2) and m is the actual period number in DBRs. n1 and n2 are the refractive index of two materials in DBR, respectively. In the following we will numerically investigate the phase and energy penetration depth according to the above definitions.

3.1 Reflection and the phase shift

First, we utilize FDTD simulation method to get R and Φr of these dielectric mirrors. A Gaussian temporal pulse excitation is used to simulate the reflectivity R. In order to calculate Φr, a continuous plane wave of a single wavelength (λ) is vertically incident on the dielectric mirrors and Φr is extracted from the stable reflected field. Here the calculated Φr is set in the range of [0, 2π]. In order to validate our numerical simulation method, we compare the simulated R and Φr of the top DBR based on FDTD with the theoretical calculated values according to the multiple thin film matrix theory [22

22. M. Born, E. Wolf, and A. Bhatia, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light: Cambridge Univ Pr, 1999.

]. The calculated results completely overlap with the theoretical ones.

Plotted in Fig. 2(a)
Fig. 2 Calculated reflection R (blue solid and black dash lines) and phase shift Φr (red dash-dot and green dot lines) spectra of top and bottom (a) DBR reflectors, (b) 1D SWG reflectors and (c) 2D PCS reflectors.
are the simulated R and Φr values for both top and bottom DBRs. In the high reflection (R>0.95) spectral band, covering from 1.2 to 2.1μm, Φr slowly increases from 0.815π to 1.19π. Shown in Fig. 2(b) are the simulated R and Φr spectra of top and bottom 1D SWG reflectors. The high reflection TM (R>0.95) spectral band spans from 1.1 to 2.06μm. Although both DBR and 1D SWG reflector have similar broad high reflection spectra bands, their Φr changes are very different. Φr of 1D SWG reflector rapidly varies in the range of [0, 2π], much faster than those of DBRs. The calculated R and Φr spectra of for the top and bottom 2D PCS reflectors are plotted in Fig. 2(c), where the overlapping high reflection spectra band covers from 1.49 to 1.58μm. Owing to the different air fill factors (rt = 0.26Λ < r b = 0.28Λ), high reflection spectral band for the bottom mirror is narrower than that of the top mirror. In this high reflection band range, the phase shift of top mirror Φrt varies in range of (0.91π, 1.17π), and the phase shift of bottom mirror Φrb changes in range of (0.7π, 1.14π). For all these three types of mirrors, Φr varies over the high reflection spectral band at drastically different change rates, which can be found by comparing the phase penetration depth Lp.

3.2 Phase penetration depth and energy penetration depth

The phase penetration depth Lp can then be calculated based on Eq. (1) and the simulated Φr shown in Fig. 2 The results are plotted in Fig. 3(a)
Fig. 3 (a) The phase penetration depths for three types of bottom reflectors; (b) Reflected field at λ = 1.540μm changes as function of time, for 1D SWG reflectors (top), 2D PCS reflectors (middle), and DBR reflectors (bottom).
, for the bottom DBR, 1D SWG, and 2D PCS reflectors in the wavelength range of 1.5-1.6μm, denoted as Lp, DBR, Lp, 1DSWG, Lp, 2DPCS, respectively. It can be seen that Lp, 2DPCS (~2.1μm) and Lp, 1DSWG (~3.2μm) are one order larger than Lp, DBR (~0.22μm). Additionally, while Lp, DBR and Lp, 1DSWG remains largely unchanged over the spectral range, Lp, 2DPCS does change significantly over different spectral locations. This not only results in a much longer reflection delay time in PC mirrors, it also leads to different resonance cavity locations. Such a long phase delay may be due to the guided mode excitation, even if 2D PCS reflectors are very thin. To verify this point, we record the dynamic process of the reflected field with λ = 1.540μm at one monitor above the 2P PCS reflector and the DBR, as shown in Fig. 3(b). One can find, for the PCS reflector based on guided mode Fano resonance, the reflected field reaches stable condition only after a long 200fs period, while it only takes about 60fs for DBR to reach the stable condition. So, it is very clear these two different reflection mechanisms result in very different Lp values in PC reflectors and DBRs.

4. FP cavities and field distributions

Finally, for the resonant FP cavity modes, we can investigate the field distribution properties in these FP cavities shown in Fig. 1. To have resonance cavity modes with similar spectral locations, the cavity lengths are chosen as Lc1 = 5.4μm, Lc2 = 5.2 and 5.4μm, Lc3 = 5.4μm in three different cavities. We chose the two resonant modes in each cavity: λ = 1.629μm and 1.510μm in Cavity I, λ = 1.504μm and 1.540μm in Cavity II, λ = 2.076μm and 1.540μm in Cavity III.

5. Conclusion

In conclusion, we have numerically investigated phase and energy penetration depths, and field distributions of 1D SWG and 2D PCS reflectors based on Fano or guided mode resonances. Comparing to the DBR reflectors, these new types of single layer ultra-compact broadband reflectors can have more complicated larger phase delays and smaller energy penetration properties, which can be engineered via dispersion engineering for large spectral dependent phase delays, and ultra-small energy penetration depths. The work reported here is mostly based on one set of optimized design parameters for 1550nm band reflectors. Following similar procedures, other design parameters can be used for reflectors with different reflection requirements, as well as different phase delays, energy penetrations, and field distributions. All the results and conclusions can be very helpful for the design of resonant cavities for a wide range of photonic applications.

Acknowledgments

DZ appreciates the help from Dr. Zexuan Qiang. This work is supported in part by US Air Force Office of Scientific Research (AFOSR) MURI program under Grant FA9550-08-1-0337, by AFOSR under grant FA9550-09-C-0200, and in part by US Army Research Office (ARO) under Grant W911NF-09-1-0505.

References and links

1.

S. Boutami, B. Benbakir, X. Letartre, J. L. Leclercq, P. Regreny, and P. Viktorovitch, “Ultimate vertical Fabry-Perot cavity based on single-layer photonic crystal mirrors,” Opt. Express 15(19), 12443–12449 (2007). [CrossRef] [PubMed]

2.

M. Sagawa, S. Goto, K. Hosomi, T. Sugawara, T. Katsuyama, and Y. Arakawa, “40-Gbit/s Operation of Ultracompact Photodetector-Integrated Dispersion Compensator Based on One-Dimensional Photonic Crystals,” Jpn. J. Appl. Phys. 47(8), 6672–6674 (2008). [CrossRef]

3.

A. Chutinan, N. P. Kherani, and S. Zukotynski, “High-efficiency photonic crystal solar cell architecture,” Opt. Express 17(11), 8871–8878 (2009). [CrossRef] [PubMed]

4.

O. Kilic, M. Digonnet, G. Kino, and O. Solgaard, “External fibre Fabry-Perot acoustic sensor based on a photonic-crystal mirror,” Meas. Sci. Technol. 18(10), 3049–3054 (2007). [CrossRef]

5.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

6.

U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

7.

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022 (1992). [CrossRef]

8.

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]

9.

D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters,” Appl. Opt. 41(7), 1241–1245 (2002). [CrossRef] [PubMed]

10.

S. T. Thurman and G. M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. 42(16), 3225–3233 (2003). [CrossRef] [PubMed]

11.

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broadband mirror (1.12-1.62 µm) using single-layer sub-wavelength grating,” IEEE Photon. Technol. Lett. 16(7), 1676–1678 (2004). [CrossRef]

12.

W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84(24), 4905 (2004). [CrossRef]

13.

S. Boutami, B. B. Bakir, H. Hattori, X. Letartre, J.-L. Leclercq, P. Rojo-Rome, M. Garrigues, C. Seassal, and P. Viktorovitch, “Broadband and compact 2-D photonic crystal reflectors with controllable polarization dependence,” IEEE Photon. Technol. Lett. 18(7), 835–837 (2006). [CrossRef]

14.

Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12(23), 5661–5674 (2004). [CrossRef] [PubMed]

15.

M. L. Wu, Y. C. Lee, C. L. Hsu, Y. C. Liu, and J. Y. Chang, “Experimental and Theoretical demonstration of resonant leaky-mode in grating waveguide structure with a flattened passband,” Jpn. J. Appl. Phys. 46(No. 8B), 5431–5434 (2007). [CrossRef]

16.

R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008). [CrossRef] [PubMed]

17.

W. Zhou, Z. Ma, H. Yang, Z. Qiang, G. Qin, H. Pang, L. Chen, W. Yang, S. Chuwongin, and D. Zhao, “Flexible photonic-crystal Fano filters based on transferred semiconductor nanomembranes,” J. Phys. D. 42(23), 234007 (2009). [CrossRef]

18.

L. Coldren, and S. Corzine, Diode lasers and photonic integrated circuits, (Wiley New York, 1995).

19.

J. H. Kim, L. Chrostowski, E. Bisaillon, and D. V. Plant, “DBR, Sub-wavelength grating, and Photonic crystal slab Fabry-Perot cavity design using phase analysis by FDTD,” Opt. Express 15(16), 10330–10339 (2007). [CrossRef] [PubMed]

20.

D. Babic and S. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28(2), 514–524 (1992). [CrossRef]

21.

C. Sauvan, J. Hugonin, and P. Lalanne, “Difference between penetration and damping lengths in photonic crystal mirrors,” Appl. Phys. Lett. 95(21), 211101 (2009). [CrossRef]

22.

M. Born, E. Wolf, and A. Bhatia, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light: Cambridge Univ Pr, 1999.

OCIS Codes
(050.5080) Diffraction and gratings : Phase shift
(230.0230) Optical devices : Optical devices
(140.3948) Lasers and laser optics : Microcavity devices

ToC Category:
Photonic Crystals

History
Original Manuscript: May 6, 2010
Revised Manuscript: June 9, 2010
Manuscript Accepted: June 12, 2010
Published: June 16, 2010

Citation
Deyin Zhao, Zhenqiang Ma, and Weidong Zhou, "Field penetrations in photonic crystal Fano reflectors," Opt. Express 18, 14152-14158 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-14152


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References

  1. S. Boutami, B. Benbakir, X. Letartre, J. L. Leclercq, P. Regreny, and P. Viktorovitch, “Ultimate vertical Fabry-Perot cavity based on single-layer photonic crystal mirrors,” Opt. Express 15(19), 12443–12449 (2007). [CrossRef] [PubMed]
  2. M. Sagawa, S. Goto, K. Hosomi, T. Sugawara, T. Katsuyama, and Y. Arakawa, “40-Gbit/s Operation of Ultracompact Photodetector-Integrated Dispersion Compensator Based on One-Dimensional Photonic Crystals,” Jpn. J. Appl. Phys. 47(8), 6672–6674 (2008). [CrossRef]
  3. A. Chutinan, N. P. Kherani, and S. Zukotynski, “High-efficiency photonic crystal solar cell architecture,” Opt. Express 17(11), 8871–8878 (2009). [CrossRef] [PubMed]
  4. O. Kilic, M. Digonnet, G. Kino, and O. Solgaard, “External fibre Fabry-Perot acoustic sensor based on a photonic-crystal mirror,” Meas. Sci. Technol. 18(10), 3049–3054 (2007). [CrossRef]
  5. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).
  6. U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]
  7. R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022 (1992). [CrossRef]
  8. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]
  9. D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters,” Appl. Opt. 41(7), 1241–1245 (2002). [CrossRef] [PubMed]
  10. S. T. Thurman and G. M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. 42(16), 3225–3233 (2003). [CrossRef] [PubMed]
  11. C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broadband mirror (1.12-1.62 µm) using single-layer sub-wavelength grating,” IEEE Photon. Technol. Lett. 16(7), 1676–1678 (2004). [CrossRef]
  12. W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84(24), 4905 (2004). [CrossRef]
  13. S. Boutami, B. B. Bakir, H. Hattori, X. Letartre, J.-L. Leclercq, P. Rojo-Rome, M. Garrigues, C. Seassal, and P. Viktorovitch, “Broadband and compact 2-D photonic crystal reflectors with controllable polarization dependence,” IEEE Photon. Technol. Lett. 18(7), 835–837 (2006). [CrossRef]
  14. Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12(23), 5661–5674 (2004). [CrossRef] [PubMed]
  15. M. L. Wu, Y. C. Lee, C. L. Hsu, Y. C. Liu, and J. Y. Chang, “Experimental and Theoretical demonstration of resonant leaky-mode in grating waveguide structure with a flattened passband,” Jpn. J. Appl. Phys. 46(No. 8B), 5431–5434 (2007). [CrossRef]
  16. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008). [CrossRef] [PubMed]
  17. W. Zhou, Z. Ma, H. Yang, Z. Qiang, G. Qin, H. Pang, L. Chen, W. Yang, S. Chuwongin, and D. Zhao, “Flexible photonic-crystal Fano filters based on transferred semiconductor nanomembranes,” J. Phys. D. 42(23), 234007 (2009). [CrossRef]
  18. L. Coldren, and S. Corzine, Diode lasers and photonic integrated circuits, (Wiley New York, 1995).
  19. J. H. Kim, L. Chrostowski, E. Bisaillon, and D. V. Plant, “DBR, Sub-wavelength grating, and Photonic crystal slab Fabry-Perot cavity design using phase analysis by FDTD,” Opt. Express 15(16), 10330–10339 (2007). [CrossRef] [PubMed]
  20. D. Babic and S. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28(2), 514–524 (1992). [CrossRef]
  21. C. Sauvan, J. Hugonin, and P. Lalanne, “Difference between penetration and damping lengths in photonic crystal mirrors,” Appl. Phys. Lett. 95(21), 211101 (2009). [CrossRef]
  22. M. Born, E. Wolf, and A. Bhatia, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light: Cambridge Univ Pr, 1999.

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