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Optimized optical nonreciprocal transmission in defective one-dimensional photonic bandgap structures with weak asymmetry
Optics Communications 450 (2019) 322–328
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Optimized optical nonreciprocal transmission in defective one-dimensional photonic bandgap structures with weak asymmetry Juan Zhang a ,∗, Pengxiang Wang a , Yipeng Ding b , Yang Wang c ,∗ a
Key laboratory of Specialty Fiber Optics and Optical Access Networks, Joint International Research Laboratory of Specialty Fiber Optics and Advanced Communication, Shanghai Institute for Advanced Communication and Data Science, School of Communication and Information Engineering, Shanghai University, Shanghai 200444, China b School of Physical Science and Electronics, Central South University, Changsha 410012, China c Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
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Keywords: Nonreciprocal transmission Kerr defect Photonic crystal Weak asymmetry
ABSTRACT A novel and compact one-dimensional (1D) geometry for optical nonreciprocal transmission was proposed in this paper. Different from previous designs with high spatial asymmetry, by simply introducing a few dielectric periods of (BA) and (AB) into a typical nonlinear Fabry–Perot resonator with symmetric Bragg mirrors (AB)N D(BA)N , excellent optical nonreciprocal transmission properties can be achieved in the optical communication wavelength range. The optimized structure can be generally expressed as: (AB)N D(BA)M (AB)L , where M = N + 1, L = 1; M = N + 2, L = 2; M = N + 3, L = 3;. . . . . . and N≥6. The relationship between the bistability and the local field distribution of the Kerr defect layer D and the optical nonreciprocal transmission of the whole 1D system was analyzed theoretically. The direction- and incident wavelength-dependent shift of defect mode are confirmed to be responsible for the optimized optical nonreciprocity. The proposed optimized structures (AB)6 D(BA)8 (AB)2 and (AB)7 D(BA)8 (AB)1 are shown to have advantages over previously reported 1D designs with wide unidirectional transmission range (UTR) width (>3.1nm), high transmission for the forward incidence (>88%) and low transmission for the backward incidence (<1.22%) at lower input intensity (10 MW/cm2 ) with fewer layers (33 layers). Interestingly, the asymmetry degree of the proposed structures can be as low as 11.12%, which is only 1/4 or less of that of previous designs. The effects of the incident angle and polarization on the nonreciprocal transmission properties were also discussed for practical applications. These results are of great importance to the design of high-performance optical nonreciprocal devices, such as all-optical diodes, isolators and limiters.
1. Introduction Optical non-reciprocal transmission is fundamental to realize optical diodes, isolators, and circulators, which are indispensible components in optical communication and signal processing systems [1–9]. The standard approach to break optical reciprocity is to use magnetooptical materials that possess an asymmetric permittivity tensor [1]. However, due to the shortage of magneto-optical materials, such as bulky, costly, and hard to be integrated, there has been significant interest in developing magnetless approaches to non-reciprocity over the past few years [9]. Although super-high contrast effectiveness (100% or close to this figure) can be achieved in some two-dimensional (2D) structures, such as by use of nonlinear Fano resonance for the directional waveguides in 2D photonic crystals (PhCs) with single aside cavity [10] or by spontaneous symmetry breaking in two mode nonlinear cavities inserted into waveguide [11,12], one-dimensional (1D) nonlinear PhCs (or quasicrystals) deserve special attention because ∗
of their simple structure, high reliability, and easy integration, which reveals their potential for compact, low-energy, integrated micro/nanophotonic devices or on-chip systems [13–24]. Optical nonreciprocity within such 1D system is often based on the direction-dependent resonance shift [14,19]. One needs to increase the spatial asymmetry or the optical nonlinearity of the structure to increase the structure’s sensitivity to the direction of incidence. Traditionally, complex multilayer structures with some distributed nonlinearity are used. And this approach works only for optical signals of sufficiently high intensity. To obtain optimized design which exhibits better optical non-reciprocal performance (higher nonreciprocal transmission ratio, lower insert loss, and wider working wavelength range) and higher feasibility (lower input intensity and fewer layers) is still challenging. Started from a simple 1D geometry of Fabry–Perot resonator with symmetric Bragg mirrors (AB)N D(BA)N , the structure optimization for nonreciprocal transmission in the optical communication wavelength
Corresponding authors. E-mail addresses: [email protected] (J. Zhang), [email protected] (Y. Wang).
https://doi.org/10.1016/j.optcom.2019.06.029 Received 11 March 2019; Accepted 11 June 2019 Available online 13 June 2019 0030-4018/© 2019 Published by Elsevier B.V.
J. Zhang, P. Wang, Y. Ding et al.
Optics Communications 450 (2019) 322–328
Fig. 2. Typical transmission spectra of the nonlinear 1D systems for optical nonreciprocal transmission.
Fig. 1. Schematic structures of the nonlinear 1D systems for optical nonreciprocal transmission. (a) an initial symmetrical geometry of (AB)N D(BA)N (N = 7), (b) an optimized structure of (AB)N D(BA)M (AB)L (N = 7, M = 8, L = 1).
(UTR), high-quality nonreciprocal (unidirectional) transmission can be expected when the values of a and b (maximum and minimum transmittance for forward incidence) are high enough (near 1) and that of c (transmittance for backward incidence) is small enough (near 0) at the same time. The value of (1-a) is proportional to the insertion loss, and that of (b-c) is often defined as nonreciprocal transmission ratio (in dB). Moreover, generally, the width of the UTR reflects the applicability of the non-reciprocal devices. However, owing to the design principles of such nonlinear systems, a dramatic trade-off between the nonreciprocal transmission ratio and the insertion loss has been experienced. The transmission characteristics of this kind of structure can be calculated out by using transfer matrix method (TMM) [25]. When electromagnetic wave propagates through the nonlinear 1D system, the electric field in the ith layer can be described as:
range was carried out theoretically in this paper. Pronounced improvement in optical nonreciprocal transmission can be achieved by introducing a few dielectric periods of (BA) and (AB) on the right. The asymmetry degree of the optimized structures can be lower than 12%. The relationship between the bistability and local field distribution of the Kerr defect layer D and the optical nonreciprocal transmission of the whole 1D system was analyzed in detail. The direction- and incident wavelength-dependent shift of defect mode was proved to be responsible for the optimized optical non-reciprocity. The proposed geometry is shown to have advantages over previously reported designs with less number of layers. For optimal structures (AB)6 D(BA)8 (AB)2 and (AB)7 D(BA)8 (AB)1 , larger than 88% transmission for left-to-right incidence and near zero transmission for the other can be obtained in a 3.1 nm wavelength range at an input intensity of 10 MW/cm2 . The influence of incident angle and polarization on the nonreciprocal transmission properties was discussed as well for the consideration of fault tolerance in practical applications.
𝐸𝑖 = 𝐴𝑖 exp(𝑖𝑘𝑖𝑧) + 𝐵𝑖 exp(−𝑖𝑘𝑖𝑧),
(1)
where 𝑘𝑖 is propagation wave number in the ith layer, 𝐴𝑖 and 𝐵𝑖 are amplitudes of forward and backward waves, respectively. When the intensity of incident light is high, the refractive index of Kerr nonlinear layer varies with increase of the intensity of local light, the amplitude of the forward and backward waves in the medium is not constant, therefore the traditional TMM is not suitable for this case. The incident light can be calculated reversely by output light based on reverse recursive algorithm [26,27]. The principle of reverse recursive algorithm is to divide the nonlinear layer into many sub-layers, as long as the number of sub-layers is large enough, each sub-layer can be regarded as a linear layer with a fixed refractive index. Therefore the nonlinear layer can be considered as a combination of multiple linear layers with different refractive index. The amplitude of the forward and backward waves in two adjacent sub-layers can be related by electromagnetic field boundary conditions, and expressed in transfer matrix form as: ) ) ( ( ⎡ ⎤ 𝑛 𝑛 1 [ ] [ ] ⎢ 1 1 + 𝑖−1 ] 1 − 𝑖−1 ⎥ [ 𝐴𝑖 exp(𝑖𝛽𝑖 ) 0 𝑛𝑖 2 𝑛𝑖 ⎥ 𝐴𝑖−1 ⎢2 = ( ) ( ) ⎥ 𝐵𝑖−1 𝐵𝑖 0 exp(−𝑖𝛽𝑖 ) ⎢ 1 𝑛 𝑛 1 ⎢ 1 − 𝑖−1 1 + 𝑖−1 ⎥ ⎣2 𝑛𝑖 2 𝑛𝑖 ⎦ [ ] 𝐴 = 𝑄𝑖 𝑖−1 , (2) 𝐵𝑖−1
2. Theoretical model and design The initial geometry (AB)N D(BA)N (N = 7) for structure optimization was shown in Fig. 1(a). It is a typical 1D Fabry–Perot resonator with symmetric Bragg mirrors. Where A and B indicate the low and high refractive index material SiO2 and TiO2 with the refractive index 𝑛A = 1.45 and 𝑛B = 2.3, respectively. D is the nonlinear Kerr medium polydiacetylene 9-BCMU with the linear refractive index 𝑛D0 = 1.55 and the third-order nonlinear coefficient 𝜒 (3) = 2.5×10−5 cm2 /MW, respectively. The optical thicknesses of A, B layers is 𝑛A 𝑑A = 𝑛B 𝑑B = 𝜆0 /4, and that of the D layer is 𝑛D0 𝑑D = 𝜆0 , where 𝜆0 = 1550 nm is the center wavelength. High reflection can be obtained for both forward (left-to-right) and backward (right-to-left) incidences regardless of the light intensity. When the spatial symmetry was broken by introducing a few additional dielectric periods of (BA) and/or (AB) into the structure on one side (such as (AB)7 D(BA)8 (AB)1 shown in Fig. 1(b)), optical nonreciprocal transmission can be obtained due to the different optical nonlinear response at different directions of incidence. Fig. 2 shows the typical transmission spectra of this kind of asymmetric 1D system. Divergent bistable responses for forward and backward incidences can be observed. As shown in Fig. 2, there is a wavelength range (the gray shaded area) where a high-transmission state only exists for one direction of incidence (forward incidence in this case). In this range
where 𝛽𝑖 = 𝑘𝑖 𝑑𝑖 , 𝑑𝑖 = 𝑑𝐷 ∕𝑚 is the thickness of the ith sub-layer, m is the number of sub-layers of the nonlinear layer, 𝑛𝑖 and 𝑛𝑖−1 are the refractive index of the ith and the (𝑖 − 1)th sub-layer, respectively. The 323
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Optics Communications 450 (2019) 322–328 Table 1 Detailed unidirectional transmission parameters (defined in Fig. 2) of different structures. The wavelength of incident light is 1563 nm and the intensity of it is 10 MW/cm𝟐 .
amplitudes of forward and backward waves in the (𝑖 − 1)th sub-layer can be calculated from that of the ith sub-layer as: [ ] [ ] 𝐴𝑖−1 𝐴𝑖 = 𝑄−1 , (3) 𝑖 𝐵𝑖−1 𝐵𝑖 𝑄−1 𝑖
where is the inverse matrix of 𝑄𝑖 . For simplicity, neglecting the saturation of absorption as well as the nonlinear saturation in the paper and considering the phase variation due to the nonlinear refraction is very small compared to the wave vector inside the sub-layer [28,29], the refractive index of (𝑖−1)th sub-layer can be approximately described as: 𝑛𝑖−1
𝜒 (3) | 2 = 𝑛𝐷0 + 𝐴 + 𝐵𝑖−1 || . 2𝑛𝐷0 | 𝑖−1
(4)
(5)
where 𝑀𝑖 (𝑖 = 1, 2, … , 𝑚) is the transfer matrix of the ith sub-layer. The transfer matrix of the structure (AB)N D(BA)M (AB)L can thus be described as: [ ] 𝑀11 𝑀12 𝑀 = 𝑀(𝐴𝐵)𝑁 ⋅ 𝑀𝐷 ⋅ 𝑀(𝐵𝐴)𝑀 ⋅ 𝑀(𝐴𝐵)𝐿 = , (6) 𝑀21 𝑀22 where 𝑀(𝐴𝐵)𝑁 and 𝑀(𝐵𝐴)𝑀 , 𝑀(𝐴𝐵)𝐿 are the transfer matrices of the stacks on the left and right side of the nonlinear layer D, respectively. Therefore the transmission of the structure can be expressed as: 𝑇 =
a
b
c
UTR width (×𝜆0 )
(AB)6 D(BA)7 (AB)1 (AB)7 D(BA)8 (AB)1 (AB)6 D(BA)8 (AB)2 (AB)7 D(BA)9 (AB)2
99.99% 99.47% 97.97% 97.91%
99.14% 90.45% 88.49% 85.03%
2.23% 0.28% 1.22% 0.33%
0.00013 0.00200 0.00210 0.00200
with that of (AB)7 D(BA)8 (AB)1 . Larger than 85% transmission for forward incidence and less than 1.22% transmission for the backward incidence can be obtained in a 0.002𝜆0 , i.e. 3.1 nm wavelength range at an input intensity of 10 MW/cm2 . More complex structures such as (AB)7 D(BA)10 (AB)3 can achieve better nonreciprocal transmission properties at the same input intensity, while the performance will deteriorate by using the structures with similar asymmetry but fewer layers such as (AB)5 D(BA)8 (AB)3 . The transmission spectra of these structures are not shown here due to the limited space. The optimized structures for nonreciprocal transmission can be expressed as: (AB)N D(BA)M (AB)L where M = N+1, L = 1; M = N+2, L = 2; M = N+3, L = 3; . . . and N ≥ 6. Pronounced improvement in optical nonreciprocal transmission can be simply achieved by introducing a few dielectric periods of (BA) and (AB) on one side of a typical Fabry–Perot resonator with symmetric Bragg mirrors (AB)N D(BA)N . The number of N should be larger than 6 to ensure sufficient nonreciprocal transmission ratio (higher forward transmission and lower backward transmission). The optimal operating region of input light intensity should be determined for a specific structure to ensure the high performance and operability at the same time. Fig. 6 shows the short, long wavelength limit, and the minimum transmission for the forward incidence in UTR with the input intensities of the structure (AB)6 D(BA)8 (AB)2 . 10 MW/cm2 seems to be an appropriate value to obtain relatively wide UTR and high forward transmission. Several key parameters of the optimized structure (AB)7 D(BA)8 (AB)1 are listed in Table 2 and show obvious advantages in unidirectional transmission when compared with that of several typical 1D nonlinear diodes reported previously. The contrast C is defined as C = (Tfor -Tback )/(Tfor +Tback ), where Tfor and Tback indicates the transmission for the forward and backward incidences, respectively. Asymmetry degree, which is defined as the ratio of the numbers of asymmetrical layers to the total layer numbers according to the metric function of mirror symmetry, is also calculated out for comparison. Interestingly, the asymmetry degree of the proposed structures can be as low as 11.12%, which is only 1/4 or less of that of previous designs.
The refractive index and the corresponding transfer matrix of each sublayer can be obtained backward recursively from the given intensity of output light. The whole transfer-matrix of the nonlinear layer can be expressed as: 𝑀𝐷 = 𝑀1 ⋅ 𝑀2 ⋯ 𝑀𝑖 ⋯ 𝑀𝑚 ,
Structures
|2 𝐼𝑜𝑢𝑡 || 𝐸𝑜𝑢𝑡 ||2 || 2 ⋅ 𝑛𝑖𝑛 | , (7) =| =| | | 𝐼𝑖𝑛 | 𝐸𝑖𝑛 | | (𝑀11 + 𝑀12 ⋅ 𝑛𝑜𝑢𝑡 ) ⋅ 𝑛𝑖𝑛 + (𝑀21 + 𝑀22 ⋅ 𝑛𝑜𝑢𝑡 ) |
where 𝐼𝑖𝑛 and 𝐼𝑜𝑢𝑡 are the intensities of incident and output light respectively, and 𝑛𝑖𝑛 and 𝑛𝑜𝑢𝑡 are the refractive indices of the incident and output environmental media, respectively. 3. Structure optimization A few additional dielectric periods of (BA) is firstly introduced into the initial structure on the right side to break the spatial symmetry of the 1D system. Fig. 3(a) shows the transmission spectra of the structure (AB)7 D(BA)N with N = 8, 9 and 10 respectively. It can be seen that the transmission is more dependent on the direction of incidence with the increase of N, which results in the increase of the width of UTR. However, the maximum transmittance decreases sharply with the increase of N. The attempt to enhance the maximum transmission by increasing the nonlinearity is confirmed to be not successful. Fig. 3(b) shows the transmission spectra of (AB)7 D(BA)8 at different input intensities. The maximum transmission keeps nearly unchanged with the increase of the input intensity although the transmission is more dependent on the direction of incidence with it. In order to enhance the maximum transmission by further increasing the asymmetry, a dielectric bi-layer (AB) is introduced into the structure on the right side. The periodicity of the right Bragg mirror is disturbed by a reversely arranged unit. Fig. 4 compares the transmittance of the structures (AB)7 D(BA)8 and (AB)7 D(BA)8 (AB)1 at different input intensities for forward and backward incidences. We can see that by inserting the unit (AB), the maximum transmissions for both forward and backward incidences increases simultaneously. High-performance nonreciprocal transmission can also be achieved in similar structures as shown in Fig. 5. The corresponding unidirectional transmission parameters (defined in Fig. 2) are listed in Table 1 in detail. Obvious trade-off between the forward transmission and the width of UTR can be observed. For the structure (AB)6 D(BA)7 (AB)1 , high forward transmission (near 100%) can only be obtained in a very narrow UTR. The optimized structures (AB)6 D(BA)8 (AB)2 and (AB)7 D(BA)9 (AB)2 have similar nonreciprocal transmission properties
4. Operation mechanism analysis The above-mentioned optical nonreciprocal transmission is originated from the nonlinear effect of light-matter interaction. A highintensity light, when traveling in a nonlinear medium, will induce an obvious varying refractive index of the medium due to the optical Kerr effect. This variation in refractive index will conversely modulate the field intensity distribution inside the medium. That is the optical Kerr effect manifests the matter and light temporally in refractive index and field intensity until a steady state is formed. Fig. 7(a) shows the distribution of the electric field intensity of the structure (AB)6 D(BA)8 (AB)2 for the forward and backward incidences. Fig. 7(b) shows the refractive index distribution in the Kerr defective layer D for both incidences. It can be seen that the electric field is anisotropically localized in different locations for the forward and backward incidences. For the forward incidence, the electric field is mainly 324
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Optics Communications 450 (2019) 322–328
Fig. 3. Transmission spectra of (a) (AB)7 D(BA)N with N = 8, 9 and 10 at the input intensity of 3 MW/cm2 and (b) (AB)7 D(BA)8 at different input intensities. Solid and dashed lines correspond to the forward and backward incidence respectively.
Fig. 4. Transmittance at different input intensities for forward (solid) and backward (dash) incidence for the structures (a) (AB)7 D(BA)8 and (b) (AB)7 D(BA)8 (AB)1 . The inserts are the enlarged top parts of the transmission curves. The wavelength of incident light is 1563 nm.
Fig. 5. Nonlinear transmission spectra for the forward (solid) and backward (dash) incidence of the structures (a) (AB)6 D(BA)7 (AB)1 ; (b) (AB)7 D(BA)8 (AB)1 ; (c) (AB)6 D(BA)8 (AB)2 ; and (d) (AB)7 D(BA)9 (AB)2 . The wavelength of incident light is 1563 nm and the intensity of it is 10 MW/cm2 . The UTR is denoted as the gray region in the figures.
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Table 2 Comparison of several key parameters in unidirectional transmission for different 1D nonlinear structures. References
Total layer numbers
Number of nonlinear layers
Laser power (MW/cm2 )
a (%)
b (%)
c (%)
UTR
C (%)
Asymmetry degree
This work [20] [13] [22] [23] [24] [21]
33 40 24 128 56 43 24
1 20 12 64 28 20 12
10 50 13 10 9.5 0.43 10
99.47 ∼100 ∼25 ∼98 92.05 ∼100 75
90.45 80 / ∼91 / ∼65 /
0.28 ∼0 ∼5 ∼5 22.7 <3 2.5
0.0019𝜔0 0.0033𝜔0 / ∼ 0.0010𝛺0 / 0.0036𝜔0 /
>99.38 >80 66.67 ∼90 60.4 >91.18 93.55
11.12% 50% 100% 50% 50% 45.46% 54.55%
Notice: ‘‘/’’ means not exist for single wavelength case. 𝜔0 is the central frequency.
The non-coincidence of the defect modes for forward and backward incidences will result in the optical nonreciprocal transmission in a certain wavelength range. The four defect modes on the right are corresponding to the input light with the wavelengths of 1560.3 nm, 1561 nm, 1562 nm, and 1563.6 nm (corresponding to the normalized wavelength of 1.0066, 1.0071, 1.0077 and 1.0088) respectively for the forward incidence. The UTR is also shown in the figure as the gray region. The transmittance at the four specified incident wavelengths is indicated as the black dots in the figure. The curve connecting the four dots forms the dispersion curve in the UTR (see the inset of Fig. 8(b)), which is corresponding to the upper part of the nonlinear (bistable) transmission spectra in the UTR shown in Fig. 5(c). 5. Angular and polarization All the analyses above are for normal incidences. Analyzing the angular dependency is of importance for practical applications especially for the multilayer resonators, since the lateral dimensions of an incident beam are always finite and thus such beam consists of multiple angular components. Fig. 9 shows the nonlinear transmission spectra of the optimized structure (AB)6 D(BA)8 (AB)2 at incident angles of 15◦ and 30◦ . The corresponding spectrum at normal incidence (0◦ ) is also shown in the same figure for comparison. It can be seen that the spectra shift to the short wavelength direction with the increase of incident angle for both polarizations. The deviation between the spectra of forward and backward incidences increases for TE polarization, whereas decreases for TM polarization with the increase of incident angle. The maximum transmission decreases with the increase of incident angle for TE polarization and is even not at the peak position due to the great bending of the spectrum at a larger incident angle (such as 30◦ ). Fig. 10 shows the short, long wavelength limit, and the minimum transmission (value b in Fig. 2) in UTR with the variation of incident angle for the forward incidence of the optimized structure (AB)6 D(BA)8 (AB)2 . It can be observed that both the short and long wavelength limits shift to the short wavelength direction with the increase of incident angle. For TE polarization, the UTR is broadened with the increase of incident angle due to the faster change of the
Fig. 6. Short, long wavelength limit, and the minimum transmission for the forward incidence in UTR with input intensities of the structure (AB)6 D(BA)8 (AB)2 .
localized in the nonlinear defective layer and a unique double-peak distribution can be observed. The different distributed field intensities inside the nonlinear layer for forward and backward incidences result in the great change of the refractive index of the layer due to its intensity dependency. Similar double-peak distribution of refractive index can be observed for the forward incident, while the refractive index nearly remains unchanged for the backward incident. Furthermore, the input lights with different wavelengths will have different refractive index distribution inside the nonlinear layer for the forward incidence due to the dispersion effect, whereas that for backward incident nearly remains the constant value of the linear refractive index at different wavelengths. Thus the location of the defect mode in the bandgap is different for the forward and backward incidences and will shift to the long wavelength direction with the increase of incident wavelength for the forward incidence, while it is almost fixed for the backward incidence, as shown in Fig. 8.
Fig. 7. Distribution of the electric field intensity (a) and refractive index in the nonlinear layer D (b) for the forward (black) and backward (red) incidences of the structure (AB)6 D(BA)8 (AB)2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Band gaps (a) and defect modes (b) of the structure (AB)6 D(BA)8 (AB)2 for the input lights with different wavelengths for the forward and backward incidences. UTR is denoted as the gray region. The transmittance at four specified wavelengths is indicated as the black dots.
Fig. 9. Nonlinear transmission spectra of the structure (AB)6 D(BA)8 (AB)2 at the incident angles of 15◦ (a and b) and 30◦ (c and d). The corresponding spectra at normal incidence (0◦ ) are also shown for comparison.
optimized structure can be generally expressed as: (AB)N D(BA)M (AB)L , where M = N+1, L = 1; M = N+2, L = 2; M = N+3, L = 3; . . . and N ≥ 6. The proposed geometry is shown to have advantages over previously reported designs. Interestingly, the asymmetry degree of the proposed structures can be only 1/4 or less of that of previous designs. Optimized structures (AB)7 D(BA)8 (AB)1 and (AB)6 D(BA)8 (AB)2 have been obtained numerically, which have excellent optical nonreciprocal transmission characteristics of wide UTR width (>3.1 nm), high transmission for the forward incidence (>88%) and low transmission for the backward incidence (<1.22%) at lower input intensity (10 MW/cm2 ) with fewer layers (33 layers). This optimal optical nonreciprocal transmission is originated from the nonlinear effect of light-matter interaction. The different distributed field intensities and refractive index inside the nonlinear layer D for the forward and backward incidences result in
short wavelength limit. The minimum transmission for the forward incidence in the UTR decreases with the increase of incident angle. Whereas, for TM polarization, the UTR becomes narrower with the increase of incident angle due to faster change of the long wavelength limit until it disappears at a certain incident angle (∼34◦ ). There is almost no significant change for the minimum transmission for the forward incidence in the UTR in a large incident angle range (0–24◦ ). 6. Conclusion In this paper, a novel and compact 1D geometry for optical nonreciprocal transmission was proposed based on the typical nonlinear Fabry–Perot resonator with symmetric Bragg mirrors (AB)N D(BA)N by introducing a few dielectric periods of (BA) and (AB) on one side. The 327
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Optics Communications 450 (2019) 322–328
Fig. 10. Short, long wavelength limit, and the minimum transmission in UTR with the variation of incident angle for the forward incidence at TE (a) and TM (b) polarizations of the structure (AB)6 D(BA)8 (AB)2 .
the non-coincidence of the defect modes in the bandgap, which causes the optical nonreciprocal transmission in a certain wavelength range. Take the optimized structure (AB)6 D(BA)8 (AB)2 as an example, the effects of the incident angle and polarization were also studied. For TE polarization, the UTR is broadened with the increase of incident angle and the minimum transmission for the forward incidence in the UTR decreases with it. Whereas, for TM polarization, the UTR becomes narrower with the increase of incident angle until it disappears at a certain incident angle (∼34◦ ) and there is almost no significant change for the minimum transmission for the forward incidence in the UTR in a large incident angle range (0–24◦ ). These results are of great importance to the design of high-performance optical nonreciprocal devices, such as all-optical diodes, isolators and limiters.
[11] E.N. Bulgakov, A.F. Sadreev, All-optical diode based on dipole modes of kerr microcavity in asymmetric L-shaped photonic crystal waveguide, Opt. Lett. 39 (7) (2014) 1787–1790. [12] T. Sato, S. Makino, T. Fujisawa, K. Saitoh, Design of a reflection-suppressed alloptical diode based on asymmetric L-shaped nonlinear photonic crystal cavity, J. Opt. Soc. Am. B. 33 (1) (2016) 54–61. [13] M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, C.M. Bowden, Thin-film nonlinear optical diode, Appl. Phys. Lett. 66 (18) (1995) 2324–2326. [14] I.S. Fogel, J.M. Bendickson, M.D. Tocci, M.J. Bloemer, M. Scalora, C.M. Bowden, J.P. Dowling, Spontaneous emission and nonlinear effects in photonic bandgap materials, Pure Appl. Opt.: J. Eur. Opt. Soc. Part A 7 (2) (1998) 393. [15] M.W. Feise, I.V. Shadrivov, Y.S. Kivshar, Bistable diode action in left-handed periodic structures, Phys. Rev. E 71 (3) (2005) 037602. [16] J. Hwang, M.H. Song, B. Park, S. Nishimura, T. Toyooka, J.W. Wu, Y. Takanishi, K. Ishikawa, H. Takezoe, Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions, Nat. Mater. 4 (5) (2005) 383. [17] J.Y. Chen, L.W. Chen, Color separating with integrated photonic band-gap optical diodes: a numerical study, Opt. Express 14 (22) (2006) 10733–10739. [18] F. Biancalana, All-optical diode action with quasiperiodic photonic crystals, J. Appl. Phys. 104 (9) (2008) 093113. [19] S.V. Zhukovsky, Perfect transmission and highly asymmetric light localization in photonic multilayers, Phys. Rev. A 81 (5) (2010) 053808. [20] Kazem Jamshidi-Ghaleh, Zeinab Safari, Fatemeh Moslemi, Angular and polarization dependence of all optical diode in one-dimensional photonic crystal, Eur. Phys. J. D. 69 (2015) 1–5. [21] Chong Li, Xiaoyong Hu, Hong Yang, Qihuang Gong, Unidirectional transmission in 1d nonlinear photonic crystal based on topological phase reversal by optical nonlinearity, AIP Adv. 7 (2017) 025203. [22] V. Grigoriev, Biancalana, F, Bistability, multistability and non-reciprocal light propagation in thue–morse multilayered structuresm, New J. Phys. 12 (5) (2010) 053041. [23] S.V. Zhukovsky, A.G. Smirnov, All-optical diode action in asymmetric nonlinear photonic multilayers with perfect transmission resonances, Phys. Rev. A 83 (2) (2011) 023818. [24] K. Jamshidi-Ghaleh, F. Moslemi, Electrically tunable all-optical diode in a one-dimensional photonic crystal structure, Appl. Opt. 56 (14) (2017) 4146–4152. [25] M. Born, E. Wolf, Principles of Optics, seventh (expanded) ed., Cambridge U. Press, Cambridge, UK, 1999. [26] J. Zhang, R. Zhang, Y. Wang, Enhanced temperature sensing based on subthreshold nonlinear spectra of one-dimensional photonic crystal with a Kerr defect layer, J. Appl. Phys. 116 (18) (2014) 183104. [27] J. Zhang, R. Zhang, Y. Wang, VO2 -like thermo-optical switching effect in onedimensional nonlinear defective photonic crystals, J. Appl. Phys. 117 (21) (2015) 213101. [28] J. He, M. Cada, Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices, Appl. Phys. Lett. 61 (18) (1992) 2150–2152. [29] J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, R. Reinisch, Dispersive optical bistability in stratified structures, Phys. Rev. B 44 (15) (1991) 8214.
Acknowledgments This work was partially supported by the National Natural Science Foundation of China (51472258), National Key Research and Development Program of China (2016YFE0110600), and Innovation Program of Shanghai Municipal Education Commission, China (15ZZ045). References [1] R.J. Potton, Reciprocity in optics, Rep. Progr. Phys. 67 (5) (2004) 717. [2] Z. Yu, S. Fan, Complete optical isolation created by indirect interband photonic transitions, Nat. photonics 3 (2) (2009) 91. [3] L. Bi, J. Hu, P. Jiang, D.H. Kim, G.F. Dionne, L.C. Kimerling, C.A. Ross, On-chip optical isolation in monolithically integrated non-reciprocal optical resonators, Nat. Photonics 5 (12) (2011) 758. [4] L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A.M. Weiner, M. Qi, An all-silicon passive optical diode, Science 335 (6067) (2011) 447–450. [5] D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J.D. Joannopoulos, M. Vanwolleghem, C.R. Doerr, M. Vanwolleghem, What is—and what is not—an optical isolator, Nat. Photonics 7 (8) (2013) 579. [6] L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, M. Xiao, Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators, Nat. Photonics 8 (7) (2014) 524. [7] Y. Shi, Z. Yu, S. Fan, Limitations of nonlinear optical isolators due to dynamic reciprocity, Nat. Photonics 9 (6) (2015) 388. [8] A.M. Mahmoud, A.R. Davoyan, N. Engheta, All-passive nonreciprocal metastructure, Nat. Commun. 6 (2015) 8359. [9] D.L. Sounas, A. Alù, Non-reciprocal photonics based on time modulation, Nat. Photonics 11 (12) (2017) 774. [10] W. Ding, B. Luk’yanchuk, C.-W. Qiu, Ultrahigh-contrast-ratio silicon Fano diode, Phys. Rev. A. 85 (2012) 025806.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Optimized optical nonreciprocal transmission in defective one-dimensional photonic bandgap structures with weak asymmetry Juan Zhang a ,∗, Pengxiang Wang a , Yipeng Ding b , Yang Wang c ,∗ a
Key laboratory of Specialty Fiber Optics and Optical Access Networks, Joint International Research Laboratory of Specialty Fiber Optics and Advanced Communication, Shanghai Institute for Advanced Communication and Data Science, School of Communication and Information Engineering, Shanghai University, Shanghai 200444, China b School of Physical Science and Electronics, Central South University, Changsha 410012, China c Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
ARTICLE
INFO
Keywords: Nonreciprocal transmission Kerr defect Photonic crystal Weak asymmetry
ABSTRACT A novel and compact one-dimensional (1D) geometry for optical nonreciprocal transmission was proposed in this paper. Different from previous designs with high spatial asymmetry, by simply introducing a few dielectric periods of (BA) and (AB) into a typical nonlinear Fabry–Perot resonator with symmetric Bragg mirrors (AB)N D(BA)N , excellent optical nonreciprocal transmission properties can be achieved in the optical communication wavelength range. The optimized structure can be generally expressed as: (AB)N D(BA)M (AB)L , where M = N + 1, L = 1; M = N + 2, L = 2; M = N + 3, L = 3;. . . . . . and N≥6. The relationship between the bistability and the local field distribution of the Kerr defect layer D and the optical nonreciprocal transmission of the whole 1D system was analyzed theoretically. The direction- and incident wavelength-dependent shift of defect mode are confirmed to be responsible for the optimized optical nonreciprocity. The proposed optimized structures (AB)6 D(BA)8 (AB)2 and (AB)7 D(BA)8 (AB)1 are shown to have advantages over previously reported 1D designs with wide unidirectional transmission range (UTR) width (>3.1nm), high transmission for the forward incidence (>88%) and low transmission for the backward incidence (<1.22%) at lower input intensity (10 MW/cm2 ) with fewer layers (33 layers). Interestingly, the asymmetry degree of the proposed structures can be as low as 11.12%, which is only 1/4 or less of that of previous designs. The effects of the incident angle and polarization on the nonreciprocal transmission properties were also discussed for practical applications. These results are of great importance to the design of high-performance optical nonreciprocal devices, such as all-optical diodes, isolators and limiters.
1. Introduction Optical non-reciprocal transmission is fundamental to realize optical diodes, isolators, and circulators, which are indispensible components in optical communication and signal processing systems [1–9]. The standard approach to break optical reciprocity is to use magnetooptical materials that possess an asymmetric permittivity tensor [1]. However, due to the shortage of magneto-optical materials, such as bulky, costly, and hard to be integrated, there has been significant interest in developing magnetless approaches to non-reciprocity over the past few years [9]. Although super-high contrast effectiveness (100% or close to this figure) can be achieved in some two-dimensional (2D) structures, such as by use of nonlinear Fano resonance for the directional waveguides in 2D photonic crystals (PhCs) with single aside cavity [10] or by spontaneous symmetry breaking in two mode nonlinear cavities inserted into waveguide [11,12], one-dimensional (1D) nonlinear PhCs (or quasicrystals) deserve special attention because ∗
of their simple structure, high reliability, and easy integration, which reveals their potential for compact, low-energy, integrated micro/nanophotonic devices or on-chip systems [13–24]. Optical nonreciprocity within such 1D system is often based on the direction-dependent resonance shift [14,19]. One needs to increase the spatial asymmetry or the optical nonlinearity of the structure to increase the structure’s sensitivity to the direction of incidence. Traditionally, complex multilayer structures with some distributed nonlinearity are used. And this approach works only for optical signals of sufficiently high intensity. To obtain optimized design which exhibits better optical non-reciprocal performance (higher nonreciprocal transmission ratio, lower insert loss, and wider working wavelength range) and higher feasibility (lower input intensity and fewer layers) is still challenging. Started from a simple 1D geometry of Fabry–Perot resonator with symmetric Bragg mirrors (AB)N D(BA)N , the structure optimization for nonreciprocal transmission in the optical communication wavelength
Corresponding authors. E-mail addresses: [email protected] (J. Zhang), [email protected] (Y. Wang).
https://doi.org/10.1016/j.optcom.2019.06.029 Received 11 March 2019; Accepted 11 June 2019 Available online 13 June 2019 0030-4018/© 2019 Published by Elsevier B.V.
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Optics Communications 450 (2019) 322–328
Fig. 2. Typical transmission spectra of the nonlinear 1D systems for optical nonreciprocal transmission.
Fig. 1. Schematic structures of the nonlinear 1D systems for optical nonreciprocal transmission. (a) an initial symmetrical geometry of (AB)N D(BA)N (N = 7), (b) an optimized structure of (AB)N D(BA)M (AB)L (N = 7, M = 8, L = 1).
(UTR), high-quality nonreciprocal (unidirectional) transmission can be expected when the values of a and b (maximum and minimum transmittance for forward incidence) are high enough (near 1) and that of c (transmittance for backward incidence) is small enough (near 0) at the same time. The value of (1-a) is proportional to the insertion loss, and that of (b-c) is often defined as nonreciprocal transmission ratio (in dB). Moreover, generally, the width of the UTR reflects the applicability of the non-reciprocal devices. However, owing to the design principles of such nonlinear systems, a dramatic trade-off between the nonreciprocal transmission ratio and the insertion loss has been experienced. The transmission characteristics of this kind of structure can be calculated out by using transfer matrix method (TMM) [25]. When electromagnetic wave propagates through the nonlinear 1D system, the electric field in the ith layer can be described as:
range was carried out theoretically in this paper. Pronounced improvement in optical nonreciprocal transmission can be achieved by introducing a few dielectric periods of (BA) and (AB) on the right. The asymmetry degree of the optimized structures can be lower than 12%. The relationship between the bistability and local field distribution of the Kerr defect layer D and the optical nonreciprocal transmission of the whole 1D system was analyzed in detail. The direction- and incident wavelength-dependent shift of defect mode was proved to be responsible for the optimized optical non-reciprocity. The proposed geometry is shown to have advantages over previously reported designs with less number of layers. For optimal structures (AB)6 D(BA)8 (AB)2 and (AB)7 D(BA)8 (AB)1 , larger than 88% transmission for left-to-right incidence and near zero transmission for the other can be obtained in a 3.1 nm wavelength range at an input intensity of 10 MW/cm2 . The influence of incident angle and polarization on the nonreciprocal transmission properties was discussed as well for the consideration of fault tolerance in practical applications.
𝐸𝑖 = 𝐴𝑖 exp(𝑖𝑘𝑖𝑧) + 𝐵𝑖 exp(−𝑖𝑘𝑖𝑧),
(1)
where 𝑘𝑖 is propagation wave number in the ith layer, 𝐴𝑖 and 𝐵𝑖 are amplitudes of forward and backward waves, respectively. When the intensity of incident light is high, the refractive index of Kerr nonlinear layer varies with increase of the intensity of local light, the amplitude of the forward and backward waves in the medium is not constant, therefore the traditional TMM is not suitable for this case. The incident light can be calculated reversely by output light based on reverse recursive algorithm [26,27]. The principle of reverse recursive algorithm is to divide the nonlinear layer into many sub-layers, as long as the number of sub-layers is large enough, each sub-layer can be regarded as a linear layer with a fixed refractive index. Therefore the nonlinear layer can be considered as a combination of multiple linear layers with different refractive index. The amplitude of the forward and backward waves in two adjacent sub-layers can be related by electromagnetic field boundary conditions, and expressed in transfer matrix form as: ) ) ( ( ⎡ ⎤ 𝑛 𝑛 1 [ ] [ ] ⎢ 1 1 + 𝑖−1 ] 1 − 𝑖−1 ⎥ [ 𝐴𝑖 exp(𝑖𝛽𝑖 ) 0 𝑛𝑖 2 𝑛𝑖 ⎥ 𝐴𝑖−1 ⎢2 = ( ) ( ) ⎥ 𝐵𝑖−1 𝐵𝑖 0 exp(−𝑖𝛽𝑖 ) ⎢ 1 𝑛 𝑛 1 ⎢ 1 − 𝑖−1 1 + 𝑖−1 ⎥ ⎣2 𝑛𝑖 2 𝑛𝑖 ⎦ [ ] 𝐴 = 𝑄𝑖 𝑖−1 , (2) 𝐵𝑖−1
2. Theoretical model and design The initial geometry (AB)N D(BA)N (N = 7) for structure optimization was shown in Fig. 1(a). It is a typical 1D Fabry–Perot resonator with symmetric Bragg mirrors. Where A and B indicate the low and high refractive index material SiO2 and TiO2 with the refractive index 𝑛A = 1.45 and 𝑛B = 2.3, respectively. D is the nonlinear Kerr medium polydiacetylene 9-BCMU with the linear refractive index 𝑛D0 = 1.55 and the third-order nonlinear coefficient 𝜒 (3) = 2.5×10−5 cm2 /MW, respectively. The optical thicknesses of A, B layers is 𝑛A 𝑑A = 𝑛B 𝑑B = 𝜆0 /4, and that of the D layer is 𝑛D0 𝑑D = 𝜆0 , where 𝜆0 = 1550 nm is the center wavelength. High reflection can be obtained for both forward (left-to-right) and backward (right-to-left) incidences regardless of the light intensity. When the spatial symmetry was broken by introducing a few additional dielectric periods of (BA) and/or (AB) into the structure on one side (such as (AB)7 D(BA)8 (AB)1 shown in Fig. 1(b)), optical nonreciprocal transmission can be obtained due to the different optical nonlinear response at different directions of incidence. Fig. 2 shows the typical transmission spectra of this kind of asymmetric 1D system. Divergent bistable responses for forward and backward incidences can be observed. As shown in Fig. 2, there is a wavelength range (the gray shaded area) where a high-transmission state only exists for one direction of incidence (forward incidence in this case). In this range
where 𝛽𝑖 = 𝑘𝑖 𝑑𝑖 , 𝑑𝑖 = 𝑑𝐷 ∕𝑚 is the thickness of the ith sub-layer, m is the number of sub-layers of the nonlinear layer, 𝑛𝑖 and 𝑛𝑖−1 are the refractive index of the ith and the (𝑖 − 1)th sub-layer, respectively. The 323
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Optics Communications 450 (2019) 322–328 Table 1 Detailed unidirectional transmission parameters (defined in Fig. 2) of different structures. The wavelength of incident light is 1563 nm and the intensity of it is 10 MW/cm𝟐 .
amplitudes of forward and backward waves in the (𝑖 − 1)th sub-layer can be calculated from that of the ith sub-layer as: [ ] [ ] 𝐴𝑖−1 𝐴𝑖 = 𝑄−1 , (3) 𝑖 𝐵𝑖−1 𝐵𝑖 𝑄−1 𝑖
where is the inverse matrix of 𝑄𝑖 . For simplicity, neglecting the saturation of absorption as well as the nonlinear saturation in the paper and considering the phase variation due to the nonlinear refraction is very small compared to the wave vector inside the sub-layer [28,29], the refractive index of (𝑖−1)th sub-layer can be approximately described as: 𝑛𝑖−1
𝜒 (3) | 2 = 𝑛𝐷0 + 𝐴 + 𝐵𝑖−1 || . 2𝑛𝐷0 | 𝑖−1
(4)
(5)
where 𝑀𝑖 (𝑖 = 1, 2, … , 𝑚) is the transfer matrix of the ith sub-layer. The transfer matrix of the structure (AB)N D(BA)M (AB)L can thus be described as: [ ] 𝑀11 𝑀12 𝑀 = 𝑀(𝐴𝐵)𝑁 ⋅ 𝑀𝐷 ⋅ 𝑀(𝐵𝐴)𝑀 ⋅ 𝑀(𝐴𝐵)𝐿 = , (6) 𝑀21 𝑀22 where 𝑀(𝐴𝐵)𝑁 and 𝑀(𝐵𝐴)𝑀 , 𝑀(𝐴𝐵)𝐿 are the transfer matrices of the stacks on the left and right side of the nonlinear layer D, respectively. Therefore the transmission of the structure can be expressed as: 𝑇 =
a
b
c
UTR width (×𝜆0 )
(AB)6 D(BA)7 (AB)1 (AB)7 D(BA)8 (AB)1 (AB)6 D(BA)8 (AB)2 (AB)7 D(BA)9 (AB)2
99.99% 99.47% 97.97% 97.91%
99.14% 90.45% 88.49% 85.03%
2.23% 0.28% 1.22% 0.33%
0.00013 0.00200 0.00210 0.00200
with that of (AB)7 D(BA)8 (AB)1 . Larger than 85% transmission for forward incidence and less than 1.22% transmission for the backward incidence can be obtained in a 0.002𝜆0 , i.e. 3.1 nm wavelength range at an input intensity of 10 MW/cm2 . More complex structures such as (AB)7 D(BA)10 (AB)3 can achieve better nonreciprocal transmission properties at the same input intensity, while the performance will deteriorate by using the structures with similar asymmetry but fewer layers such as (AB)5 D(BA)8 (AB)3 . The transmission spectra of these structures are not shown here due to the limited space. The optimized structures for nonreciprocal transmission can be expressed as: (AB)N D(BA)M (AB)L where M = N+1, L = 1; M = N+2, L = 2; M = N+3, L = 3; . . . and N ≥ 6. Pronounced improvement in optical nonreciprocal transmission can be simply achieved by introducing a few dielectric periods of (BA) and (AB) on one side of a typical Fabry–Perot resonator with symmetric Bragg mirrors (AB)N D(BA)N . The number of N should be larger than 6 to ensure sufficient nonreciprocal transmission ratio (higher forward transmission and lower backward transmission). The optimal operating region of input light intensity should be determined for a specific structure to ensure the high performance and operability at the same time. Fig. 6 shows the short, long wavelength limit, and the minimum transmission for the forward incidence in UTR with the input intensities of the structure (AB)6 D(BA)8 (AB)2 . 10 MW/cm2 seems to be an appropriate value to obtain relatively wide UTR and high forward transmission. Several key parameters of the optimized structure (AB)7 D(BA)8 (AB)1 are listed in Table 2 and show obvious advantages in unidirectional transmission when compared with that of several typical 1D nonlinear diodes reported previously. The contrast C is defined as C = (Tfor -Tback )/(Tfor +Tback ), where Tfor and Tback indicates the transmission for the forward and backward incidences, respectively. Asymmetry degree, which is defined as the ratio of the numbers of asymmetrical layers to the total layer numbers according to the metric function of mirror symmetry, is also calculated out for comparison. Interestingly, the asymmetry degree of the proposed structures can be as low as 11.12%, which is only 1/4 or less of that of previous designs.
The refractive index and the corresponding transfer matrix of each sublayer can be obtained backward recursively from the given intensity of output light. The whole transfer-matrix of the nonlinear layer can be expressed as: 𝑀𝐷 = 𝑀1 ⋅ 𝑀2 ⋯ 𝑀𝑖 ⋯ 𝑀𝑚 ,
Structures
|2 𝐼𝑜𝑢𝑡 || 𝐸𝑜𝑢𝑡 ||2 || 2 ⋅ 𝑛𝑖𝑛 | , (7) =| =| | | 𝐼𝑖𝑛 | 𝐸𝑖𝑛 | | (𝑀11 + 𝑀12 ⋅ 𝑛𝑜𝑢𝑡 ) ⋅ 𝑛𝑖𝑛 + (𝑀21 + 𝑀22 ⋅ 𝑛𝑜𝑢𝑡 ) |
where 𝐼𝑖𝑛 and 𝐼𝑜𝑢𝑡 are the intensities of incident and output light respectively, and 𝑛𝑖𝑛 and 𝑛𝑜𝑢𝑡 are the refractive indices of the incident and output environmental media, respectively. 3. Structure optimization A few additional dielectric periods of (BA) is firstly introduced into the initial structure on the right side to break the spatial symmetry of the 1D system. Fig. 3(a) shows the transmission spectra of the structure (AB)7 D(BA)N with N = 8, 9 and 10 respectively. It can be seen that the transmission is more dependent on the direction of incidence with the increase of N, which results in the increase of the width of UTR. However, the maximum transmittance decreases sharply with the increase of N. The attempt to enhance the maximum transmission by increasing the nonlinearity is confirmed to be not successful. Fig. 3(b) shows the transmission spectra of (AB)7 D(BA)8 at different input intensities. The maximum transmission keeps nearly unchanged with the increase of the input intensity although the transmission is more dependent on the direction of incidence with it. In order to enhance the maximum transmission by further increasing the asymmetry, a dielectric bi-layer (AB) is introduced into the structure on the right side. The periodicity of the right Bragg mirror is disturbed by a reversely arranged unit. Fig. 4 compares the transmittance of the structures (AB)7 D(BA)8 and (AB)7 D(BA)8 (AB)1 at different input intensities for forward and backward incidences. We can see that by inserting the unit (AB), the maximum transmissions for both forward and backward incidences increases simultaneously. High-performance nonreciprocal transmission can also be achieved in similar structures as shown in Fig. 5. The corresponding unidirectional transmission parameters (defined in Fig. 2) are listed in Table 1 in detail. Obvious trade-off between the forward transmission and the width of UTR can be observed. For the structure (AB)6 D(BA)7 (AB)1 , high forward transmission (near 100%) can only be obtained in a very narrow UTR. The optimized structures (AB)6 D(BA)8 (AB)2 and (AB)7 D(BA)9 (AB)2 have similar nonreciprocal transmission properties
4. Operation mechanism analysis The above-mentioned optical nonreciprocal transmission is originated from the nonlinear effect of light-matter interaction. A highintensity light, when traveling in a nonlinear medium, will induce an obvious varying refractive index of the medium due to the optical Kerr effect. This variation in refractive index will conversely modulate the field intensity distribution inside the medium. That is the optical Kerr effect manifests the matter and light temporally in refractive index and field intensity until a steady state is formed. Fig. 7(a) shows the distribution of the electric field intensity of the structure (AB)6 D(BA)8 (AB)2 for the forward and backward incidences. Fig. 7(b) shows the refractive index distribution in the Kerr defective layer D for both incidences. It can be seen that the electric field is anisotropically localized in different locations for the forward and backward incidences. For the forward incidence, the electric field is mainly 324
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Fig. 3. Transmission spectra of (a) (AB)7 D(BA)N with N = 8, 9 and 10 at the input intensity of 3 MW/cm2 and (b) (AB)7 D(BA)8 at different input intensities. Solid and dashed lines correspond to the forward and backward incidence respectively.
Fig. 4. Transmittance at different input intensities for forward (solid) and backward (dash) incidence for the structures (a) (AB)7 D(BA)8 and (b) (AB)7 D(BA)8 (AB)1 . The inserts are the enlarged top parts of the transmission curves. The wavelength of incident light is 1563 nm.
Fig. 5. Nonlinear transmission spectra for the forward (solid) and backward (dash) incidence of the structures (a) (AB)6 D(BA)7 (AB)1 ; (b) (AB)7 D(BA)8 (AB)1 ; (c) (AB)6 D(BA)8 (AB)2 ; and (d) (AB)7 D(BA)9 (AB)2 . The wavelength of incident light is 1563 nm and the intensity of it is 10 MW/cm2 . The UTR is denoted as the gray region in the figures.
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Table 2 Comparison of several key parameters in unidirectional transmission for different 1D nonlinear structures. References
Total layer numbers
Number of nonlinear layers
Laser power (MW/cm2 )
a (%)
b (%)
c (%)
UTR
C (%)
Asymmetry degree
This work [20] [13] [22] [23] [24] [21]
33 40 24 128 56 43 24
1 20 12 64 28 20 12
10 50 13 10 9.5 0.43 10
99.47 ∼100 ∼25 ∼98 92.05 ∼100 75
90.45 80 / ∼91 / ∼65 /
0.28 ∼0 ∼5 ∼5 22.7 <3 2.5
0.0019𝜔0 0.0033𝜔0 / ∼ 0.0010𝛺0 / 0.0036𝜔0 /
>99.38 >80 66.67 ∼90 60.4 >91.18 93.55
11.12% 50% 100% 50% 50% 45.46% 54.55%
Notice: ‘‘/’’ means not exist for single wavelength case. 𝜔0 is the central frequency.
The non-coincidence of the defect modes for forward and backward incidences will result in the optical nonreciprocal transmission in a certain wavelength range. The four defect modes on the right are corresponding to the input light with the wavelengths of 1560.3 nm, 1561 nm, 1562 nm, and 1563.6 nm (corresponding to the normalized wavelength of 1.0066, 1.0071, 1.0077 and 1.0088) respectively for the forward incidence. The UTR is also shown in the figure as the gray region. The transmittance at the four specified incident wavelengths is indicated as the black dots in the figure. The curve connecting the four dots forms the dispersion curve in the UTR (see the inset of Fig. 8(b)), which is corresponding to the upper part of the nonlinear (bistable) transmission spectra in the UTR shown in Fig. 5(c). 5. Angular and polarization All the analyses above are for normal incidences. Analyzing the angular dependency is of importance for practical applications especially for the multilayer resonators, since the lateral dimensions of an incident beam are always finite and thus such beam consists of multiple angular components. Fig. 9 shows the nonlinear transmission spectra of the optimized structure (AB)6 D(BA)8 (AB)2 at incident angles of 15◦ and 30◦ . The corresponding spectrum at normal incidence (0◦ ) is also shown in the same figure for comparison. It can be seen that the spectra shift to the short wavelength direction with the increase of incident angle for both polarizations. The deviation between the spectra of forward and backward incidences increases for TE polarization, whereas decreases for TM polarization with the increase of incident angle. The maximum transmission decreases with the increase of incident angle for TE polarization and is even not at the peak position due to the great bending of the spectrum at a larger incident angle (such as 30◦ ). Fig. 10 shows the short, long wavelength limit, and the minimum transmission (value b in Fig. 2) in UTR with the variation of incident angle for the forward incidence of the optimized structure (AB)6 D(BA)8 (AB)2 . It can be observed that both the short and long wavelength limits shift to the short wavelength direction with the increase of incident angle. For TE polarization, the UTR is broadened with the increase of incident angle due to the faster change of the
Fig. 6. Short, long wavelength limit, and the minimum transmission for the forward incidence in UTR with input intensities of the structure (AB)6 D(BA)8 (AB)2 .
localized in the nonlinear defective layer and a unique double-peak distribution can be observed. The different distributed field intensities inside the nonlinear layer for forward and backward incidences result in the great change of the refractive index of the layer due to its intensity dependency. Similar double-peak distribution of refractive index can be observed for the forward incident, while the refractive index nearly remains unchanged for the backward incident. Furthermore, the input lights with different wavelengths will have different refractive index distribution inside the nonlinear layer for the forward incidence due to the dispersion effect, whereas that for backward incident nearly remains the constant value of the linear refractive index at different wavelengths. Thus the location of the defect mode in the bandgap is different for the forward and backward incidences and will shift to the long wavelength direction with the increase of incident wavelength for the forward incidence, while it is almost fixed for the backward incidence, as shown in Fig. 8.
Fig. 7. Distribution of the electric field intensity (a) and refractive index in the nonlinear layer D (b) for the forward (black) and backward (red) incidences of the structure (AB)6 D(BA)8 (AB)2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Band gaps (a) and defect modes (b) of the structure (AB)6 D(BA)8 (AB)2 for the input lights with different wavelengths for the forward and backward incidences. UTR is denoted as the gray region. The transmittance at four specified wavelengths is indicated as the black dots.
Fig. 9. Nonlinear transmission spectra of the structure (AB)6 D(BA)8 (AB)2 at the incident angles of 15◦ (a and b) and 30◦ (c and d). The corresponding spectra at normal incidence (0◦ ) are also shown for comparison.
optimized structure can be generally expressed as: (AB)N D(BA)M (AB)L , where M = N+1, L = 1; M = N+2, L = 2; M = N+3, L = 3; . . . and N ≥ 6. The proposed geometry is shown to have advantages over previously reported designs. Interestingly, the asymmetry degree of the proposed structures can be only 1/4 or less of that of previous designs. Optimized structures (AB)7 D(BA)8 (AB)1 and (AB)6 D(BA)8 (AB)2 have been obtained numerically, which have excellent optical nonreciprocal transmission characteristics of wide UTR width (>3.1 nm), high transmission for the forward incidence (>88%) and low transmission for the backward incidence (<1.22%) at lower input intensity (10 MW/cm2 ) with fewer layers (33 layers). This optimal optical nonreciprocal transmission is originated from the nonlinear effect of light-matter interaction. The different distributed field intensities and refractive index inside the nonlinear layer D for the forward and backward incidences result in
short wavelength limit. The minimum transmission for the forward incidence in the UTR decreases with the increase of incident angle. Whereas, for TM polarization, the UTR becomes narrower with the increase of incident angle due to faster change of the long wavelength limit until it disappears at a certain incident angle (∼34◦ ). There is almost no significant change for the minimum transmission for the forward incidence in the UTR in a large incident angle range (0–24◦ ). 6. Conclusion In this paper, a novel and compact 1D geometry for optical nonreciprocal transmission was proposed based on the typical nonlinear Fabry–Perot resonator with symmetric Bragg mirrors (AB)N D(BA)N by introducing a few dielectric periods of (BA) and (AB) on one side. The 327
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Fig. 10. Short, long wavelength limit, and the minimum transmission in UTR with the variation of incident angle for the forward incidence at TE (a) and TM (b) polarizations of the structure (AB)6 D(BA)8 (AB)2 .
the non-coincidence of the defect modes in the bandgap, which causes the optical nonreciprocal transmission in a certain wavelength range. Take the optimized structure (AB)6 D(BA)8 (AB)2 as an example, the effects of the incident angle and polarization were also studied. For TE polarization, the UTR is broadened with the increase of incident angle and the minimum transmission for the forward incidence in the UTR decreases with it. Whereas, for TM polarization, the UTR becomes narrower with the increase of incident angle until it disappears at a certain incident angle (∼34◦ ) and there is almost no significant change for the minimum transmission for the forward incidence in the UTR in a large incident angle range (0–24◦ ). These results are of great importance to the design of high-performance optical nonreciprocal devices, such as all-optical diodes, isolators and limiters.
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