We theoretically investigate the feasibility of constructing compact and highly efficient all-optical diodes (AODs) based on light tunneling mechanism in heterostructures. Due to light tunneling behaviors in heterostructures with one-dimensional photonic crystals (1D PC) and lossy metallic film, not only very large nonlinear permittivity of metal can be utilized sufficiently but also the structures with strongly nonreciprocal electric field distributions can be constructed. Finally we design a composite structure consisting of 1D PC-metal heterostructures to achieve the optimal unidirectional light transmission with 0.984 transmission contrasts, 42% transmission and operating light power at working wavelength 557.2nm.
© 2010 OSA
An all-optical diode (AOD) is a spatially nonreciprocal device that offers unidirectional transmission of optical signals for specific wavelengths and power densities. Similar to electronic diodes, AODs possess the unique property of photonic rectification and play key roles in all-optical signal processing. Thus, how to construct highly efficient AODs has become an attractive topic to researchers in recent years. The first study of working mechanism for AODs was implemented by Scalora et al in 1994 [1,2]. They constructed a ramped-index nonlinear PC and found out that there is a striking contrast between the distributions of electric field near the band edge for incidence of different direction. Consequently such ramped-index nonlinear PC can offer nonreciprocal propagation of electromagnetic (EM) waves due to forward/backward different dynamic nonlinear shift of the band edges. Later Gallo et al defined the transmission contrast as to characterize efficiency of AODs , and investigated nonreciprocal transmission of fundamental-frequency light based on second-harmonic generation theoretically  and experimentally . In 2002 Mingaleev and Kivshar discussed nonreciprocal transmission of PC waveguides with embedded nonlinear defects . The bistable diode action in an asymmetric multilayer structure consisting of two materials with positive and negative refractive indices was investigated in 2005 . The nonreciprocal transmission behavior based on asymmetric nonlinear absorption was studied numerically and verified experimentally in 2007 . Very recently, Biancalana demonstrated AOD action in Thue-Moise superlattices . These investigations are very significant and provide important reference for the design of AODs. However, the transmission contrasts in above schemes are lower than 0.9 and hence are hardly available for practical applications.
In order to improve the transmission contrast, the nonlinear effect in the asymmetric structure for AODs should be governed effectively. However, the complexity of the nonlinear processes makes it difficult. In 2005 Maes et al demonstrated optical bistable switching with high contrast and low threshold in coupled nonlinear PC resonators . In 2006 this method of governing optical bistability was used for design of AODs by researchers [10–12]. They proposed a PC waveguide with asymmetric defect pair and found out that the transmission contrast for different incident directions can be enhanced with an increase of discrepancy between resonant wavelengths of two defect modes. However, they also indicated that a large separation between two defect modes leads to a rapid decrease of total transmission. Besides, small nonlinear permittivity in the PC waveguide results in high operating light intensity. For example, when the maximum transmission contrast of ~0.96 is achieved, corresponding transmission is only 0.13 and its operating light intensity is over . These disadvantages restrict the practical application of AODs greatly.
In this paper we discuss a different type of AOD design based on light tunneling in heterostructures. Metals are the unique materials possessing very large nonlinear permittivity (several orders of magnitude larger than those of dielectrics) and having certain loss, but opaque below its plasma frequency. In order to utilize much larger nonlinear permittivity of metal than typical dielectric mediums, two types of composite structures, i.e., metal-dielectric PCs based on the resonant nature of multiple Bragg reflections [13–15] and 1D PC-metal heterostructures based on light tunneling of evanescent wave [16–19], were investigated. By comparison, due to light tunneling behaviors in heterostructures, the electric field in the thick metal film is enhanced greatly and hence the 1D PC-metal heterostructures possess much larger nonlinear effect. On the other hand, the loss properties of metals lead to a decrease of total transmission in structures. Therefore, the loss properties of metals are commonly regarded as the disadvantage in researches. However, in 2001 researcher pointed out that the presence of absorption in the materials can give rise to nonreciprocal reflection , which indicates that the loss properties of metals can be available for constructing the structures with strongly nonreciprocal electric field distributions though it is disadvantage for total transmission. Therefore, it is anticipated that 1D PC-metal heterostructures can be employed to build compact and highly efficient AODs. The paper is organized as follows. In section 2 we study the nonreciprocal transmission behaviors in 1D PC-metal heterostructures. Then in section 3 we propose a type of composite structures with 1D PC-metal heterostructures, numerically investigate their linear and nonlinear transmission behaviors and achieve the performance of AODs. Finally, a brief summary is given in section 4.
2. Nonreciprocal transmission behaviors in 1D PC-metal heterostructures
Firstly we consider a heterostructure composed of a 1D SiO2/TiO2 PC and a thick silver layer. The heterostructure is denoted by , where A represents silver with thickness , C and D denote SiO2 and TiO2 with refractive indices , , thicknesses , and , respectively, and is the periodic number. The linear permittivity of silver is described by the Drude model 22]. Silver is an opaque material below the plasma frequency. First we consider a linear case corresponding to the low-intensity response of the structure. Supposing light was normally incident on the structure in z direction, we use the transfer-matrix method  to calculate the transmitted spectrum. Due to light tunneling in 1D PC-metal heterostructure, a tunneling mode with wavelength appears and its transmittance is 0.623, as is shown in Fig. 1(a) . It is shown that the linear transmission of a multilayer structure is independent of the direction of propagation of the light, regardless of whether or not the layers are absorbing. However, for the multilayer structure with absorbing layer, the reflections of two directions of propagation are different . In other words, the presence of absorption in the metallic film results in nonreciprocal reflection. There is negligible reflection for left incidence of light but obvious reflection for right incidence. Therefore it is anticipated that the electric field in the thick silver layer for left incidence will be stronger than that for right incidence, as is illustrated in Fig. 1(b).
Due to large nonlinearity of silver and its loss properties, optical nonlinearity enhancement in the heterostructure will result in nonreciprocal transmission of EM waves, as is exhibited in Fig. 1(c). For the sake of convenience, we emphasize two regions of input intensity in Fig. 1(c). In region I, the structure exhibits optical bistable behavior for left incidence. In region II, the transmission of left incident light is at the upper branches due to its intensity beyond bistability threshold, whereas the transmission of right incident light is at the lower branches due to its intensity below bistability threshold. Thus the nonreciprocal transmission in region II can be useful for AODs. However, the maximum transmission contrast in region II is only about 0.74, which is not large enough. It can be found that it is the result of rapid falling of the upper branches of optical bistability in region I. It is well known that optical bistability is cause by optical feedback in the nonlinear structures . The gap-edge wavelength will shift dynamically since the nonlinear dielectric constant varies with the field. When the transmitted spectrum is a narrowband one, such dynamic shift is sensitive to the input intensity of light, which give rise to rapid falling of the upper branches of optical bistability.
3. Construction of the composite structure for a highly efficient AOD
The key of constructing a highly efficient AOD is to enhance nonlinear response for left incidence and to restrict nonlinear response for right incidence simultaneously. We shall achieve this objective in two stages. At the first step, we consider a PC-metal-PC sandwich structure consisting of two identical PC-metal heterostructures , where A represents silver, C denotes SiO2 with thickness , D denotes TiO2 with thickness , and m is period of PC (m from 3 to 6). The thickness of A is chosen according to the value of m. For a perfect-matching heterostructure , corresponding to , respectively. Notably, the sandwich structure is symmetric, so the nonreciprocal transmission doesn’t appear in the structure. The purpose that we propose such a sandwich structure is to achieve effective control of optical bistability and considerable unidirectional transmission simultaneously.
The sandwich structure exhibits a broadband transmitted spectrum due to the coupling of two interface mode localized at the interface between PC and the silver film, as is shown in Fig. 2(a) . Since these two interface mode are identical, the coupling between them doesn’t lead to obvious decrease of transmission. A typical example of nonlinear transmission in the sandwich structure is illustrated in Fig. 2(b). It can be found out that the nonlinear response in the structure is shown as optical multistability. It is because the broadband transmitted spectrum is a double-peak one. In region I the maximum transmission of upper branches of optical multistability is 0.655. When input optical intensity is beyond multistability threshold, as is expected, corresponding nonlinear transmission in region II still reaches 0.635.
At the second step, we construct a composite structure with two different PC-metal-PC sandwich structures. We fix the period of PC in the left sandwich structure as six, and thus the structure is denoted by , where m is period of PC (m from 3 to 5). The thicknesses of A(A’) are and corresponding to , respectively. It can be found out from Fig. 2(a) that the transmitted band of covers that of completely. Therefore there is almost no different between the transmitted spectrum of the composite structure consistent for different value of m and their transmission are still rather high, as is illustrated in Fig. 3(a) .
Notably there are two layers of silver film in the structures. But the roles of these layers of silver film are totally different. On the one hand, due to , the electric field intensity localized in the silver film of the left sandwich structure is larger than that localized in the silver film of the right one. Consequently, the nonlinear effect in the silver film of the left sandwich structure is excited at lower intensities. In other words, the left silver film plays the key role in nonlinear enhancement. On the other hand, due to the lossy of the silver film, the transmission of the right sandwich structure is less than 1, and hence the electric field intensity localized in the left silver film for left incidence is large than that for right incidence. Accordingly it can be anticipated that the nonlinear effect in the structure for left incidence is excited at lower intensities than that for right incidence. That is to say, the right silver film plays the key role in tuning the efficiency of AODs. A typical example of the electric field distribution for different incident direction is given in Fig. 3(b).
The nonlinear transmission behaviors in the composite structure are exhibited in Fig. 4 . The pump wavelength is 552.5nm, 551.9nm and 553.3nm corresponding to , respectively, and their linear transmission is 0.012. It can be found from Fig. 4 that nonlinear transmission for incidence of different direction shows a striking contrast. In Fig. 4(a), for example, the nonlinear transmission for left incidence jumps to the upper branches and reaches the maximum 0.423 when the input intensity is beyond the multistability threshold . By contrast, the nonlinear transmission for right incidence has been always keeping a rather low value in the whole range of the considered input intensity. Thus for the maximum transmission contrast can be obtained. Similarly, the maximum transmission contrast reaches for in Fig. 4(b) and for in Fig. 4(c). The results show that the composite structure is suitable for design of AODs very much.
Moreover, other significant results also can be found from Fig. 4. In the case of , the nonlinear transmission at the input intensity for right incidence is only . It is even lower than corresponding linear transmission 0.012. It indicates that in the case of the optimal transmission contrast can be obtained. In addition, low operating light intensity is necessary for practical application. The case of exhibits relatively low operating light intensity . Nevertheless, lower operating light intensity can be obtained when the transmission contrast is sacrificed properly. As for the cases of and , the working light intensity is only and , respectively. Simultaneously, their maximum transmission contrasts still reach 0.934 and 0.936 even though the values are lower than that for the case of . These significant results provide diverse selection for potential application.
Our analysis above has shown the feasibility of the composite structure as AODs. The optimal performance of the composite structure as AODs can be achieved at and corresponding . On the basis of dynamic shift mechanism of the band edges in nonlinear medium, it is not difficult to understand that the nonreciprocal transmission behavior in the composite structure also depends on the pump wavelength. With pump wavelength away from the broadband transmitted peak, as is shown in Fig. 5 , the maximum transmission contrast increases monotonously, and even reaches ~0.988 at working wavelength . In addition, the maximum transmission for left incidence always is larger than 0.4 when working wavelength but decrease sharply when . Therefore, can be considered as the optimal working wavelength for the designed AOD with corresponding and . Numerical result also shows that corresponding operating light intensity is . Nonreciprocal transmission in the composite structure exhibits AODs with high performance.
In summary, we have studied the linear and nonlinear transmission properties of a composite structure consisting of 1D PC-metal heterostructures. Because metal is transparent in light-tunneling heterostructures, its tailored absorption and very large nonlinear permittivity in combination with spatial asymmetry can be used to design AODs. In the designed structure, the optimal transmission contrast up to 0.984 is achieved because the nonlinear response for left incidence is governed effectively, whereas the one for right incidence is suppressed intensely. However, the unidirectional transmission still exceeds 42%, and at the same time, the operating light intensity is only , which is the other merit of the structure for design of AODs. Moreover, the structure is rather compact and can be implemented with the current fabrication technologies . Our design is rather robust and will provide significant application for all-optical signal processing in the future.
The authors acknowledge the financial support from the National Basic Research Program of China (grant 2006CB921701), from the National Natural Science Foundation of China (NNSFC) (grants 10634050 and 10704055), and from the Program for Key Basic Research of the Shanghai Science and Technology Committee (grant 08dj1400301).
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