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Design of ultrahigh-contrast all-optical diodes based on coupled nonlinear photonic crystal defects
Optics Communications 285 (2012) 1959–1963
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Design of ultrahigh-contrast all-optical diodes based on coupled nonlinear photonic crystal defects Xuhong Cai, Xiaofan Wang, Shaohui Li ⁎ Department of physics, Shantou University, Shantou 515063, P.R. China
a r t i c l e
i n f o
Article history: Received 6 July 2011 Received in revised form 5 December 2011 Accepted 7 December 2011 Available online 20 December 2011 Keywords: All-optical diodes Optical bistability Photonic crystal Coupled cavities
a b s t r a c t We design and investigate the unidirectional transmission behavior of coupled photonic crystal defects with instantaneous Kerr nonlinearity. Theoretical analysis based on coupled-mode theory and numerical simulations based on finite-difference time-domain techniques indicate that ultrahigh-contrast all-optical diodes can be achieved by properly misaligning the frequencies of the coupled cavities. The transmission contrasts of such configurations composed of three and four cavities are demonstrated to be as high as 518 and 7015, with 42.8% and 46.3% forward transmissions, respectively. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In recent years, all-optical diodes have attracted much attention for their potential application in integrated photonic circuits, alloptical signal processing, and telecommunications in the future. Since Scalora et al. suggested using one-dimensional nonlinear photonic crystals (PCs) with a spatial graduation in the linear refractive index to realize unidirectional propagation of signals in 1994 [1], various schemes have been proposed to construct compact and highly efficient all-optical diodes [2–9], For example, Feise et al. demonstrated the bistable diode action in an asymmetric multilayer structure consisting of left-handed materials [3]. Philip et al. studied numerically and verified experimentally the passive all-optical diode behavior utilizing asymmetric nonlinear absorption [4]. Configurations of a PC waveguide with embedded nonlinear PC defects and with asymmetric defect pair are found to display nonreciprocal effects as well [5–8]. However, the achieved transmittance contrasts of all-optical diodes in above schemes are all lower than 100. In order to improve the transmission contrast, very recently, Xue et al. suggested using one-dimensional PC-metal heterostructures to achieve highly efficient all-optical diode action [10], and Hu et al. proposed a strategy for obtaining ultrahigh-contrast all-optical diodes based on tunable surface Plasmon polaritons [11]. However, the transmission contrast of the former one is only about 124, still not high enough. For the latter one, although the transmission contrast
⁎ Corresponding author. Tel.: + 86 754 82902067; fax: + 86 754 82902767. E-mail address: [email protected] (S. Li). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.034
ratio can be as high as 2166, the forward transmittance is only 0.06, which is too low to be used in the practical applications. Recently, we have studied the transmission property of two directly coupled nonlinear defects and proposed that it can function as an all-optical bistable switch or diode with high transmission contrast if we deliberately and properly misalign the resonant frequencies of the two defects [12]. The application of this kind of configuration in optical switching has already been demonstrated [13]. In this paper, we design photonic crystal structures composed of three or four nonlinear defects. We prove that all-optical diodes can be realized for this configuration and the transmission contrast can be further increased by selecting the resonant frequencies of the constitutional defects properly. 2. Unidirectional transmission behavior of coupled cavities For an ideal diode, its forward transmission should be 100%, while it vanishes for backward propagation. From the view point of practical application, the key to achieve a highly nonreciprocal transmission diode is to make its backward transmission as low as possible, and meanwhile, keep the forward transmission at a relatively high level. We first pay attention to the forward transmission of the coupled defects. For a PC molecule composed of two identical PC defects, the energy trapped in the defect region of the first PC defect which is adjacent to the input port is always larger than that of the second one [12]. Therefore, the shift of the resonant frequency will be larger for the first defect compared to that of the second one because the shift of the resonant mode is proportional to the energy in the defect [14]. But if we misaligned the resonant frequencies by changing the defect sizes appropriately, setting the frequency of the
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first defect a little larger than that of the second one, it is possible to make the resonant frequencies of the two PC atoms shift to a same value, i.e., a new common resonant frequency is reached for the two PC defects. By this way, the nonlinear transmission spectrum will change from multistability to the bistability, and a relatively high transmittance can be reached [12,13]. This method can be used to a structure composed of three or four defects. Fig. 1 shows the linear transmission spectra of a structure composed of three identical defects with an arbitrary given resonant frequency ω0 and a linewidth γ. The results are obtained from coupled-mode theory (CMT) [15], which has been proved to be an efficient and reliable way dealing with the coupling of defects [16]. It can be seen clearly that for such a configuration, the transmission spectra depend on the phase shift φ of the incident signal between two adjacent defects. If the phase shift is chosen to be 90°, its transmission spectrum has a quasi-flat top pass band, similar to that of two-defect structures [13]. This indicates that for a configuration consisting of three defects with different resonant frequencies, if their resonant frequencies are properly arranged and can be moved to a same position under certain incident power in nonlinear situation, the nonlinear transmission spectrum could possess the property of bistability, similar to that of the two-defect structures. Then, a relatively high transmission can be expected. Consider now the backward transmission of such three coupled defects. For the structure of two coupled defects, we know that if the incident signal is from the opposite direction, the low defect mode is near the input port and the larger shift of this mode making the deviation between the resonant frequencies of the two cavities enlarged. Enlarged difference of the resonant frequencies will lead to a lower transmission of the structure. In Fig. 2, the theoretical transmission spectra of both a two-defect and a three-defect coupling structures are presented. As an example, the resonant modes of the two-defect system are set to be ω1, 2 = ω0 ± γ and with a same linewidth γ, i.e., the difference of the resonant frequencies are chosen to be 2γ. The phase shift φ is set to be 90°. The corresponding three-defect system is constructed by inserting a defect with resonant frequency ω0 in between the two defects. Compared with the twodefect structure, we can see that although the three-defect structure has a near 100% transmission at the frequency ω0, its transmission decreases sharply and becomes much lower than that of the two-defect structure outside the central frequency region. This transmission characteristic keeps almost unchanged when we adjust the relative value of the three resonant frequencies in a limited range. We know that for a PC optical diode, to ensure a high forward transmission, the incident frequency should be lower than the resonant frequencies of all the defects, which means that the backward transmission of the three-defect structure will be much lower than that of the two-defect structure as shown in Fig. 2. Therefore, a much higher transmission contrast can be expected.
Fig. 1. Theoretical linear transmission spectra of a PC structure consisting of three identical defects for some different phase shifts.
Fig. 2. Theoretical linear transmission spectra of PC structures consisting of two or three defects with different resonant frequencies but same linewidths.
3. Numerical verification of all-optical diode performances We now design a nonlinear PC diode with three defects based on above analysis. To realize all-optical diode operation, the resonant frequencies of the constitutional defects should shift to a same position with increasing input power in nonlinear regime. Since the shift of the resonant frequencies is proportional to the energy trapped in the defects which can be obtained from the CMT, the relative position of the original resonant frequencies of the defects can be roughly determined [14]. The structure of such a diode is schematically shown in Fig. 3(a). Without loss of generality, we use a two-dimensional (2D) waveguide structure which is commonly used to study the transmission property of a PC device theoretically and experimentally [17,18]. The 2D waveguide plate is made of a material with Kerr nonlinearity. The linear refractive index and the nonlinear coefficient for the Kerr material are set to be no = 3.37 and n2 = 1.5 × 10 − 5μm 2/W, which is close to that of GaAs. Eighteen identical air holes are drilled in the 2D waveguide, the lattice constant, i.e., the distance between
Fig. 3. (a) Geometry of a PC diode configuration composed of three defects. (b) Linear transmission spectra of three single defects A1, A2, and A3, respectively. (c) Linear transmission spectra of the coupled defects composed of defects A1 + A1 + A1, A2 + A2 + A2, and A3 + A3 + A3, respectively.
X. Cai et al. / Optics Communications 285 (2012) 1959–1963
the neighboring air holes is a = 0.38μm. The radius of the air holes is 0.25a and the width of the waveguide is a. The three defects which are denoted as defects A1, A2, and A3 are introduced by increasing the separation between the third and fourth air holes and that between the ninth and the tenth air holes, as well as that between the fifteen and sixteen air holes. The size of the three defects is selected to be d1 = 1.300a, d2 = 1.314a and d3 = 1.324a, respectively, as marked in Fig. 3(a). The finite-difference time-domain (FDTD) technique is employed to simulate the linear and nonlinear transmission behavior (http://www.rsoftdesign.com). In the numerical simulations, we choose to simulate the transverse magnetic (TM) mode of the 2D structure. The simulation domain is set to be 3a in the lateral direction, and to cover the whole waveguide in the propagation direction. The grid size used for both directions and the boundary width of the perfectly matched layer are set to be a/16 and a, respectively. The type of the excitation continuous wave (CW) source is slab mode and the width is equal to the width of the waveguide. The transmission is defined as the ratio of the output power to the input power. The linear transmission spectra of the three defects A1, A2, and A3 and of the coupling defect structures consisting of Ai + Ai + Ai (i = 1,2,3) obtained by FDTD are shown in Fig. 3(b) and (c), respectively. Due to the difference of the sizes, the resonant frequencies of the defects are located at ω1 = 0.24124(2πc/a), ω2 = 0.23962(2πc/a) and ω3 = 0.23872(2πc/a), respectively, and the linewidths of them are measured to be γ1 ≈ γ2 ≈ γ3 ≈ 0.00132(2πc/a). The transmission spectra in Fig. 3(c) reveal that the three structures all have a quasiflat top transmission spectrum, indicating that the phase shift between any two adjacent defects is nearly 90°, the influence of small difference of the defect sizes can be ignored. Then, for the structure composed of defects A1 + A2 + A3 as shown in Fig. 3(a), if the resonant frequencies of the defects can be moved to a same position in the nonlinear situation, its forward transmission (rightward) can be expected to have a high transmission. When the signal incidents from right to left (leftward), the defect A3 is adjacent to the input port, more energy will be trapped in the defect compared with other two defects, which will make the resonant frequency of defect A3 shift more than that of defects A1 and A2, and the different shift of the resonant mode will lead to the increase of ω2 − ω3 and ω1 − ω3. From Fig. 3(b) we know that the original difference of the resonant frequencies of the three defects is ω1 − ω2 = 1.23γ, ω2 − ω3 = 0.68γ, and ω1 − ω3 = 1.91γ, respectively, the increase of ω2 − ω3 combined with the increase of ω1 − ω3, will further lower the transmission in the backward direction. Therefore, the PC structure in Fig. 3(a) is expected to have a good performance as an alloptical diode. The FDTD simulation results verify above analysis. In Fig. 4, the nonlinear transmission spectra of the system are presented. It demonstrated very clearly that the three-defect structure displays an excellent unidirectional transmission. For three incident frequencies
Fig. 4. FDTD simulation results of the nonlinear transmission spectra of Fig. 3(a) for some different incident frequencies. The solid and open squares (circles, triangles) denote the rightward and leftward transmission, respectively.
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ω = 0.23731, 0.23712 and 0.23693(2πc/a), at incident power Pin = 35.5, 38.0, and 40.5W/μm, which is often called the threshold power, the rightward transmissions experience an up-jump transition, respectively. After the transition, the transmission changes from very low values to 0.473, 0.452 and 0.428, respectively. For the leftward propagation, the transmissions of the three incident frequencies at the threshold powers are 1.98 × 10 − 3, 1.40 × 10 − 3, and 8.25 × 10 − 4, respectively. The transmission contrasts for the three incident frequencies are calculated to be C = 237, 321 and 518, respectively. We have also studied the transmission as a function of incident frequency. In Fig. 5, the transmittances in both the rightward and leftward directions at a fixed input power P = 35W/μm are presented. For the rightward transmission, the transmittance transits from a much lower value to 0.422 at an incident frequency ω = 0.2370(2πc/a), and then decreases slowly with the increasing frequency. The leftward transmittance maintains at a very low level and increases slowly with the increasing frequency until it transits to a relatively higher value at frequency ω = 0.2377(2πc/a). The transmission contrasts at the two up-jump transition frequencies are 382 and 103, respectively. In-between the frequencies ω = 0.2370 and 0.2377(2πc/a), the PC structure can possess a high forward transmittance and a large transmission contrast simultaneously. If we alter the input power, the corresponding frequency region, the forward transmittance and the contrast will change accordingly. Compared with the transmission of a CW, the transmission of a short pulse through the PC configuration is more practical and convincing. The curves in Fig. 6 show the output pulses of an input Gaussian pulse in two launch directions. The central frequency, width and peak power of the incident Gaussian pulse are selected to be 0.2375(2πc/a), 3 ps and 35W/μm, respectively. It can be seen clearly that transmission of the short pulse is similar to that of a CW. For the rightward transmission, only when the power in the leading edge of the input pulse reaches the threshold power, the resonant frequencies of the three defects are shifting near a same position, does the transmittance increases sharply. In the trailing edge of the input pulse, with the decrease of input power, the resonant frequencies return to their original values and separate again, the transmittance falls sharply accordingly. Therefore, the output pulse becomes narrower in pulse width compared with the input pulse. On the contrary, the input pulse in the leftward direction enlarges the difference of the resonant frequencies of the three defects, which lowers the leftward transmittance and makes the output width to be enlarged. If we define the ratio of the peak power of the two output pulses in two launch direction as the transmission contrast, the contrast is calculated to be 157, which is similar to that in CW case. We have tried to extend this method to four-defect PC structure. In Fig. 7, we show the transmission results of a similar PC coupling structure as Fig. 3(a) but with four defects. The sizes of the four constitutional defects from left to right are selected to be d1 = 1.296a,
Fig. 5. Dependence of the transmittance on the incident frequency for two launch directions.
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Fig. 6. Output pulses for two launch directions of an input Gaussian pulse with central frequency, pulse width and peak power being 0.2375(2πc/a), 3ps and 35W/μm, respectively.
d2 = 1.307a, d3 = 1.311a and d4 = 1.317a, respectively. The defect sizes are similar to that in Fig. 3(a), therefore, the four defects should have different resonant modes but almost the same linewidths, and the phase shift between any two adjacent defects of the system can be taken as near 90°. For this system, the forward transmission of an incident frequency ω = 0.23750(2πc/a) transits at threshold power Pin = 38.4W/μm, and the transmission jumps to 0.463. Meanwhile, the backward transmission at the threshold power is only 6.6 × 10 − 5, the corresponding transmission contrast is as high as 7015. From above simulation results, we have verified that excellent performance as an optical diode can be achieved for a PC structure composed of three or four properly selected defects. The combination of the defects of such PC structure is not unique. We found that in a small range of the defect size, the resonant frequency has almost a linear relationship with the defect size. The coupled defects can also function as a diode if we make a similar alteration to the resonant frequencies by changing the sizes of the defects. The transmission contrasts may change a little due to the difference of the structure, but they are larger than that of the two-defect PC coupling structure and that of other schemes. In many cases, the enhancement of transmission contrast is achieved by sacrificing the transmission, just as the case of [11]. If we use the product of transmission contrast and the threshold transmission as a figure of merit to evaluate the performance of an optical diode as suggested in Ref. [6], the figure of merit in our case is calculated to be 222 and 3248 for the three-defect structure and four-defect structure, respectively. It reveals clearly that the efficiency of the diode action is greatly enhanced as compared with 12.67, the figure of merit of the configuration of an asymmetric defect pair [6], 52 of the one-dimensional PC-metal heterostructures [10], and 130 of the structure using tunable surface Plasmon polaritons in a silver grating coated with a nonlinear organic material in [11].
It should be pointed out that the high transmission contrast of the multi-defect PC structure is realized via the shift of the resonant frequencies of the defects, the threshold power of the multi-defect structure is mainly determined by the constitutional defects. We know that the shift of the resonant frequency of a nonlinear defect is due to the change of the refractive index in the defect, which is caused by the electric field trapped in the cavity, and can be expressed to be Δn ∝ n2E 2. Here, the n2 and E are the nonlinear coefficient of the material and the amplitude of the electric field in the cavity, respectively. Therefore, there are two ways to lower the threshold power, and accordingly, lower the light intensity. One is to utilize a material with higher nonlinear coefficient, the other is to use defects with high quality factor Q. For a same variation of the refractive index, a high-Q defect needs less input power compared with that of a low-Q defect. And furthermore, since the quality factor Q is in inverse proportion to the linewidth of the linear transmission spectrum of the defect, higher Q indicates less linewidth. In general, the shift of the resonant frequency of the defect in nonlinear situation is several linewidths of the defect, less linewidth means less shift of the resonant frequency, and accordingly, less variation of the refractive index in the defect is needed for high-Q defect, which will finally result in further decrease of the required input power. It has been demonstrated experimentally that with high-Q photonic crystal nanocavities, the required power density or energy to realize the bistable action can be greatly reduced [19,20]. Because there is no contradiction between the method of using coupled defects to increase the transmission contrast and the method of using high-Q to decrease the required input power, it is reasonable to believe that if high-Q defects are employed in the configuration, low threshold power density and high transmission contrast can be realized simultaneously. 4. Conclusion In summary, we have investigated the unidirectional transmission behavior of PC structures composed of three or four nonlinear defects. Theoretical analysis based on CMT indicates that the transmission contrast of such a three-defect structure can be an order higher than that of a two-defect structure. The FDTD simulation results show that the contrast of a three-defect structure can be as high as 518, which is in good agreement with the theoretical prediction. Furthermore, the simulation result of a four-defect structure reveals that the transmission contrast up to 7015 can be achieved with the forward transmission exceeds 46%. When the input power exceeds the threshold power for a given incident frequency, the forward transmittance maintains at a relatively higher level and decreases slowly with the input power, which is beneficial to the transmission of intensity-modulated signals. Except the excellent performances of diode action in CW case, the coupled structure exhibits unidirectional transmission for a short pulse as well. The coupled defect structure of our design is compact and we expect it to be useful in the application of all-optical signal processing. Acknowledgements This work was supported by the program of Research Foundation of Shantou University under Grant No. YR08005 and No. NTF10020. References
Fig. 7. FDTD simulation results of the nonlinear transmission spectra of a system composed of four distinct defects.
[1] M. Scalora, J.P. Dowling, C.M. Bowden, M.J. Bloemer, Journal of Applied Physics 76 (1994) 2023. [2] K. Gallo, G. Assanto, K.R. Parameswaran, M.M. Fejer, Applied Physics Letters 79 (2001) 314. [3] M.W. Feise, I.V. Shadrivov, Y.S. Kivshar, Physical Review E 71 (2005) 037602. [4] R. Philip, M. Anija, C.S. Yelleswarapu, D.V.G.L.N. Rao, Applied Physics Letters 91 (2007) 141118. [5] S.F. Mingaleev, Y.S. Kivshar, Journal of the Optical Society of America B: Optical Physics 19 (2002) 2241.
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[15] H.A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, Englewood Cliffs, N. J., 1984 [16] M.F. Yanik, S. Fan, M. Soljačić, Applied Physics Letters 83 (2003) 2739. [17] M. Belotti, J.F. Galisteo-Lόpez, S.D. Angelis, M. Galli, I. Maksymov, L.C. Andreani, D. Peyrade, Y. Chen, Optics Express 16 (2008) 11624. [18] J.S. Foresi, P.R. Villeneuve, J. Ferrera, E.R. Thoen, G. Steinmeyer, S. Fan, J.D. Joannopoulos, L.C. Kimerling, H.I. Smith, E.P. Ippen, Nature 390 (1997) 143. [19] M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, T. Yanabe, Optics Express 13 (2005) 2678. [20] X.D. Yang, C. Husko, C.W. Wong, M.B. Yu, D.L. Kwong, Applied Physics Letters 91 (2007) 051113.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Design of ultrahigh-contrast all-optical diodes based on coupled nonlinear photonic crystal defects Xuhong Cai, Xiaofan Wang, Shaohui Li ⁎ Department of physics, Shantou University, Shantou 515063, P.R. China
a r t i c l e
i n f o
Article history: Received 6 July 2011 Received in revised form 5 December 2011 Accepted 7 December 2011 Available online 20 December 2011 Keywords: All-optical diodes Optical bistability Photonic crystal Coupled cavities
a b s t r a c t We design and investigate the unidirectional transmission behavior of coupled photonic crystal defects with instantaneous Kerr nonlinearity. Theoretical analysis based on coupled-mode theory and numerical simulations based on finite-difference time-domain techniques indicate that ultrahigh-contrast all-optical diodes can be achieved by properly misaligning the frequencies of the coupled cavities. The transmission contrasts of such configurations composed of three and four cavities are demonstrated to be as high as 518 and 7015, with 42.8% and 46.3% forward transmissions, respectively. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In recent years, all-optical diodes have attracted much attention for their potential application in integrated photonic circuits, alloptical signal processing, and telecommunications in the future. Since Scalora et al. suggested using one-dimensional nonlinear photonic crystals (PCs) with a spatial graduation in the linear refractive index to realize unidirectional propagation of signals in 1994 [1], various schemes have been proposed to construct compact and highly efficient all-optical diodes [2–9], For example, Feise et al. demonstrated the bistable diode action in an asymmetric multilayer structure consisting of left-handed materials [3]. Philip et al. studied numerically and verified experimentally the passive all-optical diode behavior utilizing asymmetric nonlinear absorption [4]. Configurations of a PC waveguide with embedded nonlinear PC defects and with asymmetric defect pair are found to display nonreciprocal effects as well [5–8]. However, the achieved transmittance contrasts of all-optical diodes in above schemes are all lower than 100. In order to improve the transmission contrast, very recently, Xue et al. suggested using one-dimensional PC-metal heterostructures to achieve highly efficient all-optical diode action [10], and Hu et al. proposed a strategy for obtaining ultrahigh-contrast all-optical diodes based on tunable surface Plasmon polaritons [11]. However, the transmission contrast of the former one is only about 124, still not high enough. For the latter one, although the transmission contrast
⁎ Corresponding author. Tel.: + 86 754 82902067; fax: + 86 754 82902767. E-mail address: [email protected] (S. Li). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.034
ratio can be as high as 2166, the forward transmittance is only 0.06, which is too low to be used in the practical applications. Recently, we have studied the transmission property of two directly coupled nonlinear defects and proposed that it can function as an all-optical bistable switch or diode with high transmission contrast if we deliberately and properly misalign the resonant frequencies of the two defects [12]. The application of this kind of configuration in optical switching has already been demonstrated [13]. In this paper, we design photonic crystal structures composed of three or four nonlinear defects. We prove that all-optical diodes can be realized for this configuration and the transmission contrast can be further increased by selecting the resonant frequencies of the constitutional defects properly. 2. Unidirectional transmission behavior of coupled cavities For an ideal diode, its forward transmission should be 100%, while it vanishes for backward propagation. From the view point of practical application, the key to achieve a highly nonreciprocal transmission diode is to make its backward transmission as low as possible, and meanwhile, keep the forward transmission at a relatively high level. We first pay attention to the forward transmission of the coupled defects. For a PC molecule composed of two identical PC defects, the energy trapped in the defect region of the first PC defect which is adjacent to the input port is always larger than that of the second one [12]. Therefore, the shift of the resonant frequency will be larger for the first defect compared to that of the second one because the shift of the resonant mode is proportional to the energy in the defect [14]. But if we misaligned the resonant frequencies by changing the defect sizes appropriately, setting the frequency of the
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first defect a little larger than that of the second one, it is possible to make the resonant frequencies of the two PC atoms shift to a same value, i.e., a new common resonant frequency is reached for the two PC defects. By this way, the nonlinear transmission spectrum will change from multistability to the bistability, and a relatively high transmittance can be reached [12,13]. This method can be used to a structure composed of three or four defects. Fig. 1 shows the linear transmission spectra of a structure composed of three identical defects with an arbitrary given resonant frequency ω0 and a linewidth γ. The results are obtained from coupled-mode theory (CMT) [15], which has been proved to be an efficient and reliable way dealing with the coupling of defects [16]. It can be seen clearly that for such a configuration, the transmission spectra depend on the phase shift φ of the incident signal between two adjacent defects. If the phase shift is chosen to be 90°, its transmission spectrum has a quasi-flat top pass band, similar to that of two-defect structures [13]. This indicates that for a configuration consisting of three defects with different resonant frequencies, if their resonant frequencies are properly arranged and can be moved to a same position under certain incident power in nonlinear situation, the nonlinear transmission spectrum could possess the property of bistability, similar to that of the two-defect structures. Then, a relatively high transmission can be expected. Consider now the backward transmission of such three coupled defects. For the structure of two coupled defects, we know that if the incident signal is from the opposite direction, the low defect mode is near the input port and the larger shift of this mode making the deviation between the resonant frequencies of the two cavities enlarged. Enlarged difference of the resonant frequencies will lead to a lower transmission of the structure. In Fig. 2, the theoretical transmission spectra of both a two-defect and a three-defect coupling structures are presented. As an example, the resonant modes of the two-defect system are set to be ω1, 2 = ω0 ± γ and with a same linewidth γ, i.e., the difference of the resonant frequencies are chosen to be 2γ. The phase shift φ is set to be 90°. The corresponding three-defect system is constructed by inserting a defect with resonant frequency ω0 in between the two defects. Compared with the twodefect structure, we can see that although the three-defect structure has a near 100% transmission at the frequency ω0, its transmission decreases sharply and becomes much lower than that of the two-defect structure outside the central frequency region. This transmission characteristic keeps almost unchanged when we adjust the relative value of the three resonant frequencies in a limited range. We know that for a PC optical diode, to ensure a high forward transmission, the incident frequency should be lower than the resonant frequencies of all the defects, which means that the backward transmission of the three-defect structure will be much lower than that of the two-defect structure as shown in Fig. 2. Therefore, a much higher transmission contrast can be expected.
Fig. 1. Theoretical linear transmission spectra of a PC structure consisting of three identical defects for some different phase shifts.
Fig. 2. Theoretical linear transmission spectra of PC structures consisting of two or three defects with different resonant frequencies but same linewidths.
3. Numerical verification of all-optical diode performances We now design a nonlinear PC diode with three defects based on above analysis. To realize all-optical diode operation, the resonant frequencies of the constitutional defects should shift to a same position with increasing input power in nonlinear regime. Since the shift of the resonant frequencies is proportional to the energy trapped in the defects which can be obtained from the CMT, the relative position of the original resonant frequencies of the defects can be roughly determined [14]. The structure of such a diode is schematically shown in Fig. 3(a). Without loss of generality, we use a two-dimensional (2D) waveguide structure which is commonly used to study the transmission property of a PC device theoretically and experimentally [17,18]. The 2D waveguide plate is made of a material with Kerr nonlinearity. The linear refractive index and the nonlinear coefficient for the Kerr material are set to be no = 3.37 and n2 = 1.5 × 10 − 5μm 2/W, which is close to that of GaAs. Eighteen identical air holes are drilled in the 2D waveguide, the lattice constant, i.e., the distance between
Fig. 3. (a) Geometry of a PC diode configuration composed of three defects. (b) Linear transmission spectra of three single defects A1, A2, and A3, respectively. (c) Linear transmission spectra of the coupled defects composed of defects A1 + A1 + A1, A2 + A2 + A2, and A3 + A3 + A3, respectively.
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the neighboring air holes is a = 0.38μm. The radius of the air holes is 0.25a and the width of the waveguide is a. The three defects which are denoted as defects A1, A2, and A3 are introduced by increasing the separation between the third and fourth air holes and that between the ninth and the tenth air holes, as well as that between the fifteen and sixteen air holes. The size of the three defects is selected to be d1 = 1.300a, d2 = 1.314a and d3 = 1.324a, respectively, as marked in Fig. 3(a). The finite-difference time-domain (FDTD) technique is employed to simulate the linear and nonlinear transmission behavior (http://www.rsoftdesign.com). In the numerical simulations, we choose to simulate the transverse magnetic (TM) mode of the 2D structure. The simulation domain is set to be 3a in the lateral direction, and to cover the whole waveguide in the propagation direction. The grid size used for both directions and the boundary width of the perfectly matched layer are set to be a/16 and a, respectively. The type of the excitation continuous wave (CW) source is slab mode and the width is equal to the width of the waveguide. The transmission is defined as the ratio of the output power to the input power. The linear transmission spectra of the three defects A1, A2, and A3 and of the coupling defect structures consisting of Ai + Ai + Ai (i = 1,2,3) obtained by FDTD are shown in Fig. 3(b) and (c), respectively. Due to the difference of the sizes, the resonant frequencies of the defects are located at ω1 = 0.24124(2πc/a), ω2 = 0.23962(2πc/a) and ω3 = 0.23872(2πc/a), respectively, and the linewidths of them are measured to be γ1 ≈ γ2 ≈ γ3 ≈ 0.00132(2πc/a). The transmission spectra in Fig. 3(c) reveal that the three structures all have a quasiflat top transmission spectrum, indicating that the phase shift between any two adjacent defects is nearly 90°, the influence of small difference of the defect sizes can be ignored. Then, for the structure composed of defects A1 + A2 + A3 as shown in Fig. 3(a), if the resonant frequencies of the defects can be moved to a same position in the nonlinear situation, its forward transmission (rightward) can be expected to have a high transmission. When the signal incidents from right to left (leftward), the defect A3 is adjacent to the input port, more energy will be trapped in the defect compared with other two defects, which will make the resonant frequency of defect A3 shift more than that of defects A1 and A2, and the different shift of the resonant mode will lead to the increase of ω2 − ω3 and ω1 − ω3. From Fig. 3(b) we know that the original difference of the resonant frequencies of the three defects is ω1 − ω2 = 1.23γ, ω2 − ω3 = 0.68γ, and ω1 − ω3 = 1.91γ, respectively, the increase of ω2 − ω3 combined with the increase of ω1 − ω3, will further lower the transmission in the backward direction. Therefore, the PC structure in Fig. 3(a) is expected to have a good performance as an alloptical diode. The FDTD simulation results verify above analysis. In Fig. 4, the nonlinear transmission spectra of the system are presented. It demonstrated very clearly that the three-defect structure displays an excellent unidirectional transmission. For three incident frequencies
Fig. 4. FDTD simulation results of the nonlinear transmission spectra of Fig. 3(a) for some different incident frequencies. The solid and open squares (circles, triangles) denote the rightward and leftward transmission, respectively.
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ω = 0.23731, 0.23712 and 0.23693(2πc/a), at incident power Pin = 35.5, 38.0, and 40.5W/μm, which is often called the threshold power, the rightward transmissions experience an up-jump transition, respectively. After the transition, the transmission changes from very low values to 0.473, 0.452 and 0.428, respectively. For the leftward propagation, the transmissions of the three incident frequencies at the threshold powers are 1.98 × 10 − 3, 1.40 × 10 − 3, and 8.25 × 10 − 4, respectively. The transmission contrasts for the three incident frequencies are calculated to be C = 237, 321 and 518, respectively. We have also studied the transmission as a function of incident frequency. In Fig. 5, the transmittances in both the rightward and leftward directions at a fixed input power P = 35W/μm are presented. For the rightward transmission, the transmittance transits from a much lower value to 0.422 at an incident frequency ω = 0.2370(2πc/a), and then decreases slowly with the increasing frequency. The leftward transmittance maintains at a very low level and increases slowly with the increasing frequency until it transits to a relatively higher value at frequency ω = 0.2377(2πc/a). The transmission contrasts at the two up-jump transition frequencies are 382 and 103, respectively. In-between the frequencies ω = 0.2370 and 0.2377(2πc/a), the PC structure can possess a high forward transmittance and a large transmission contrast simultaneously. If we alter the input power, the corresponding frequency region, the forward transmittance and the contrast will change accordingly. Compared with the transmission of a CW, the transmission of a short pulse through the PC configuration is more practical and convincing. The curves in Fig. 6 show the output pulses of an input Gaussian pulse in two launch directions. The central frequency, width and peak power of the incident Gaussian pulse are selected to be 0.2375(2πc/a), 3 ps and 35W/μm, respectively. It can be seen clearly that transmission of the short pulse is similar to that of a CW. For the rightward transmission, only when the power in the leading edge of the input pulse reaches the threshold power, the resonant frequencies of the three defects are shifting near a same position, does the transmittance increases sharply. In the trailing edge of the input pulse, with the decrease of input power, the resonant frequencies return to their original values and separate again, the transmittance falls sharply accordingly. Therefore, the output pulse becomes narrower in pulse width compared with the input pulse. On the contrary, the input pulse in the leftward direction enlarges the difference of the resonant frequencies of the three defects, which lowers the leftward transmittance and makes the output width to be enlarged. If we define the ratio of the peak power of the two output pulses in two launch direction as the transmission contrast, the contrast is calculated to be 157, which is similar to that in CW case. We have tried to extend this method to four-defect PC structure. In Fig. 7, we show the transmission results of a similar PC coupling structure as Fig. 3(a) but with four defects. The sizes of the four constitutional defects from left to right are selected to be d1 = 1.296a,
Fig. 5. Dependence of the transmittance on the incident frequency for two launch directions.
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X. Cai et al. / Optics Communications 285 (2012) 1959–1963
Fig. 6. Output pulses for two launch directions of an input Gaussian pulse with central frequency, pulse width and peak power being 0.2375(2πc/a), 3ps and 35W/μm, respectively.
d2 = 1.307a, d3 = 1.311a and d4 = 1.317a, respectively. The defect sizes are similar to that in Fig. 3(a), therefore, the four defects should have different resonant modes but almost the same linewidths, and the phase shift between any two adjacent defects of the system can be taken as near 90°. For this system, the forward transmission of an incident frequency ω = 0.23750(2πc/a) transits at threshold power Pin = 38.4W/μm, and the transmission jumps to 0.463. Meanwhile, the backward transmission at the threshold power is only 6.6 × 10 − 5, the corresponding transmission contrast is as high as 7015. From above simulation results, we have verified that excellent performance as an optical diode can be achieved for a PC structure composed of three or four properly selected defects. The combination of the defects of such PC structure is not unique. We found that in a small range of the defect size, the resonant frequency has almost a linear relationship with the defect size. The coupled defects can also function as a diode if we make a similar alteration to the resonant frequencies by changing the sizes of the defects. The transmission contrasts may change a little due to the difference of the structure, but they are larger than that of the two-defect PC coupling structure and that of other schemes. In many cases, the enhancement of transmission contrast is achieved by sacrificing the transmission, just as the case of [11]. If we use the product of transmission contrast and the threshold transmission as a figure of merit to evaluate the performance of an optical diode as suggested in Ref. [6], the figure of merit in our case is calculated to be 222 and 3248 for the three-defect structure and four-defect structure, respectively. It reveals clearly that the efficiency of the diode action is greatly enhanced as compared with 12.67, the figure of merit of the configuration of an asymmetric defect pair [6], 52 of the one-dimensional PC-metal heterostructures [10], and 130 of the structure using tunable surface Plasmon polaritons in a silver grating coated with a nonlinear organic material in [11].
It should be pointed out that the high transmission contrast of the multi-defect PC structure is realized via the shift of the resonant frequencies of the defects, the threshold power of the multi-defect structure is mainly determined by the constitutional defects. We know that the shift of the resonant frequency of a nonlinear defect is due to the change of the refractive index in the defect, which is caused by the electric field trapped in the cavity, and can be expressed to be Δn ∝ n2E 2. Here, the n2 and E are the nonlinear coefficient of the material and the amplitude of the electric field in the cavity, respectively. Therefore, there are two ways to lower the threshold power, and accordingly, lower the light intensity. One is to utilize a material with higher nonlinear coefficient, the other is to use defects with high quality factor Q. For a same variation of the refractive index, a high-Q defect needs less input power compared with that of a low-Q defect. And furthermore, since the quality factor Q is in inverse proportion to the linewidth of the linear transmission spectrum of the defect, higher Q indicates less linewidth. In general, the shift of the resonant frequency of the defect in nonlinear situation is several linewidths of the defect, less linewidth means less shift of the resonant frequency, and accordingly, less variation of the refractive index in the defect is needed for high-Q defect, which will finally result in further decrease of the required input power. It has been demonstrated experimentally that with high-Q photonic crystal nanocavities, the required power density or energy to realize the bistable action can be greatly reduced [19,20]. Because there is no contradiction between the method of using coupled defects to increase the transmission contrast and the method of using high-Q to decrease the required input power, it is reasonable to believe that if high-Q defects are employed in the configuration, low threshold power density and high transmission contrast can be realized simultaneously. 4. Conclusion In summary, we have investigated the unidirectional transmission behavior of PC structures composed of three or four nonlinear defects. Theoretical analysis based on CMT indicates that the transmission contrast of such a three-defect structure can be an order higher than that of a two-defect structure. The FDTD simulation results show that the contrast of a three-defect structure can be as high as 518, which is in good agreement with the theoretical prediction. Furthermore, the simulation result of a four-defect structure reveals that the transmission contrast up to 7015 can be achieved with the forward transmission exceeds 46%. When the input power exceeds the threshold power for a given incident frequency, the forward transmittance maintains at a relatively higher level and decreases slowly with the input power, which is beneficial to the transmission of intensity-modulated signals. Except the excellent performances of diode action in CW case, the coupled structure exhibits unidirectional transmission for a short pulse as well. The coupled defect structure of our design is compact and we expect it to be useful in the application of all-optical signal processing. Acknowledgements This work was supported by the program of Research Foundation of Shantou University under Grant No. YR08005 and No. NTF10020. References
Fig. 7. FDTD simulation results of the nonlinear transmission spectra of a system composed of four distinct defects.
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