Improving sideband of chiral photonic crystal based on PSTD approach

Optics Communications 279 (2007) 43–49 www.elsevier.com/locate/optcom

Improving sideband of chiral photonic crystal based on PSTD approach Lei Jin, Junqing Li *, Hongming Li, Quanfen Lin, Chunfei Li Department of Physics, Harbin Institute of Technology, 92 Xidazhi Street, Nangang District, Harbin 150001, China Received 11 December 2006; received in revised form 13 June 2007; accepted 19 June 2007

Abstract A pseudospectral time-domain approach (PSTD) is proposed for the first time to treat one-dimensional photonic crystal with bi-isotropic chiral media. It is proved that PSTD method has the computational advantages over finite-difference time-domain (FDTD) method. The switching schemes based on the defect and the sidebands are summarized, and the idea of utilizing the smoothed sideband is also introduced. Compared with several apodization types, the envelop functions are emphasized, which benefits a switch with enough contrast below a certain limited threshold. Ó 2007 Elsevier B.V. All rights reserved.

1. Introduction Photonic crystal (PC) is a new kind of artificial microstructures, the electromagnetic parameters of which present some regularity in the given directions. Generally, the materials composing PC can be dielectrics and semiconductors, or even the metals. One may use different PC structures to design photonic devices, such as filter [1], low threshold semiconductor laser [2], optical diode [3,4], alloptical switch (AOS) [5,6] and so on. Particularly, the ultrafast optical switch is urgently needed for the next generations of all optical communication, and the PC-based switch can be compact and fast, so that it can be easily integrated to future photonic circuit. The one-dimensional (1D) PC switch is considered as the simplest structure that can be easily fabricated. It has been suggested that the Kerr nonlinearity (even two-photon absorption, TPA [7]) can be introduced into a specific PC structures (e.g. the ones with defect) to realize some device applications such as AOS, or optical bistable devices. Self-phase modulation (SPM) and cross-phase modulation (XPM) are the two basic mechanisms that can be used in the nonlinear AOS scheme. In SPM, the sig*

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0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.06.042

nal is modulated by itself. While in XPM, the signal is modulated by the pump. However, it is necessary to discriminate them at the output, e.g. utilizing orthogonal polarization configuration. The introduction of chirality into PC makes it possible to realize the polarization depended devices, such as polarization filter, polarization converter [8], and sideband laser [9], even controllable AOS. Using a finite-different time-domain (FDTD) method, Tran et al. incorporated the chirality into PC switch to relax the requirement of using a polarized beam splitter at the output [10,11]. We also proposed an AOS scheme based on the 1D chiral PC [12]. A typical band structure of PC with defect is shown in Fig. 1, based on which there are two schemes to realize the polarization filter and AOS. After introducing nonlinearity, area A (the defect mode) and area B (the sideband) will shift with the change of light intensity. The light at some frequency in both areas will experience a transition between the upper and the lower state. In fact, there will be some problems if one only uses the two schemes. In area A, for instance, the defect mode can be used to operate switching. The issues lie in the narrowness of the operating band and the difficulty in controlling the upper state. Suppose that the signal is initially located at the lower state. The defect mode is so narrow that it is very hard to confine the signal just at the higher state; if we generate the signal

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In our simulation, the material is chosen as bi-isotropic chiral medium, of which the constitutive equations can be written as: pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi D ¼ eE  jj l0 e0 H; B ¼ jj leE þ lH; ð1Þ where j is the chiral parameter, e and l are the permittivity and permeability, respectively. In such a medium, left-hand circularly polarized (LCP, denoted by ‘‘+’’) light and righthand circularly polarized (RCP, denoted by ‘‘’’) light are the two eigenmodes. The total fields can be decomposed as E = E+ + E and H = H+ + H. The constitutive relations can be derived as: D ¼ e E ;

B ¼ l H ;

ð2Þ

with e± = e(1 ± j/n0) and l± = l(1 ± j/n0). Thus we can get the Maxwell equations that only have E and H [15]: Fig. 1. A typical band structure of PC with defect.

at the higher state first, then after nonlinearity action, the defect mode will move away, which will bring about a falling of the signal into the lower state easily. But it is difficult to fabricate a defect PC switch of which the defect wavelength or frequency coincide with the operation ones. Also, because of the straitness of the defect mode, it is difficult to contain much more channels in future DWDW systems. Although it seems that the side band in area B may overcome the disadvantages above, due to the existence of sidelobes, the upper state still has uncertainty, which hinders implementation of the scheme. So we must suppress the sidelobes to realize high contrast switching. In this paper, to improve the smoothness of sideband, we will make use of the apodization technique, which is applied in filter tailoring of fiber Bragg grating (FBG). A more efficient approach, PSTD method, is utilized, which possesses merits in the speed of iteration computing. We will attempt to use PSTD to treat the 1D PC with bi-isotropic chiral layers, and hope it will be helpful for exploiting the chiral PC devices. 2. Algorithm Since the last decades, the FDTD method [13] has been widely used in electromagnetics and optics. However, to get a higher precision, FDTD needs much more cells per minimum wavelength computing large-scale or fine structure, which results in much more time and memory consumption. The PSTD method, which is based on pseudospectral approach to solve time-domain questions, was first proposed by Liu [14]. With the help of Fourier transform and inverse Fourier transform, PSTD takes advantage of space domain integral transform and inverse spectral domain transform instead of space difference quotient. Because the integral function covers all the space and has no approximation caused by difference quotient like in FDTD, one can get the accuracy as high as expected.

r  H ¼ e

o E ; ot

r  E ¼ l

o H : ot

ð3Þ

To solve Eq. (3) using PSTD method, the spatial derivatives of E and H are obtained from fast Fourier transform (FFT) and inverse fast Fourier transform (iFFT). So we have     n h io 1 1 1 ðxÞ ðxÞ ðyÞ H n þ ¼ H n   iFFT jk z FFT E ðnÞ ; 2 2 l ð4aÞ     n h io 1 1 1 ðyÞ ðyÞ ðxÞ ¼ H n  þ iFFT jk z FFT E ðnÞ ; H n þ 2 2 l ð4bÞ   1 1 ðxÞ ðxÞ ðyÞ ; E ðn þ 1Þ ¼ E ðnÞ  iFFT jk z FFT H  n þ e 2 



ð4cÞ    1 1 ðyÞ ðyÞ ðxÞ : E ðn þ 1Þ ¼ E ðnÞ þ iFFT jk z FFT H  n þ e 2 

ð4dÞ We use the initial-condition technique to model the wave propagation. Suppose that the input wave is Gaussian pulse before the structures with an analytical form at initial time as:

( ðiic Þ2 exp n ðic  ip Þ 6 i 6 ðic þ ip Þ E 0 2 ip E0 ðiÞ ¼ ; 0 others

( ðiic 0:5Dx=cDtÞ2 E0 exp n ðic  ip Þ 6 i 6 ðic þ ip Þ 2 1=2 ip ; H ðiÞ ¼ Z 0 0 others where E0 is the amplitude of the pulse, Z0 is the impedance in free space, ic and ip are the location of the peak and the location where the peak value falls to the value of 0.001E0 (n = ln(0.001)). The term (0.5Dx/cDt) indicates the magnetic field delays behind the electric field by a half time step. It should be noticed that, for the chiral medium, the absorption boundary condition is different from that in the achiral case. We must use the bi-perfect matched layer

L. Jin et al. / Optics Communications 279 (2007) 43–49

(BI-PML) boundary condition to deal with the bi-isotropic chiral medium [16]. 3. Apodization In order to validate our PSTD results on chiral PC, we first use FDTD to compute the 1D chiral PC with 28 layers as a comparison. All the high index layers consist of the

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chiral material whose chiral parameters are with the value j = 0.04. The structure contains 4 layers with high refractive index in the centre as the defect. The average refractive index of the high and the low refractive index is 1.5 and 1.0, respectively. The low index layer’s length is k0/4, and the other layer’s is kn/4(kn = k0/1.5, where k0 = 1550 nm is the reference wavelength). Fig. 2 illustrates the band structures of chiral PC without and with defect, and the field

Fig. 2. The comparison of chiral PC (j = 0.04) between FDTD and PSTD. The solid curve and the hexagrams correspond to the cases of FDTD and PSTD respectively. (a)/(b) and (c)/(d) correspond to the band structure of LCP/RCP cases without and with defect. x/x0 is the ratio of the frequency located in a certain place (corresponding to k) to the reference frequency (corresponding to k). (e)/(f) correspond to the field intensity inside the PC with defect.

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Fig. 3. Bandgap structure of chiral PC with stair apodization.

distribution inside corresponding to the case with defect. The solid lines and the hexagrams in Fig. 2a–d correspond to the calculated results by FDTD and by PSTD, respectively. It can be seen that the deviation of the results in PSTD from that in FDTD is small enough. From the figures, the bandgap structure separation for RCP and LCP is observed clearly in both simulations. We find that when the time step is fixed, both methods will give almost the same results, if the cells per minimum wavelength are chosen as 16 for FDTD and 2 for PSTD, respectively. It is estimated that with the same accuracy, the cells needed in PSTD are about 1/8 of that in FDTD. Therefore, the PSTD method may occupy about 1/8D memory and CPU time (D is the dimension of the issue, e.g for two-dimensional case, D = 2) to accomplish the same work that FDTD does. All the investigations below will be based on PSTD method. In general, the transmission fluctuating with frequency in the sidebands (sidelobes) originates from the weak reflection formed by paired layers at two ends, at which the refractive index changes sharply compared to the region outside PC. To remove the sidelobes, we will utilize an apodization technique such as enveloping refractive index to smooth the sidebands. All the solid and dashed curves in the figures below indicate the bandgap structures for

LCP and RCP. The basic parameters, such as indices and chiral parameters are the same with those in Fig. 2. Firstly, we treat the PC of 28-layer with defect, which has been mentioned in our recent work where FDTD is used [12]. A simple way is to use stair apodization, as shown in Fig. 3a. The apodization is only imposed on the first three and last three high index layers. In Fig. 3b–d, the refractive index distributions in the apodized areas are 1.15/1.3/1.45, 1.1/1.2/1.3 and 1.2/1.3/ 1.4. It seems that the apodization acts on the sideband of low frequency better than that of high frequency if we treat PC symmetrically. In comparison, the effect suppressing sidelobes in Fig. 3b is the best from the three, which makes the low frequency sideband quite flattened. In spite of the effectiveness, the stair apodization does not appear powerful enough to suppress the sidelobes. As one of the alternative methods, some specific function which is enveloping apodized area may be more efficient. In this case the high refractive index varies gradually according to a specific function. The line function is a simpler one to apodize the PC, as is seen in the Fig. 4. However, it is important to know how many apodized layers are suitable for a 28-layer PC. We choose the first and last 1, 2, 3 and 4 layers to compare the apodization effects. Compared with unapodizated cases, a more or fewer num-

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Fig. 4. The bandgap of linear apodized PC, where the apodization function is line function: nap(z) = 1 + 0.5(z  z0)/(z1  z0) on the left and nap(z) = 1.5 + 0.5(z  z2)/(z2  z3) on the right.

ber of apodized layers will lead to undesirable effects. More apodized layers will benefit the smoothness of sideband, but it will lead the deterioration of the transmission gradient between the stopband and the sideband. On the contrary, the fewer apodized layers make the gradient steeper, but the sidelobes are not smoothed completely. There is a tradeoff between the smoothness and the gradient. It should be emphasized that for AOS, the gradient is of great importance. The larger the gradient, the lower the switching threshold. From a comprehensive consideration, 3 apodized layers are suitable for a 28-layer PC (see the PC structure in Fig. 4e). We can also use the nonlinear functions to apodize. Fig. 5 describes the cases of choosing elliptic function (a quarter of the circumference) and Gaussian function as envelope, in which the apodization area still involves three layers at two PC ends. From the figures, it can be found that the sidelobes are suppressed further.

4. Discussion From the simulations above, the apodization not only suppresses the sidelobes but also influences the defect mode’s full-width at half-maximum. Whichever function is chosen, the defect mode is broadened to a certain degree. It does not matter if we only use the sideband to realize the switching. More importantly, in the switching scheme based on the apodized sideband, the defect will react on the apodization effect, such as on suppressing sidelobes and sharpening gradient. The existence of defect may sharpen the gradient but weaken the apodization. In practice, a switching with enough contrast is easy to realize even when there exists a weak switching fluctuation caused by the incompleteness in apodization of the upper state. Therefore sometimes we can sacrifice the smoothness of sideband to ensure enough gradient in order to keep the switching threshold within a certain limit.

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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 5. The bandgap of quarter circumferenceffi apodized PC, where (a) the apodization function is: nap ðzÞ ¼ 1:5  0:5 1  ðz  z0 Þ2 =ðz1  z0 Þ2 on the left qaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

and nap ðzÞ ¼ 1:5  0:5 1  ðz  z3 Þ2 =ðz2  z3 Þ2 on the right, and (b) is the corresponding bandgap structure. The bandgap of Gaussian apodized PC, where (c) the apodization function is: nL(z) = 1 + 0.5 exp[2(1.5 + (z  z0)/(z1  z0))2] on the left and nR(z) = 1 + 0.5 exp[2(1.5 + (z  z3)/(z2  z3))2] on the right (d) is the corresponding bandgap structure.

For illustrative purposes, we also give the bandgap structures of the PC without defect apodized by the four types of apodizing functions corresponding to the case of stair apodization, linear function apodization, elliptic function apodization and Gaussian function apodization (Fig. 6). One can see that in spite of the excellent smoothness in the case of the PC without defect, the gradient worsens, which does not benefit the switching. So we prefer using the PC with defect to realize AOS. Fig. 6 The case of 28-layer without defect, where in (a)/(b)/(c)/(d) corresponds the bandgap structure of chiral PC with stair apodization/linear function apodization/ quarter elliptic function apodization/Gaussian function apodization. In application, there are many techniques can be used to realize the continuous changes in refractive index, such as glancing angle deposition and matrix-assisted pulsed laser evaporation [17,18]. However, in our cases, for the solid media doped with the chiral molecules, it seems that one can control the gradient of dopant concentration to make the refractive index changing according to a certain spatial function.

5. Conclusion In summary, we have proposed a novel approach, PSTD method, to treat the 1D PC with bi-isotropic chiral media. It is proved that this method is more practical and effective compared with traditional FDTD method. At the same accuracy PSTD reduces the computing time by almost an order of magnitude. Several apodization treatments, such as stair apodization and function envelop apodization, have been implemented to suppressing the sidelobes that do not benefit the switching at the sideband. It can be concluded that a proper apodization function can be chosen to improve the sideband, which is helpful to realize a switching with enough contrast while keep the switching threshold within a certain limit. Acknowledgements The authors appreciate the support from the Natural Science Foundation of Heilongjiang Province under Grant No. A200507.

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Fig. 6. The case of 28-layer without defect, where in (a)/(b)/(c)/(d) corresponds the bandgap structure of chiral PC with stair apodization/linear function apodization/quarter elliptic function apodization/Gaussian function.

References [1] H.L. Ren, C. Jiang, W.S. Hu, M.Y. Gao, J.Y. Wang, Opt. Commun. 266 (2006) 342. [2] K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, K. Busch, Appl. Phys. B 82 (2006) 539. [3] M.W. Feise, I.V. Shadrivov, Y.S. Kivshar, Appl. Phys. Lett. 85 (2004) 1451. [4] M.W. Feise, I.V. Shadrivov, Y.S. Kivshar, Phys. Rev. E 71 (2005) 037602. [5] F. Cuesta-Soto, A. Martinez, J. Garcia, F. Ramos, P. Sanchis, J. Blasco, J. Marti, Opt. Express 12 (2004) 161. [6] G. Priem, I. Notebaert, B. Maes, P. Bienstman, G. Morthier, R. Baets, IEEE J. Sel. Top. Quant. Electron. 10 (2004) 1070. [7] G. Priem, P. Bienstman, G. Morthier, R. Baets, J. Appl. Phys. 99 (2006) 063103. [8] H. Satoh, N. Yoshida, Y. Miyanaga, Electr. Eng. Jpn. 152 (2005) 7.

[9] V.I. Kopp, Z.Q. Zhang, A.Z. Genack, Prog. Quant. Electron. 27 (2003) 369. [10] P. Tran, J. Opt. Soc. Am. B 16 (1999) 70. [11] L. Gilles, P. Tran, J. Opt. Soc. Am. B 19 (2002) 630; W.S. Weiglhofer, A. Lakhtakia, M.W. McCall, J. Opt. Soc. Am. B 19 (2002) 3042. [12] J.Q. Li, L. Jin, L. Li, C.F. Li, IEEE Photo. Technol. Lett. 18 (2006) 1261. [13] K.S. Kunz, R.J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC Press, Boca Raton, FL, 1993. [14] Q.H. Liu, Microw. Opt. Technol. Lett. 15 (1997) 158. [15] A. Akyurtlu, D.H. Werner, K. Aydin, Microw. Opt. Technol. Lett. 26 (2000) 239. [16] S.G. Garcia, I.V. Perez, R. Martin, B. Olmedo, IEEE Microw. Guieded W. Lett. 8 (1998) 297. [17] S.R. Kennedy, M.J. Brett, Appl. Opt. 42 (2003) 4573. [18] K. Rodrigo, P. Czuba, B. Toftmann, J. Schou, R. Pedrys, Appl. Surf. Sci. 252 (2006) 4824.