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We realize all-optical sensitive phase shifting based on nonlinear out-of-plane coupling to a slab waveguide through Fano resonance of a slab 1-D photonic crystal (PhC). We use a graphene layer as the nonlinear material and change its refractive index by the input light intensity through Kerr nonlinear effect to obtain a shift in the Fano resonance frequency. The Fano resonance and self-focusing effect lead to light-intensity enhancement on the graphene in the PhC, reinforcing the nonlinear effect of refractive index in the graphene. Through finite-difference time-domain simulation, we demonstrate that the phase changing sensitivity obtained can be 4 orders higher than that by a single graphene under the same input light intensity. Moreover the threshold pump intensity for all-optical sensitive phase shifting in the coupled light to the waveguide is as low as ~4 MW per square centimeter. The results are applicable in micro optical integrated circuits for phase shifters, phase modulators, power limiters, and phase logic elements for optical computation, digital phase shift keying in communication systems, and non-contact sensitive signal detectors.
© 2014 Optical Society of America
1. Introduction
The design of optical devices using graphene has attracted great attention in recent years due to its special ability in tuning its absorption and refractive index by electro-optical and nonlinear effects [1–6]. Another special interesting optical property of graphene is that this tuning ability can be used in a wide range of wavelength spectrum from UV to far infrared and microwave with constant absorption coefficient [7–10]. Such a favorable absorption nonlinearity of graphene makes it a good choice as saturable absorption material in mode locking and Q switching applications for lasers [10–12]. Also graphene has high Kerr nonlinearity as large as n2 = 10−7 cm2/W [13]. However, the main limitation for using the high Kerr nonlinearity of graphene in designing nonlinear optical devices is its relatively high absorption coefficient (~2.3% for each layer) and its low thickness (~0.34nm) which forces people to use a large number of graphene layers (that leads to higher absorption) or a high-intensity input wave or pump beam to get sensible Kerr nonlinear effects.
In order to overcome the mentioned limitations especially for integrated optics applications, in this paper, we propose to use Fano resonance effect in a one dimensional slab photonic crystal (PhC) with a single layer of graphene for nonlinear light coupling to a dielectric waveguide through the slab PhC.
Fano resonance is due to the surface coupling of light at the frequencies corresponding to the discrete leaky modes in the PhC [14]. At Fano resonance frequencies, the interaction of the out-of-plane incident light with PhC modes leads to two effects: light intensity enhancement due to resonance effect and sharp changing in the reflection and transmission spectrum of the PhC. Through the nonlinear graphene, with the change of refractive index (due to Kerr nonlinear effect by using different pump intensities) the frequency of the Fano resonance changes. Moreover, the sharp change in the spectrum of the Fano resonance leads to high sensitivity of reflectance and transmittance of the PhC to a small change of refractive index of materials in the PhC.
At Fano resonance it is expected that, the interaction of out-of-plane incident beam with PhC modes also leads to sharp change in the phase of the incident light in addition to its intensity change. Until now the ability of sharp change in the intensity of the incident beam at Fano resonance has been used to design all-optical switches in various nonlinear PhCs [15–18], but no report about the ability of phase changing by Fano resonance in PhCs. We show the phase changing ability by Fano resonance in one dimensional slab PhCs and its application to design all-optical phase shifters by using a single layer of graphene as nonlinear material.
In order to reduce the limitations related to the absorption of graphene, we used a single layer of graphene inside one of the cells of the PhC. For a single layer of graphene, its absorption and initial refractive index have no significant effect on the properties of the PhC.
We have used finite-difference time-domain (FDTD) method to show that for out-of-plane illumination on the PhC, the light coupled to the waveguide by the PhC with Fano resonance (Fig. 1) undergoes a sharp phase change due to the refractive index change in the single layer of graphene.
Fig. 1 Structure studied. It consists of a one dimensional PhC slab, which is illuminated vertically, and two dielectric waveguides. The PhC slab has 9 periods of Si and air with a single layer of graphene (Gr) in the middle Si layer. We consider an equivalent composite of Gr and Si with d = 0.02a at y = 0.03a. The horizontal width of the composite layer is the same as that of the silicon layer.
The property of sharp phase change can be applied in integrated optical circuits for optical phase logic devices, sensitive remote signal detection, digital phase shift keying (PSK), phase modulators, and optical limiters in optical communication systems and optical signal processing systems.
2. The structure to be studied
The structure investigated is shown in Fig. 1. It includes silicon (Si) PhC of 9 periods with a single layer of graphene in the middle Si layer. The substrate is SiO2 and the PhC is between two Si waveguides. The filling factor (F = w/a) of Si and the thickness of each Si layer in the PhC are 0.9 and h = 0.5a, respectively, where a is the period of the PhC that is considered to be a≈810nm for operation at the optical communication wavelength at λ = 1550nm. The filling factor and thickness have been optimized to have maximum quality factor or sharpness of Fano resonance and waveguide coupling coefficient for normalized frequencies (f = a/λ) in the range of [0-0.7], with changing h and F in the range of [0.05a, a] and [0.05a, 0.95a], respectively.
3. Calculation method
For the computation, we have used our two-dimensional FDTD code with uniform grid size of 0.02a and perfectly matched layer as boundary condition. The dielectric constant of Si and SiO2 are taken as 12 and 2.1, respectively. As is known, a layer of graphene is very thin. It is so thin that it will take tremendous computation time in FDTD simulations. As a practical treatment, we replace the graphene layer by an equivalent composite layer with an equivalent dielectric constant.
Graphene conductivity and its dielectric constant can be calculated as follows [7, 19, 20]:
where dg≈0.34nm is the graphene single layer thickness and EF, ε0, T, KB and ω are the graphene Fermi energy, vacuum permittivity, temperature, Boltzman constant and angular frequency, respectively. By considering EF = 0, T = 300K and λ≈1550nm we have:So, in the simulation we considerFor the light polarization parallel to the graphene plane, the dielectric constant of the equivalent composite layer of graphene and silicon can be written out as [20, 21]:
where εd, d and ρ = dg/d are the silicon dielectric constant, vertical height (thickness) of the composite layer and volume contribution of the graphene inside the composite layer. We consider d = 0.02a as the thickness of the composite layer at y = 0.03a. The composite layer is inside the middle Si layer.Since the PhC is designed for operation in a narrow bandwidth, it is sufficient to calculate the dielectric constant by using single operating wavelength (λ = 1550nm) in Eq. (6).
For the nonlinear calculation, the graphene refractive index increment can be written as a function of the intensity I:
with n2 = 10−7 cm2/W and Is = 600 MW/cm2 [13]. Then the increment of the equivalent dielectric constant of the composite layer is calculated bywhere Δεg can be calculated aswhere kg is the graphene imaginary refractive index.By considering the decreasing nature of the absorption upon increase in the intensity, one can have
Thus the maximum required threshold intensity can be calculated by considering only the real refractive index nonlinearity. Hence for further simplification, we have neglected the effect of absorption nonlinearity. Moreover, as we will show in the next section, for a single layer of graphene, the absorption has no significant effect on the designed PhC.
In order to determine the suitable PhC structure, Fano resonance frequency, and position of graphene, we first calculate the waveguide coupling spectrum (versus normalized frequency) without the graphene using linear FDTD. For this purpose, we illuminate the PhC normally (see Fig. 1) with a Gaussian beam with a full width of half maximum (FWHM) being 5a and a polarization orienting in the direction of the axis of the grooves in the PhC. We have performed such calculations for various PhC structures with different height and filling factor as described in the previous section, for finding out the best Fano resonance frequency in view of sharpness and strength of resonance mode in the coupling coefficient spectrum.
Since higher intensity of field leads to higher change in the refractive index, we set the graphene to be inside the middle Si layer in the PhC. Also, we set the graphene to be at the height (vertical position) at which the electric field takes maxima.
Finally, we have performed nonlinear FDTD simulation to calculate phase change and intensity of the coupled light inside the waveguides for frequencies around the selected Fano resonance for different intensities of pump beam.
In the calculation, the coupling coefficient is defined as
where and are respectively the input power from top, the power coupled to the left-hand side and the power coupled to right-hand side. In the one-dimensional system, the term power is an abbreviation of power per unit length in the direction normal to y-axis and the propagating vector of the wave in Fig. 1. The power has thus the unit of W/cm and is the integration of intensity over the beam width. For the structure considered with mirror symmetry about the vertical axis, we haveThe transmission coefficient is defined aswhere is the power transmitted to the bottom.4. Results and discussion
We first look at the case without the graphene layer. In this case, the coupling coefficient inside the horizontal waveguides and transmission coefficient for the vertically transmitted beam (as indicated in Fig. 1) for optimized PhC structure are shown in Fig. 2. As can be seen from Fig. 2, the coupling coefficient has a sharp peak at the frequencies that the vertical transmission coefficient has also a relatively sharp change, which is the Fano resonance. We can see that, there is a relatively sharp Fano resonance at normalized frequency of f = 0.525. For a detailed view, the distribution of the square of electric field amplitude inside the PhC at this Fano resonance frequency is shown in Fig. 3. In this Fano resonance, the maximum electric field occurs for the height of y = 0.03a (from the bottom of the Si layer). Hence, we have set the graphene layer in this position in this paper to enhance the effect of nonlinearity as large as possible.
Fig. 2 The transmission coefficient (upper plot) of power from top to bottom in the PhC without graphene (for vertical incidence of the pump) and the coupling coefficient in one of the waveguide (lower plot) versus normalized frequency (a/λ)
Fig. 3 The distribution of the square of electric field amplitude inside the PhC for f = a/λ = 0.525.
Then we look at the case when the graphene layer is inserted. In Fig. 4, the coupling coefficient of power to the horizontal waveguides and the vertical transmission through the PhC at normal incidence at the selected Fano resonance are shown for various numbers of graphene layers (N) between 0 and 10. It is evident that the absorption of graphene leads to the reduction of the Fano resonance sharpness or quality factor (and increasing of the bandwidth as well) and also reduction in the coupling coefficient. These effects will lead to the reduction of light intensity enhancement at Fano resonance and also reduction in its sensitivity to refractive index change. However, as can be seen from Fig. 4, for only a single layer of graphene, the effect of absorption is negligible.
Fig. 4 The transmission coefficient (upper plot) of the PhC (for normal incidence) and coupling coefficient in the waveguide (lower plot) for various number of graphene layers (N).
In Fig. 5, the phase change in the light coupled to the waveguides (for the case with a single layer of graphene inserted in the PhC) is plotted against the peak intensity of the input pump, for three normalized frequencies through nonlinear simulations. These frequencies are in the range of the bandwidth of the Fano resonance and near the position of the peak of coupling coefficient (corresponding to the center frequency of Fano resonance) with distances of Δf ≈0.001, 0.002 and 0.003 which correspond to f = 0.524, 0.523 and 0.522, respectively. Moreover, in Fig. 6 the coupling coefficient and average coupling intensity versus pump peak intensity for the three frequencies are shown.
Fig. 5 The phase change of the light coupled to one of the waveguides versus peak intensity of incident beam (pump) for f = a/λ = 0.522, 0.523 and 0.524.
Fig. 6 The coupling coefficient (upper plot) and the average coupling intensity Iout (lower plot) versus peak intensity of the incidence beam (pump) for f = a/λ = 0.522, 0.523 and 0.524.
From Fig. 5 we can see that the absolute value of phase change in 0-5 MW/cm2 of pump peak intensity is quite large and is more than 0.3π and 0.45π for frequencies of 0.524 and 0.522, respectively. So, this region can be applied for phase shifters. Moreover, Fig. 5 shows that the phase change sharply near 3.5 MW/cm2 (for frequency of 0.522), so it can be applied to detect the intensity change of the input power with high sensitivity. Furthermore, this can find application for phase logic elements in micro optical integrated circuits and optical PSK for communication systems.
To explain the results in Fig. 5, we need to be aware of the following three factors. First, light intensity can be enhanced due to Fano resonance that is larger for the frequencies nearer to the center of Fano resonance spectrum. This light enhancement leads to greater change of graphene refractive index and thus more shift of Fano resonance frequency. Second, the sensitivity in phase change is high near Fano resonance frequency due to sharp change of the speed of light in the structure and sharp change of mode distribution near Fano resonance frequency. And third, light intensity can be enhanced due to self-focusing effect.
The first and second factors originate from the Fano resonance property and the nonlinear property of graphene. Through the first factor, we can tune the Fano resonance frequency to the operating frequency by changing the pump intensity. However, higher pump intensity is needed for larger tuning of frequency. The second factor also means that the tuning becomes very sensitive at a certain value of pump intensity, which can be regarded as a threshold. Thus, the threshold pump intensity for different operating frequency is different and a larger threshold pump intensity is required for a greater tuning of frequency. The first and second factors together means that a sensitive phase shifting can be obtained by changing the pump intensity at the threshold pump intensity. This agrees with the result indicated in Fig. 5. At lower pump intensity, the absolute phase shift for f = 0.524 is the largest because it is the closest to the Fano resonance frequency. Moreover, as the distance of f = 0.524, 0.523, 0.522 to the Fano resonance frequency increases, the threshold pump intensity increases also (~2MW/cm2, 2.5MW/cm2 and 3.5MW/cm2, respectively for f = 0.524, 0.523 and 0.522, as shown in Fig. 5).
The third factor originates from the fact that electric field is concentrated in the high-refractive-index region in a structure. The self-focusing effect becomes stronger when the pump intensity is increased, as shown in Fig. 7 for the frequency of 0.522, which shows a strong focusing of light under a pump intensity of 4MW/cm2. Moreover, the self-focusing effect can help speeding up tuning the Fano resonance frequency. This factor together with factors 1 and 2 explain the property shown in Fig. 5 near the threshold pump intensity: as the threshold pump intensity for larger frequency distance (from the center of Fano resonance) gets higher, there is correspondingly also a higher sensitivity of phase shifting in the sensitive tuning region, or the region near the center of Fano resonance.
Fig. 7 Distribution of square of electric field amplitude in the structure for the normalized frequency of 0.522 for pump intensities of 2 MW/cm2 (a), 3 MW/cm2 (b), and 4 MW/cm2 (c).
The effects of the three factors can join up together, leading to very high sensitivity of phase shifting. For example, as can be seen from Fig. 5, for f = 0.522 we have Δφ ≈-0.35π (rad) for ΔI = 1MW/cm2 or |Δφ/ΔI| ≈0.35 (radMW−1cm2) at the intensity threshold of Ith≈3.5 MW/cm2. Such an amount of phase changing is more than one order of magnitude greater than the phase shift that can be obtained using a single layer of graphene at high intensities of about 1GW/cm2 for the input wave [13]. In another word, for the same pump input power, the maximum phase changing slope or sensitivity obtained by the structure in Fig. 1 is 3~4 orders of magnitude higher than that by a separate layer of graphene reported in [13].
Another interesting behaviour that can be seen from Figs. 5 and 6 is that the phase shift accompanies with a decrease of the coupling coefficient when the input pump power is greater than a certain value, resulting that the coupled power being approximately constant for the pump beam intensity beyond a certain value. This phenomenon is reasonable because the bandwidth of the Fano resonance is small and also because the large graphene refractive index due to high pump intensity moves the resonance frequency of the PhC further away from the operating frequency, leading to less coupling coefficient, further leading to an approximately constant coupled intensity of wave as the pump increases.
Such an effect is just the one that all-optical phase modulators require. As an example, we can see from Fig. 6 that, for f = 0.522 the phase changes about 0.4π with coupled intensity being about the same for the intensity of input pump from 3.2MW/cm2 to 4.5MW/cm2.
The coupled-power saturation property is also applicable for optical limiters in which the coupled power increases with the input power at first, but later becomes constant when the input power is greater than a certain value.
For a further point, we mention that the structure in Fig. 1 can be used as a sensor for non-contact detecting in which any variation of the input power is reflected in the phase of the coupled output wave. The phase modulated signal can be transmitted through a waveguide or a fiber to a remote place and processed there.
At last, we point out that the coupling coefficient is no more than 20% for the structure shown in Fig. 1 which corresponds to a low power efficiency. To improve the power efficiency, we can use the structures indicated in Fig. 8. In Fig. 8, the right-hand-side transmitting waves are reflected towards the left-hand side by the metal layer or by the extended region of photonic crystal with different filling factor to form band gap near the operating wavelength, so that the power transmitted to the left side can be doubled. Also it is possible to use non-vertical incident pump to increase the couple coefficient to one of the waveguides. Detailed study of the improved structures is omitted here to save pages.
Fig. 8 Improved structures that can double the coupling coefficient or power efficiency. A silver or other high reflective metal layer (blue) is set in (a) and an extended region of photonic crystal in the right hand side in (b).
5. Conclusion
All-optical sensitive phase shifting and saturable power coupling are realized based on nonlinear out-of-plane coupling to a slab waveguide through Fano resonance effect in a one- dimension slab PhC with a single layer of graphene inside the PhC.
By increasing of the intensity of pump light, the refractive index of graphene increases due to Kerr nonlinearity, resulting the shift of Fano resonance frequency in the 1-D slab PhC. At the same time, Fano resonance can enhance the nonlinear effect of graphene. In addition, in this design due to concentration of graphene in a small volume of the PhCs, field tends to focus on graphene when the refractive index of graphene becomes high, which helps to get more light intensity at graphene position. Therefore, light intensity enhancement due to Fano resonance and self-focusing add together to provide greater refractive index change at lower pump intensities.
The threshold input power of pump for sharp phase change can be in the order of MW/cm2. A phase change of about 0.4π has been obtained. Also, a slope of phase changing of about 0.3π/(MW/cm2) has been obtained that is about 4 orders of magnitude greater than the phase shift through a separate graphene layer.
Besides application in phase shifters and phase modulators, the results have also good potentials for applications in optical phase logic devices for optical computations, optical PSK in communication systems, optical limiters for optical digital signal generators, non-contact sensors, optical signal processing systems and optical detectors with integrated optical circuits.
Acknowledgments
This work was supported by the NSFC (Grant No.: 61275043, 61171006, 61107049, 60877034), the Guangdong Province NSF (Key project, Grant No.: 8251806001000004) and the Shenzhen Science Bureau (Grant No. 200805, CXB201105050064A).
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