An optical diode that uses nonlinear ring resonators in 2D photonic crystal is numerically simulated by using the finite-difference time-domain (FDTD) method. Nonlinear polystyrene is used as the Kerr medium forming ring resonators. The operating wavelength of the optical diode is considered to be the coupling wavelength at which light couples efficiently from waveguide to ring resonator, which is also equal to the average of the resonant wavelengths of the two resonators considered in the proposed structure. For both forward and backward propagation, the characteristics of the proposed optical diode are similar to those of an electronic diode. FDTD simulation is done using the MEEP package, which exhibits the desired results. © 2013 Optical Society of America OCIS codes: (160.5298) Photonic crystals; (130.4310) Nonlinear; (130.4815) Optical switching devices. http://dx.doi.org/10.1364/AO.52.008252

1. Introduction

A photonic crystal (PhC) optical diode is an essential and useful component in all-optical circuits. Various mechanisms [1,2] have been developed to realize optical diodes using a PhC structure. The diode structure maintains the unidirectional propagation of waves through the structure. That means output power should increase with input power for forward propagation and a very low output power should be maintained for the reverse direction. Some of the mechanisms use the nonlinear effect using a Kerr medium in PhC. For example, Tocci et al. [3] proposed a nonlinear asymmetric distributed Bragg reflector (1D PhC) with a gradually increasing refractive index for each period after the first. This allows unidirectional propagation of light. Wang et al. [4] incorporated a defect layer in a 1D nonlinear PhC. Light interaction with this structure is anisotropic with 1559-128X/13/348252-06$15.00/0 © 2013 Optical Society of America 8252

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respect to left and right propagation, which behaves as a diode. Another design was proposed by Gao et al. [5] using a photon tunneling effect through a nonlinear cavity sandwiched between two air layers of different thicknesses. This forms a nonsymmetrical structure and behaves as a diode in the nonlinear case. In addition to 1D PhC, many designs are based on 2D PhC. For example, Mingaleev and Kivshar [6] put four nonlinear defect rods along a line defect whose positions were asymmetric with respect to both ends of the line defect. When the input power is high enough to induce nonlinearity in the defects, the sensitivity of resonant frequency becomes asymmetric to intensity change for forward and backward propagation, which results in a unidirectional waveguide transmission. Zhou et al. [7] made use of the asymmetric coupling strength of an incident wave with an asymmetrically confined single or paired nonlinear defect in two launch directions. Recently, Fan et al. [8] used the principle of wave coupling from a Si waveguide to a Si microring to show nonreciprocal transmission due to the nonlinear effect.

In this paper we have used the technique of light coupling from a PhC waveguide to a nonlinear PhC ring resonator in a 2D PhC to design an optical diode. The structure is designed such that the coupling factor from the waveguide to the resonator is different for forward and backward propagation. Hence, this produces an asymmetric configuration that acts like a diode when the nonlinearity is switched on in the resonators. Polystyrene (PS) is used as the Kerr medium in the ring resonator. Finite-difference timedomain (FDTD) simulation is carried out using MEEP [9] software to obtain the transmission and field propagation through the structure. 2. Proposed Structure of the Diode

The diode is designed using a 2D PhC structure consisting of square lattice of cylindrical linear Si rods in air. The Si rods have a refractive index of 3.4 and radius 0.185a, a being the lattice constant. The band diagram for this structure is calculated using MIT Photonic Band Package software [10], which gives a TM bandgap of 38.1% between band-1 and band-2, as shown in Fig. 1. Two ring resonators R1 and R2 are formed (as shown in Fig. 2) by replacing Si rods with nonlinear PS rods (shown as lighter dots). PS is a nonlinear organic conjugated polymer with a linear refractive index of 1.59. It has a large third-order nonlinear susceptibility of the order of 10−12 cm2 ∕W and a fast response time of the order of femtoseconds [11], making it a suitable candidate material for the current study. The resonant frequencies and Q factors of the two resonators are given in Table 1. The frequencies are given in normalized form a∕λ, where a is the lattice constant and λ is the wavelength of light. Both resonators have almost the same resonant frequency (Table 1), the difference in the surrounding rod structure accounting for the small difference in resonant frequencies. Two waveguides are also formed (one above and one below the rings) by removing a linear sequence of Si rods. The upper one covers both rings, while the lower one covers ring R2 only. This ensures that there is no direct coupling of light between R1 and

Fig. 2. Optical diode consisting of two PhC ring resonators R1 and R2 and two PhC waveguides. Forward and backward propagation directions are shown by arrows.

the lower waveguide. Ring R1 is separated from the upper waveguide by two rows of Si rods. Ring R2 is separated from the lower and upper waveguides by one and three rows of Si rods, respectively. This unequal gap between R2 and the waveguides results in an asymmetric configuration. Each resonator consists of four extra PS rods at the corners of each ring [12], whose radius is optimized to 0.191 to give an improved drop efficiency and spectral selectivity so that light from/to the waveguide to/from the resonator can be coupled effectively. 3. Principle of Diode Operation

The forward and backward propagation directions of the diode are indicated by arrows in Fig. 2. During forward propagation, input is given at port 1 and its output from port 2 is analyzed. Input from port 1 passes through R1 first, and the transmission curve after R1 at point A is shown in Fig. 3(a), which shows a dip at frequency 0.3211. The dip indicates that there is resonance coupling of light from the upper waveguide to R1 at this frequency, so there is a reduction in power at this frequency when it reaches point A. For backward propagation (input at port 2 and output analyzed at port 1), light couples to the upper waveguide at a resonant frequency of R2. The backward transmission curve after R2 at point A is shown in Fig. 3(b), which shows a peak at frequency 0.3211. Table 1.

Resonant Frequencies and Q Factors of Resonators

Resonator Fig. 1. Band diagram of 2D square lattice (inset) of cylindrical Si rods.

R1 R2

Resonant Frequency (a∕λ)

Q Factor

0.3210 0.3212

622.74 935.38

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For diode operation we will consider this coupling frequency of a∕λ 0.3211 as the operating frequency, which is also equal to the average of the resonant frequencies of the two resonators.

The two ring resonators gather optical energy at this coupling frequency. When the energy accumulated in the ring becomes high enough to induce optical nonlinearity, it will change the refractive index of PS. As is known, the average change in refractive index of a Kerr medium due to intensity I is given by Δn n2 I, where n2 is the Kerr coefficient. To induce nonlinearity, the input intensity should be above its threshold value. For all-optical operation of the device, this nonlinearity may be switched on/off optically through an external source illuminating the resonators, or by an increase in power at the input. For optical switching applications, external source illumination may be suitable. For the proposed diode operation of the device, an input power above threshold is used to induce nonlinearity. A change in the refractive index of the material (PS) of the PhC ring will shift its coupling frequency. Since resonator R2 is coupled to the upper waveguide through a larger gap than with the lower waveguide, less energy will be available in R2 during forward propagation than during backward propagation. Therefore, the amount of resonance shift for R2 in the forward direction will be smaller than that in the backward direction. This is the underlying principle behind the working of the current optical diode configuration. A. Linear Case

When the input intensity is below its threshold value, there is no nonlinear effect in the resonators. In forward propagation, most of the power in the input light at the operational frequency of a∕λ 0.3211 couples to R1 and very little power reaches port 2. Hence we will get a dip in the transmission at port 2 at the frequency 0.3211, as shown in Fig. 3(c). In the case of backward propagation, light from port 2 couples to the upper waveguide through R2 and propagates further but gets coupled to R1 , resulting in a dip in the transmission curve at port 1, at the operational frequency. This is shown in Fig. 3(d). Since both the forward and backward transmission curves show dips at the operational frequency, in the linear case, transmission is independent of the propagation direction. Figures 4(a) and 4(d) show the corresponding field propagation in the forward and backward directions at the operational frequency of a∕λ 0.3211, which shows how light is coupled to the corresponding resonators, leading to smaller but similar outputs at the corresponding output ports. B. Nonlinear Case

Fig. 3. Transmission in the linear case. (a) Forward transmission at A, (b) backward transmission at A, (c) forward transmission at port 2, and (d) backward transmission at port 1. Parts (c) and (d) indicate that transmission in the linear case is independent of the propagation direction, because the dips for both cases are at the same frequency of 0.3211. 8254

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We show here that the structure works like a diode when the nonlinearity is switched on in the resonators. In the forward direction, when input power at port 1 is such that it induces nonlinearity in R1 , there will be a downshift of the resonance frequency of R1, which reduces the coupling of light of frequency 0.3211 to R1 , as shown in Figs. 4(b) and 4(c), where the light propagates in the upper waveguide toward R2 . However, due to power reduction in propagation

Fig. 4. (a)–(c) Field propagation in the forward direction, which shows that (a) a very negligible power reaches port 2 in the linear case, whereas (b), (c) the power at port 2 increases with increasing input power at port 1; (d)–(f) field propagation in the backward direction, showing much less power reaching port 1 with increasing input power at port 2.

and also a larger gap to R2 , the resonance frequency of R2 is unaltered, and hence light gets coupled to R2 . The light can further couple to the lower waveguide and transmit to port 2, achieving reasonable transmission as shown in Figs. 4(b) and 4(c). This increase in transmission for the forward propagation at the operational frequency is also depicted in Fig. 5(a). The dotted line corresponds to the transmission in the linear case. The solid lines (higher thickness for higher input power) are the transmission plots for the nonlinear case. The nonlinear curves are shifted leftward with respect to the linear curve due to a resonant frequency shift in the nonlinear case. Since the device is operated at a frequency of 0.3211, power at port 2 increases with input power at port 1, indicated by vertical arrows in Fig. 5(a). During backward propagation, a small increase in input power at port 2 will shift the resonant frequency of R2 because of the small gap between R2 and the lower waveguide. Hence there will be a reduction in light coupling to R2 at the operational frequency, as shown in Figs. 4(e) and 4(f), leading to negligible power at port 1. Any further increase in the input power at port 2 repeats the same process, and a dip is obtained at the operational frequency independent of the input power at port 2, as shown in Fig. 5(b). 4. Discussion

In the nonlinear case, output power increases with input power in the forward direction and remains

at low power for backward propagation. This confirms the diode-like behavior of the proposed device, whose characteristic curve is shown in Fig. 6. In that figure transmittance, which is defined as the ratio of output power to input power, is plotted against input power for both forward and backward propagation. We can get 57% contrast for forward output to backward output for the proposed diode. As shown in the figure, the curve matches a diode characteristic curve within a certain range of powers. The curve in the forward propagation case bends downward beyond a certain power (indicated by the vertical arrow). This is because as power increases beyond this limit, it will induce nonlinearity in R2 also. As a result its coupling frequency will be changed. So there will be no further coupling to R2 , and transmission at port 2 goes on decreasing. In the low power region (near the origin) the forward curve is almost horizontal (shown in the inset of Fig. 6) up to certain power. This shows that a threshold power is needed to start the nonlinear effect in the forward propagation direction. Referring to Fig. 6, this threshold power (per unit length) is around 2.5 (arbitrary unit), which corresponds to a light intensity of 2.69 GW∕cm2. This much light intensity increases the refractive index of PS (n2 0.2966 × 10−12 cm2 ∕W) in R1 by 0.0008. Up to that threshold power, there is no increase in output power during forward propagation. And because of the smaller gap between R2 and the lower waveguide, a comparatively smaller light intensity of the order of 0.036 GW∕cm2 at input port 2 is 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS

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Fig. 6. Variation of transmittance with input power for forward and backward propagation, which is similar to the characteristics of a diode. The horizontal nature of the curve in the low power region is clearly visible in the magnified image shown in the inset.

can provide less waveguide dispersion and highQ-factor resonators because of the photonic bandgap. We can tune the resonant wavelength (λ) by changing the lattice parameter (a) of the PhC structure. 5. Conclusion

Fig. 5. Forward and backward transmission spectra, where the dotted and solid lines correspond to the linear and nonlinear cases, respectively. (a) As intensity increases, the curve shifts toward the left, so the output power at 0.3211 increases as indicated by vertical arrows and (b) there is no shift of the curve with increased intensity, so the dip is always at 0.3211. The thicker solid curves indicate more intense input light.

needed to induce nonlinearity in R2 during the backward propagation direction. So the range of input power within which the structure behaves like a diode in the forward propagation case starts when nonlinearity is switched on in R1 and ends when nonlinearity is induced in R2 in the forward propagation direction. The performance of an optical diode using nonlinear material to achieve optical nonreciprocity greatly depends on the material’s nonlinear properties. In contrast to semiconductor [8] materials, our proposed diode contains organic PS as the nonlinear material. Semiconductor materials have relatively small nonlinearity (of the order of 10−14 cm2 ∕W) and very slow response time compared to PS, due to slow relaxation of carriers generated by twophoton absorption. Typically the response time for bulk Si and GaAs are of the order of nanoseconds and hundreds of picoseconds, respectively. But use of PS can give our diode a response time of the order of 10–20 fs [11]. Also, due to its larger nonlinear coefficient, PS can give the diode a lower threshold power than structures using a semiconductor as a Kerr medium. In contrast to waveguides and ring resonators made up of Si, the proposed PhC diode 8256

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In conclusion, we have demonstrated the diode behavior of our proposed structure containing PhC waveguides and nonlinear ring resonators with PS as the nonlinear material. The asymmetric coupling strength between waveguides and resonators for two propagation directions plays a key role in making the structure behave like a diode. Both the resonators R1 and R2 show nonlinear behavior when the input intensity exceeds its corresponding threshold values, so that the structure works like a diode for both forward and backward propagations, respectively. Nonlinearity in R1 is needed to shift its resonance frequency from 0.3211 (operating frequency), so that light at 0.3211 can propagate farther to port 2 without being coupled to R1 . Nonlinearity in R2 is needed to change its resonance frequency from 0.3211 so that light at this operating frequency cannot couple to R2 , and we will always get very negligible output power at port 1 even if input power is increased at port 2. In the forward propagation case the structure behaves like a diode within a range of intensities, which starts when nonlinearity is switched on in R1 and ends when nonlinearity is induced in R2 . The simulation results reveal that our optical diode characteristic is similar to that of an electronic diode. It is expected that experimental realization of the proposed structure, even though difficult, is possible because it is based on a 2D PhC structure, which is less complex to fabricate using interference lithography or electron beam lithography, and direct laser writing can be used to make defects at precise positions on the structure. References 1. D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110, 093901 (2013).

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