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Effective refractive index analysis of optical Kerr nonlinearity in photonic bandgap structures
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Optics Communications 281 (2008) 1625–1628 www.elsevier.com/locate/optcom
Effective refractive index analysis of optical Kerr nonlinearity in photonic bandgap structures Jisoo Hwang a, J.W. Wu b,* a
Division of Physical Metrology, Korea Research Institute of Standards and Science, Daejeon 305-340, Republic of Korea b Department of Physics and Division of Nano-sciences, Ewha Womans University, Seoul 120-750, Republic of Korea Received 28 June 2007; received in revised form 5 November 2007; accepted 23 November 2007
Abstract A numerical dispersion relation is employed to analyze the linear and nonlinear optical effective refractive indices of 1-D finite periodic photonic bandgap structures. A Bragg reflector (BR) and a photonic crystal microcavity (PCMC) are examined by assuming that the high indexed layer of the two constituent layers possesses a nonlinear optical response. For the BR, the singularity of refractive index, appearing at bandgap edges in a Bloch index description, is removed. In the case of the PCMC, the optical responses at a defect mode and bandgap edges are properly described, thanks to the use of the numerical dispersion relation. This also allows us to quantitatively compare the Kerr nonlinearity observed at the defect mode and the bandgap edges. The efficiencies of the BR bandgap edge and the PCMC defect mode in achieving a given transmission change are compared by calculating the required nonlinear optical refractive index change. The PCMC defect mode is found to be 1.5 times and twice more efficient than the BR bandgap edge in 20 and 10 dB transmission modulation, respectively. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.70.Qs; 42.79.Dj; 42.65.k; 42.65.Hw Keywords: Photonic bandgap structures; Bragg reflectors; Nonlinear optics; Optical Kerr effect; Dispersion relation
1. Introduction Inside a photonic bandgap (PBG) structure, the wave propagation characteristics of an electromagnetic (EM) wave is governed by the energy-momentum dispersion relation, which is determined from a spatial distribution of refractive indices of constituent periodic layers. For example, the electric field-enhancement at PBG edges is understood in terms of a low group velocity and the anomaly in the dispersion relation. An analytic expression of dispersion relation is readily available in the form of a Bloch index for an infinite periodic PBG structure. On the other hand, in the case of a finite periodic PBG structure, a dispersion relation is numerically obtained from the measured *
Corresponding author. Tel.: +82 23 277 2369; fax: +82 23 277 2372. E-mail address: [email protected] (J.W. Wu).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.067
transmission coefficient. When the PBG structure has to be designed in order to obtain highly efficient generation of harmonics, using nonlinear optical (NLO) processes, the numerical dispersion relation (NDR) can be very useful. In fact, using the NDR allows us properly tailor the PBG characteristics, in order to achieve the phase matching condition [1]. Galisteo-Lopez et al. compared the dispersion relation measured by a white light interferometry and the calculated NDR of a 3-D PBG structure, and found that the agreement was remarkable [2]. The NDR analysis was further extended to obtain an explicit relation between the real and the imaginary components of an EM wave emerging from a PBG structure by use of a Kramers– Kronig relation [3]. While all-optical switching at bandgap edges [4,5] and a defect mode [6,7], originating from optical Kerr related nonlinearity, has been demonstrated experimentally in a PBG structure, NLO analysis of the beam
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propagating properties based on the NDR was not pursued, mainly due to the difficulty in obtaining an analytical dispersion relation. In this paper, we utilize the NDR to investigate the Kerr NLO changes in two important 1-D finite periodic photonic bandgap structures, namely, a Bragg reflector (BR) and a photonic crystal microcavity (PCMC). A BR is a 1-D finite periodic photonic bandgap structure without a defect layer, and a PCMC is in a structure of two identical BRs sandwiched symmetrically with a defect layer-between, hence exhibiting an optical defect mode. We report three major findings. First, the singularity of a Bloch index, occurring at bandgap edges of a BR, is removed. Second, the optical properties of a PCMC, based on the dispersion relations of linear and nonlinear refractive indices, are properly described. In particular, optical Kerr nonlinearity is found to be more enhanced at a defect mode than at bandgap edges of a PCMC, which is understood by taking account of the density of modes (DOM) and the EM field localization. Third, the NLO transmission property associated with the optical Kerr nonlinearity is investigated and compared for the BR bandgap edge and the PCMC defect mode. The NLO refractive index changes required to achieve transmission changes from 100% to 10% and 100% to 1% are numerically simulated. 2. Effective refractive indices of a Bragg reflector Originally, the NDR has been developed to overcome the singularity problem of Bloch index occurring at band edges in description of the linear optical properties of a 1-D PBG structure [8]. The Bloch index nB refers to the refractive index of an infinite periodic structure, which is introduced through a Bloch phase b related with nB as nB ¼ bc=ðxKÞ with K the period, and can be obtained from the analytic dispersion relation in a straightforward way [3,9]. On the other hand, NDR is equivalent to introducing an effective refractive index (ERI) in the PBG structure. Once the transmission coefficient is determined for an inhomogeneous optical structure, the ERI is numerically obtained through the relation neff ðxÞ ¼ ðc=xLÞ ½tan1 ðy=xÞ þ mp ði=2Þ lnðx2 þ y 2 Þ, with x and y the real and the imaginary parts of transmission coefficients and L the sample length. In the experimental case, the transmission coefficient is determined through the measurement of both transmittance pffiffiffiffi T and phase /, since they satisfy the relation x þ iy ¼ T ei/ . Or, simply, we can determine ERI from the measured T and L in cooperation with the Kramers–Kronig relation, due to the fact that the real and imaginary parts of ERI satisfy the Kramers–Kronig relation [1]. First of all, in order to clarify the difference between the Bloch index nB and the ERI neff obtained through NDR in describing the linear optical and NLO properties of band edges, we study the transmission spectrum, linear optical and NLO refractive indices of a BR. All the calculation of transmission spectrum and ERI are performed using transfer matrix method [9].
In Fig. 1 are plotted the transmission spectra and the neff of a 10 bilayer BR and nB of an infinite periodic BR, which are composed of quarter-wave-thick layers with the refractive indices of n1 ¼ 1:42 and n2 ¼ 2:08 for low- and highindex layers, respectively. The thickness of each layers was chosen to satisfy the Bragg reflection at 780 nm. A PBG structure is seen in the transmission spectrum of Fig. 1a. The solid curve corresponds to the PBG structure of a 10 bilayer BR, while the dashed curve corresponds to that of an infinite periodic BR. In Fig. 1b and c, the real and imaginary parts of the linear nB are plotted in the dashed curves, while the solid curves correspond to the linear neff of a finite periodic BR. In both nB and neff , we observe that a PBG structure behaves as an optical layer possessing the anomalously dispersive real part and the absorptive imaginary part, which in fact arises from a multiple reflection of EM wave in a transparent periodic optical structure. We note that neff smooths out at the band edges and exhibits an oscillation outside the bandgap. Now let us turn to the NLO refractive indices. The optical Kerr nonlinearity of a BR can be described in terms of ð2Þ ð2Þ the NLO coefficients n1 and n2 of the low- and high-index layers composing the BR. For the optical Kerr nonlinearið2Þ ð2Þ ties of Dn1 ¼ n1 I ¼ 0 and Dn2 ¼ n2 I ¼ 0:05, both irradiance-dependent Bragg index change, DnB , and ERI change, Dneff , are calculated. DnB and Dneff satisfy the relað0Þ ð0Þ tions, nB ðIÞ ¼ nB þ DnB ðIÞ and neff ðIÞ ¼ neff þ Dneff ðIÞ, ð2Þ and I is the incident irradiance. The value of n2 I ¼ 0:05 11 is assumed for NLO material with n2 ¼ 10 cm2/W and
Fig. 1. Numerically calculated (a) transmission spectra, (b) real part of ð0Þ ð0Þ neff , (c) imaginary part of neff , (d) real part of Dneff , and (e) imaginary part of Dneff of BRs. The solid (dashed) curves correspond to a 10 bilayer (infinite) BR. The arrow in (a) indicates the low-frequency edge of 10 bilayer BR, x ¼ 0:844.
J. Hwang, J.W. Wu / Optics Communications 281 (2008) 1625–1628
the irradiance of I ¼ 5 GW/cm2, which is provided by focusing a beam, of a femto-second laser with 200-fs pulse width and 2.5-nJ energy at 400 kHz repetition rate, as 10 lm width. Fig. 1d and e shows that a singularity is present at PBG edges for an infinite BR (dashed curves). In order to look into the singularity of DnB ðIÞ, we get the following expression of DnB ðIÞ: ( !) ð2Þ ð2Þ ð0Þ ð0Þ ðn1 a þ n2 bÞw 1 n2 n1 DnB ðIÞ ’ þ 1þ I; ð1Þ 2 n1ð0Þ n2ð0Þ ðbð0Þ N pÞK where w is the normalized frequency deviation from the ð0Þ center of the stop-band, x0 , i.e., w ¼ n1 aðx x0 Þ=c ¼ ð0Þ n2 bðx x0 Þ=c, and a and b are the thicknesses of lowand high-index layer with indices n1 and n2 , respectively. N is an integer. From Eq. (1), we find that the nonlinear Bloch index diverges at bandgap edges, where bð0Þ ¼ N p. On the other hand, when an NDR is employed for describing a finite periodic BR, the singularity is removed (solid curves). The solid curves correspond to the effective nonlinearity Dneff with the relation. The opposite signs of Im½Dneff at the low- and high-frequency edges are due to the spectral change caused by both red shift and widening of the bandgap, coming from an increase in the optical thickness of high-index layers. 3. Effective refractive indices of a photonic crystal microcavity Next, we look at the optical Kerr response of a defect mode and bandgap edges in a PCMC. We note that the analytic dispersion relation from Bloch theorem cannot be adopted to describe the linear and NLO property of a period-modified Bragg reflector, because the Bloch phase is the eigenvalue of a unit-cell translation matrix, namely, the broken translation symmetry does not allow to adopt the Bloch phase. In dealing with a translation symmetry broken PBG structures, the tight-binding (TB) formalism, originally introduced in solid state physics, has been successfully applied to calculate the frequency spectrum of a defect mode in a Bragg reflector [10,11]. In the TB formalism, the eigenmode is expressed as a linear combination of supercell modes with the defect taken into account, and the TB parameters are obtained by comparing with the experimental dispersion relation. However, the supercells possessing a defect are assumed to be infinitely periodic in order to facilitate the incorporation of Bloch theorem. This implies that a singularity problem still exists in describing the refractive index. On the other hand, the NDR analysis can be extended to the description of the optical Kerr nonlinearity in a PCMC. The NDR derived from the transmission coefficient includes a phase change which the broken translation symmetry incurs. We numerically calculated the ERI of a finite periodic PCMC using transfer matrix method [9], which is structured as a composition of a NLO half-wave defect layer (ndefect ¼ 2:08) between two quarter-wave Bragg mirrors
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of five bilayers (n1 ¼ 1:42, n2 ¼ 2:08) satisfying the Bragg reflection at 780 nm. The solid curve in Fig. 2a shows the simulated transmission spectrum as a function of dimensionless frequency, exhibiting a PBG structure with a defect mode. The dotted curve is DOM derived from dispersion relation [8]. In Fig. 2b, the real and imaginary parts of ð0Þ neff are shown. As expected, the spectral region where the transmission is prohibited possesses the non-vanishing ð0Þ Imfneff g, and two anomalous dispersions appear in the defect mode where transmission is allowed. Dneff is calcuð2Þ lated with nonlinearities of Dn1 ¼ n1 I ¼ 0 and Dn2 ¼ ð2Þ ð2Þ n2 I ¼ Dndefect ¼ ndefect I ¼ 0:05 in Fig. 2c. We observe that the optical Kerr nonlinearity is enhanced at both lowenergy bandgap edge and defect mode. Compared with the spectral region far from the PBG location, where the field localization is not significantly achieved, the defect mode and the bandgap edges give rise to a strong enhancement in the NLO response via a field localization [12]. The enhancement factors of Re½Dneff are determined as 4.7 and 2.5 for the defect mode and the low-energy bandgap edge, respectively. Also, we find that the PCMC possesses the larger enhancement factor, when compared with the 10 bilayer BR dealt with in the previous section. The enhancement factor of Re½Dneff of the 10 bilayer BR is determined as 1.6 for the low-energy bandgap edge. Under the assumption that the field localization is well established, the effective nonlinearity Dneff can be expressed as the following proportional relation [13]: Dneff / q2x DnLoc tLoc ;
ð2Þ
where qx is DOM, and DnLoc and tLoc are Dn and the accumulated sum of thickness of the layers where the EM field is localized, respectively. For example, DnLoc is Dn of high-index (defect) layers for the low-energy bandgap edge (defect mode), and tLoc is the sum of each thickness of corresponding layers. In our case, q ¼ 5:0 and tLoc ¼ 780=2 nm for the
Fig. 2. Numerically simulated (a) transmission spectrum (solid curve) and ð0Þ density of modes (dotted curve), (b) neff , and (c) Dneff of a PCMC. The ð0Þ solid (dashed) curve is real (imaginary) part of neff and Dneff in (b and c).
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defect mode, and q ¼ 1:8 and tLoc ¼ 3900=2 nm for the lower-energy mode. DnLoc is same for both of the modes as Dndefect ¼ Dn2 ¼ 0:05. With these values, the enhancement factor of Dneff at the defect mode is determined as 4.3, which is 1.6 times larger than that of the low-energy bandgap edge. This result is in good agreement with the values determined through NDR above. We note that the enhancement factor increases more rapidly at the defect mode than at the lowenergy bandgap edge as the number of layers of two BRs in a PCMC increases, since the increase in Dneff via DOM wins over via thickness owing to the superlinear dependence of DOM on the number of layers. Similar to the BR, the sign reversal of imaginary part of Dneff in the middle of stop-band indicates that the red shift of defect mode occurs, which originates from the increase in optical thickness of high-index layers as well as a defect layer. 4. Comparison of a bandgap edge and a defect mode NLO refractive index changes required to achieve transmission changes from 100% to 10% and 100% to 1% are numerically simulated for three different 1-D PBG structures, that is, the 10 bilayer BR, the infinite periodic BR, and the PCMC, as a function of optical Kerr nonlinearity of high-index layers, Dn. We pay a particular attention to the NLO transmission changes in a BR bandgap edge and a PCMC defect mode. In Fig. 3 are plotted NLO transmissions for each structure. We find that the transmission change from 100% to 10% occurs at Dn ’ 0:08 for the low-x bandgap edge of the finite BR, and Dn ’ 0:04 for the defect mode of the PCMC, which can be understood in terms of the relation Eq. (2). Similarly, we calculate Dn required for the transmission change from 100% to 1 %, resulting Dn ’ 0:16 for the low-x bandgap edge of the finite BR and Dn ’ 0:11 for the defect mode of the PCMC. These facts lead to the statement that the defect mode of a PCMC is twice (1.5 times) more efficient than the bandgap
Fig. 3. Numerically simulated transmission changes at the low-x bandgap edge (x ¼ 0:844) of a BR and the defect mode (x ¼ 1) of a PCMC. The solid and dashed curves correspond to the defect mode of a PCMC and the low-x bandgap edge of a finite BR, respectively. As a reference, that of the infinite BR is shown as the dotted curve.
edge of a BR in the application of NLO switching for achieving 10 (20) dB modulation. 5. Conclusion In summary, we investigated optical properties of 1-D finite periodic photonic bandgap structures by introducing an effective refractive index through a numerical dispersion relation. The linear and nonlinear optical effective indices of a finite 10 bilayer BR and Bloch indices of a infinite periodic BR are calculated. It is found that ERI description removes the singularities at bandgap edges in a Bloch index description. Further, we investigated a PCMC composed of two identical BRs sandwiched symmetrically with a defect layer-between. A defect layer-between two BRs makes a defect mode appear within photonic bandgap by breaking the translation symmetry. To obtain ERIs of the PCMC, we adopted a numerical dispersion relation, and found that the optical Kerr nonlinearity is enhanced at bandgap edges and a defect mode owing to a field localization. By calculating the refractive index change required to achieve a given transmission change, the efficiencies for a NLO switching application of the BR bandgap edge and the PCMC defect mode were compared, and the PCMC defect mode is found to be twice and 1.5 times more efficient than the BR bandgap edge for 10 and 20 dB modulation, respectively. Acknowledgements This work was supported by Seoul Research and Business Development Program (10816) and the Korea Research Foundation (KRF-2005-214-C00052). References [1] M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M.J. Bloemer, C.M. Bowden, I. Nefedov, Phys. Rev. E 60 (1999) 4891. [2] J.F. Galisteo-Lo´pez, M. Galli, M. Patrini, A. Balesteri, L.C. Andreani, C. Lo´pez, Phys. Rev. B 73 (2006) 125103. [3] Q. Zou, A. Ramdane, B.-E. Benkelfat, Opt. Commun. 230 (2004) 167. [4] D.A. Mazurenko, R. Kerst, J.I. Dijkhuis, A.V. Akimov, V.G. Golubev, D.A. Kurdyukov, A.B. Pevtsov, A.V. Sel’kin, Phys. Rev. Lett. 91 (2003) 213903. [5] J. Hwang, M.J. Kim, J.W. Wu, S.M. Lee, B.K. Rhee, Opt. Lett. 31 (2006) 377. [6] X. Hu, Q. Gong, Y. Liu, B. Cheng, D. Zhang, Appl. Phys. Lett. 87 (2005) 231111. [7] G. Ma, J. Shen, Z. Zhang, Z. Hua, S.H. Tang, Opt. Express 14 (2006) 858. [8] J.M. Bendickson, J.P. Dowling, M. Scalora, Phys. Rev. E 53 (1996) 4107. [9] A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley, 1984. [10] E. Yablonovitch, T.J. Gmitter, R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, Phys. Rev. Lett. 67 (1991) 3380. [11] E. Lidorkis, M.M. Sigalas, E.N. Economou, C.M. Soukoulis, Phys. Rev. Lett. 81 (1998) 1405. [12] J.D. Joannopoulos, Photonic Crystal: Molding the Flow of Light, Princeton University Press, 1995. [13] G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J.A. Levenson, M.J. Bloemer, J.W. Hause, M. Bertolotti, Phys. Rev. E 64 (2001) 016609.
Optics Communications 281 (2008) 1625–1628 www.elsevier.com/locate/optcom
Effective refractive index analysis of optical Kerr nonlinearity in photonic bandgap structures Jisoo Hwang a, J.W. Wu b,* a
Division of Physical Metrology, Korea Research Institute of Standards and Science, Daejeon 305-340, Republic of Korea b Department of Physics and Division of Nano-sciences, Ewha Womans University, Seoul 120-750, Republic of Korea Received 28 June 2007; received in revised form 5 November 2007; accepted 23 November 2007
Abstract A numerical dispersion relation is employed to analyze the linear and nonlinear optical effective refractive indices of 1-D finite periodic photonic bandgap structures. A Bragg reflector (BR) and a photonic crystal microcavity (PCMC) are examined by assuming that the high indexed layer of the two constituent layers possesses a nonlinear optical response. For the BR, the singularity of refractive index, appearing at bandgap edges in a Bloch index description, is removed. In the case of the PCMC, the optical responses at a defect mode and bandgap edges are properly described, thanks to the use of the numerical dispersion relation. This also allows us to quantitatively compare the Kerr nonlinearity observed at the defect mode and the bandgap edges. The efficiencies of the BR bandgap edge and the PCMC defect mode in achieving a given transmission change are compared by calculating the required nonlinear optical refractive index change. The PCMC defect mode is found to be 1.5 times and twice more efficient than the BR bandgap edge in 20 and 10 dB transmission modulation, respectively. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.70.Qs; 42.79.Dj; 42.65.k; 42.65.Hw Keywords: Photonic bandgap structures; Bragg reflectors; Nonlinear optics; Optical Kerr effect; Dispersion relation
1. Introduction Inside a photonic bandgap (PBG) structure, the wave propagation characteristics of an electromagnetic (EM) wave is governed by the energy-momentum dispersion relation, which is determined from a spatial distribution of refractive indices of constituent periodic layers. For example, the electric field-enhancement at PBG edges is understood in terms of a low group velocity and the anomaly in the dispersion relation. An analytic expression of dispersion relation is readily available in the form of a Bloch index for an infinite periodic PBG structure. On the other hand, in the case of a finite periodic PBG structure, a dispersion relation is numerically obtained from the measured *
Corresponding author. Tel.: +82 23 277 2369; fax: +82 23 277 2372. E-mail address: [email protected] (J.W. Wu).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.067
transmission coefficient. When the PBG structure has to be designed in order to obtain highly efficient generation of harmonics, using nonlinear optical (NLO) processes, the numerical dispersion relation (NDR) can be very useful. In fact, using the NDR allows us properly tailor the PBG characteristics, in order to achieve the phase matching condition [1]. Galisteo-Lopez et al. compared the dispersion relation measured by a white light interferometry and the calculated NDR of a 3-D PBG structure, and found that the agreement was remarkable [2]. The NDR analysis was further extended to obtain an explicit relation between the real and the imaginary components of an EM wave emerging from a PBG structure by use of a Kramers– Kronig relation [3]. While all-optical switching at bandgap edges [4,5] and a defect mode [6,7], originating from optical Kerr related nonlinearity, has been demonstrated experimentally in a PBG structure, NLO analysis of the beam
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J. Hwang, J.W. Wu / Optics Communications 281 (2008) 1625–1628
propagating properties based on the NDR was not pursued, mainly due to the difficulty in obtaining an analytical dispersion relation. In this paper, we utilize the NDR to investigate the Kerr NLO changes in two important 1-D finite periodic photonic bandgap structures, namely, a Bragg reflector (BR) and a photonic crystal microcavity (PCMC). A BR is a 1-D finite periodic photonic bandgap structure without a defect layer, and a PCMC is in a structure of two identical BRs sandwiched symmetrically with a defect layer-between, hence exhibiting an optical defect mode. We report three major findings. First, the singularity of a Bloch index, occurring at bandgap edges of a BR, is removed. Second, the optical properties of a PCMC, based on the dispersion relations of linear and nonlinear refractive indices, are properly described. In particular, optical Kerr nonlinearity is found to be more enhanced at a defect mode than at bandgap edges of a PCMC, which is understood by taking account of the density of modes (DOM) and the EM field localization. Third, the NLO transmission property associated with the optical Kerr nonlinearity is investigated and compared for the BR bandgap edge and the PCMC defect mode. The NLO refractive index changes required to achieve transmission changes from 100% to 10% and 100% to 1% are numerically simulated. 2. Effective refractive indices of a Bragg reflector Originally, the NDR has been developed to overcome the singularity problem of Bloch index occurring at band edges in description of the linear optical properties of a 1-D PBG structure [8]. The Bloch index nB refers to the refractive index of an infinite periodic structure, which is introduced through a Bloch phase b related with nB as nB ¼ bc=ðxKÞ with K the period, and can be obtained from the analytic dispersion relation in a straightforward way [3,9]. On the other hand, NDR is equivalent to introducing an effective refractive index (ERI) in the PBG structure. Once the transmission coefficient is determined for an inhomogeneous optical structure, the ERI is numerically obtained through the relation neff ðxÞ ¼ ðc=xLÞ ½tan1 ðy=xÞ þ mp ði=2Þ lnðx2 þ y 2 Þ, with x and y the real and the imaginary parts of transmission coefficients and L the sample length. In the experimental case, the transmission coefficient is determined through the measurement of both transmittance pffiffiffiffi T and phase /, since they satisfy the relation x þ iy ¼ T ei/ . Or, simply, we can determine ERI from the measured T and L in cooperation with the Kramers–Kronig relation, due to the fact that the real and imaginary parts of ERI satisfy the Kramers–Kronig relation [1]. First of all, in order to clarify the difference between the Bloch index nB and the ERI neff obtained through NDR in describing the linear optical and NLO properties of band edges, we study the transmission spectrum, linear optical and NLO refractive indices of a BR. All the calculation of transmission spectrum and ERI are performed using transfer matrix method [9].
In Fig. 1 are plotted the transmission spectra and the neff of a 10 bilayer BR and nB of an infinite periodic BR, which are composed of quarter-wave-thick layers with the refractive indices of n1 ¼ 1:42 and n2 ¼ 2:08 for low- and highindex layers, respectively. The thickness of each layers was chosen to satisfy the Bragg reflection at 780 nm. A PBG structure is seen in the transmission spectrum of Fig. 1a. The solid curve corresponds to the PBG structure of a 10 bilayer BR, while the dashed curve corresponds to that of an infinite periodic BR. In Fig. 1b and c, the real and imaginary parts of the linear nB are plotted in the dashed curves, while the solid curves correspond to the linear neff of a finite periodic BR. In both nB and neff , we observe that a PBG structure behaves as an optical layer possessing the anomalously dispersive real part and the absorptive imaginary part, which in fact arises from a multiple reflection of EM wave in a transparent periodic optical structure. We note that neff smooths out at the band edges and exhibits an oscillation outside the bandgap. Now let us turn to the NLO refractive indices. The optical Kerr nonlinearity of a BR can be described in terms of ð2Þ ð2Þ the NLO coefficients n1 and n2 of the low- and high-index layers composing the BR. For the optical Kerr nonlinearið2Þ ð2Þ ties of Dn1 ¼ n1 I ¼ 0 and Dn2 ¼ n2 I ¼ 0:05, both irradiance-dependent Bragg index change, DnB , and ERI change, Dneff , are calculated. DnB and Dneff satisfy the relað0Þ ð0Þ tions, nB ðIÞ ¼ nB þ DnB ðIÞ and neff ðIÞ ¼ neff þ Dneff ðIÞ, ð2Þ and I is the incident irradiance. The value of n2 I ¼ 0:05 11 is assumed for NLO material with n2 ¼ 10 cm2/W and
Fig. 1. Numerically calculated (a) transmission spectra, (b) real part of ð0Þ ð0Þ neff , (c) imaginary part of neff , (d) real part of Dneff , and (e) imaginary part of Dneff of BRs. The solid (dashed) curves correspond to a 10 bilayer (infinite) BR. The arrow in (a) indicates the low-frequency edge of 10 bilayer BR, x ¼ 0:844.
J. Hwang, J.W. Wu / Optics Communications 281 (2008) 1625–1628
the irradiance of I ¼ 5 GW/cm2, which is provided by focusing a beam, of a femto-second laser with 200-fs pulse width and 2.5-nJ energy at 400 kHz repetition rate, as 10 lm width. Fig. 1d and e shows that a singularity is present at PBG edges for an infinite BR (dashed curves). In order to look into the singularity of DnB ðIÞ, we get the following expression of DnB ðIÞ: ( !) ð2Þ ð2Þ ð0Þ ð0Þ ðn1 a þ n2 bÞw 1 n2 n1 DnB ðIÞ ’ þ 1þ I; ð1Þ 2 n1ð0Þ n2ð0Þ ðbð0Þ N pÞK where w is the normalized frequency deviation from the ð0Þ center of the stop-band, x0 , i.e., w ¼ n1 aðx x0 Þ=c ¼ ð0Þ n2 bðx x0 Þ=c, and a and b are the thicknesses of lowand high-index layer with indices n1 and n2 , respectively. N is an integer. From Eq. (1), we find that the nonlinear Bloch index diverges at bandgap edges, where bð0Þ ¼ N p. On the other hand, when an NDR is employed for describing a finite periodic BR, the singularity is removed (solid curves). The solid curves correspond to the effective nonlinearity Dneff with the relation. The opposite signs of Im½Dneff at the low- and high-frequency edges are due to the spectral change caused by both red shift and widening of the bandgap, coming from an increase in the optical thickness of high-index layers. 3. Effective refractive indices of a photonic crystal microcavity Next, we look at the optical Kerr response of a defect mode and bandgap edges in a PCMC. We note that the analytic dispersion relation from Bloch theorem cannot be adopted to describe the linear and NLO property of a period-modified Bragg reflector, because the Bloch phase is the eigenvalue of a unit-cell translation matrix, namely, the broken translation symmetry does not allow to adopt the Bloch phase. In dealing with a translation symmetry broken PBG structures, the tight-binding (TB) formalism, originally introduced in solid state physics, has been successfully applied to calculate the frequency spectrum of a defect mode in a Bragg reflector [10,11]. In the TB formalism, the eigenmode is expressed as a linear combination of supercell modes with the defect taken into account, and the TB parameters are obtained by comparing with the experimental dispersion relation. However, the supercells possessing a defect are assumed to be infinitely periodic in order to facilitate the incorporation of Bloch theorem. This implies that a singularity problem still exists in describing the refractive index. On the other hand, the NDR analysis can be extended to the description of the optical Kerr nonlinearity in a PCMC. The NDR derived from the transmission coefficient includes a phase change which the broken translation symmetry incurs. We numerically calculated the ERI of a finite periodic PCMC using transfer matrix method [9], which is structured as a composition of a NLO half-wave defect layer (ndefect ¼ 2:08) between two quarter-wave Bragg mirrors
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of five bilayers (n1 ¼ 1:42, n2 ¼ 2:08) satisfying the Bragg reflection at 780 nm. The solid curve in Fig. 2a shows the simulated transmission spectrum as a function of dimensionless frequency, exhibiting a PBG structure with a defect mode. The dotted curve is DOM derived from dispersion relation [8]. In Fig. 2b, the real and imaginary parts of ð0Þ neff are shown. As expected, the spectral region where the transmission is prohibited possesses the non-vanishing ð0Þ Imfneff g, and two anomalous dispersions appear in the defect mode where transmission is allowed. Dneff is calcuð2Þ lated with nonlinearities of Dn1 ¼ n1 I ¼ 0 and Dn2 ¼ ð2Þ ð2Þ n2 I ¼ Dndefect ¼ ndefect I ¼ 0:05 in Fig. 2c. We observe that the optical Kerr nonlinearity is enhanced at both lowenergy bandgap edge and defect mode. Compared with the spectral region far from the PBG location, where the field localization is not significantly achieved, the defect mode and the bandgap edges give rise to a strong enhancement in the NLO response via a field localization [12]. The enhancement factors of Re½Dneff are determined as 4.7 and 2.5 for the defect mode and the low-energy bandgap edge, respectively. Also, we find that the PCMC possesses the larger enhancement factor, when compared with the 10 bilayer BR dealt with in the previous section. The enhancement factor of Re½Dneff of the 10 bilayer BR is determined as 1.6 for the low-energy bandgap edge. Under the assumption that the field localization is well established, the effective nonlinearity Dneff can be expressed as the following proportional relation [13]: Dneff / q2x DnLoc tLoc ;
ð2Þ
where qx is DOM, and DnLoc and tLoc are Dn and the accumulated sum of thickness of the layers where the EM field is localized, respectively. For example, DnLoc is Dn of high-index (defect) layers for the low-energy bandgap edge (defect mode), and tLoc is the sum of each thickness of corresponding layers. In our case, q ¼ 5:0 and tLoc ¼ 780=2 nm for the
Fig. 2. Numerically simulated (a) transmission spectrum (solid curve) and ð0Þ density of modes (dotted curve), (b) neff , and (c) Dneff of a PCMC. The ð0Þ solid (dashed) curve is real (imaginary) part of neff and Dneff in (b and c).
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defect mode, and q ¼ 1:8 and tLoc ¼ 3900=2 nm for the lower-energy mode. DnLoc is same for both of the modes as Dndefect ¼ Dn2 ¼ 0:05. With these values, the enhancement factor of Dneff at the defect mode is determined as 4.3, which is 1.6 times larger than that of the low-energy bandgap edge. This result is in good agreement with the values determined through NDR above. We note that the enhancement factor increases more rapidly at the defect mode than at the lowenergy bandgap edge as the number of layers of two BRs in a PCMC increases, since the increase in Dneff via DOM wins over via thickness owing to the superlinear dependence of DOM on the number of layers. Similar to the BR, the sign reversal of imaginary part of Dneff in the middle of stop-band indicates that the red shift of defect mode occurs, which originates from the increase in optical thickness of high-index layers as well as a defect layer. 4. Comparison of a bandgap edge and a defect mode NLO refractive index changes required to achieve transmission changes from 100% to 10% and 100% to 1% are numerically simulated for three different 1-D PBG structures, that is, the 10 bilayer BR, the infinite periodic BR, and the PCMC, as a function of optical Kerr nonlinearity of high-index layers, Dn. We pay a particular attention to the NLO transmission changes in a BR bandgap edge and a PCMC defect mode. In Fig. 3 are plotted NLO transmissions for each structure. We find that the transmission change from 100% to 10% occurs at Dn ’ 0:08 for the low-x bandgap edge of the finite BR, and Dn ’ 0:04 for the defect mode of the PCMC, which can be understood in terms of the relation Eq. (2). Similarly, we calculate Dn required for the transmission change from 100% to 1 %, resulting Dn ’ 0:16 for the low-x bandgap edge of the finite BR and Dn ’ 0:11 for the defect mode of the PCMC. These facts lead to the statement that the defect mode of a PCMC is twice (1.5 times) more efficient than the bandgap
Fig. 3. Numerically simulated transmission changes at the low-x bandgap edge (x ¼ 0:844) of a BR and the defect mode (x ¼ 1) of a PCMC. The solid and dashed curves correspond to the defect mode of a PCMC and the low-x bandgap edge of a finite BR, respectively. As a reference, that of the infinite BR is shown as the dotted curve.
edge of a BR in the application of NLO switching for achieving 10 (20) dB modulation. 5. Conclusion In summary, we investigated optical properties of 1-D finite periodic photonic bandgap structures by introducing an effective refractive index through a numerical dispersion relation. The linear and nonlinear optical effective indices of a finite 10 bilayer BR and Bloch indices of a infinite periodic BR are calculated. It is found that ERI description removes the singularities at bandgap edges in a Bloch index description. Further, we investigated a PCMC composed of two identical BRs sandwiched symmetrically with a defect layer-between. A defect layer-between two BRs makes a defect mode appear within photonic bandgap by breaking the translation symmetry. To obtain ERIs of the PCMC, we adopted a numerical dispersion relation, and found that the optical Kerr nonlinearity is enhanced at bandgap edges and a defect mode owing to a field localization. By calculating the refractive index change required to achieve a given transmission change, the efficiencies for a NLO switching application of the BR bandgap edge and the PCMC defect mode were compared, and the PCMC defect mode is found to be twice and 1.5 times more efficient than the BR bandgap edge for 10 and 20 dB modulation, respectively. Acknowledgements This work was supported by Seoul Research and Business Development Program (10816) and the Korea Research Foundation (KRF-2005-214-C00052). References [1] M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M.J. Bloemer, C.M. Bowden, I. Nefedov, Phys. Rev. E 60 (1999) 4891. [2] J.F. Galisteo-Lo´pez, M. Galli, M. Patrini, A. Balesteri, L.C. Andreani, C. Lo´pez, Phys. Rev. B 73 (2006) 125103. [3] Q. Zou, A. Ramdane, B.-E. Benkelfat, Opt. Commun. 230 (2004) 167. [4] D.A. Mazurenko, R. Kerst, J.I. Dijkhuis, A.V. Akimov, V.G. Golubev, D.A. Kurdyukov, A.B. Pevtsov, A.V. Sel’kin, Phys. Rev. Lett. 91 (2003) 213903. [5] J. Hwang, M.J. Kim, J.W. Wu, S.M. Lee, B.K. Rhee, Opt. Lett. 31 (2006) 377. [6] X. Hu, Q. Gong, Y. Liu, B. Cheng, D. Zhang, Appl. Phys. Lett. 87 (2005) 231111. [7] G. Ma, J. Shen, Z. Zhang, Z. Hua, S.H. Tang, Opt. Express 14 (2006) 858. [8] J.M. Bendickson, J.P. Dowling, M. Scalora, Phys. Rev. E 53 (1996) 4107. [9] A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley, 1984. [10] E. Yablonovitch, T.J. Gmitter, R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, Phys. Rev. Lett. 67 (1991) 3380. [11] E. Lidorkis, M.M. Sigalas, E.N. Economou, C.M. Soukoulis, Phys. Rev. Lett. 81 (1998) 1405. [12] J.D. Joannopoulos, Photonic Crystal: Molding the Flow of Light, Princeton University Press, 1995. [13] G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J.A. Levenson, M.J. Bloemer, J.W. Hause, M. Bertolotti, Phys. Rev. E 64 (2001) 016609.