A type of modulator based on a shallow-etched photonic crystal (PC) slab of silicon-on-insulator material with electro-optic (EO) polymer cladding is designed and investigated. The transmission spectra of the PC slab with the EO polymer are calculated using a finite-difference time-domain method. The band structure and the field distribution of the guided mode resonance are calculated and analyzed. The modulation voltage and bandwidth of the hybrid modulator are simulated. It is shown that flexible designs of low-voltage modulation (0.2 V) or high-bandwidth modulation (62 GHz) can be obtained with the hybrid modulator.
© 2014 Optical Society of America
Modulators are key components in information processing and optical communication fields. For planar modulators based on silicon-on-insulator (SOI) material, there are three main types: micro-ring modulators, Mach-Zehnder interferometers (MZIs) and electro-absorption modulators . For micro-ring resonance modulators, the speed can be as high as 40 Gb/s , but their performance is highly sensitive to ambient temperature. The MZI modulators based on SOI materials can reach a high speed, up to 60 Gb/s . However, the MZI modulators are of sizes of the order of millimeters, which are too large for photonic integration. Modulators based on photonic crystal (PC) structures can reduce device size. Moreover, the slow-light effect in PCs can enhance the non-linear effects of the modulators . Recently, Hong C. Nguyen et al. experimentally demonstrated a 10 Gb/s modulation in a PC waveguide-based MZI modulator with 200 arms . Silicon-based PN junction modulators use the plasma dispersion effect to control the optical waveguide index. The speed of this effect is limited to dozens of GHz by carrier mobility. Although good efforts have been made, it is still difficult to obtain higher modulation speeds (100 Gb/s) using silicon material alone.
Electro-optic (EO) polymers with large second-order nonlinearity coefficients have attracted increasing attention because of their applications in high-speed EO modulation. W. Freude et al. proposed a 78 GHz compact silicon-based MZI modulator with a slotted PC waveguide arm that was infiltrated with EO polymer . EO modulation in slotted PC waveguides based on SOI substrates covered and infiltrated with EO polymer has also been experimentally demonstrated . Che-Yun Lin et al. reported a similar structure with 23 dB EO coefficient enhancement using the PC slow light effect . Up to THz modulation effects in EO modulators have been reported . The value of the second-order nonlinearity of a recently reported EO polymer is several times higher than that of the commercial modulator material . Moreover, the EO polymer is easy to integrate with the SOI devices, and is compatible with standard semiconductor processing.
Interest in research of guided resonance in PC slabs is increasing. Guided resonance modes in PC slabs were first investigated by S. H. Fan et al. . These guided resonance modes not only can be confined to the slab, but also can couple to external radiation. For light incident from free space to the PC slab, the guided resonance modes can strongly affect the transmission and reflection spectra. Fano modes will appear in these spectra, and the Fano line shape is asymmetric. The transmission varying between the minimum and maximum with this Fano line shape has a very narrow line width, which is suitable for applications in narrow-band filters, modulators, etc. Qianfan Xu et al. demonstrated a compact spatial light modulator by using cavity modes in an SOI slab with perturbation-base gratings . To our knowledge, an optical modulator based on guided resonance modes has not been investigated until now. Here, we study modulators based on the EO polymer and PC slabs with Fano line shape transmissions. These modulators will have potential applications in imaging, display, holography, metrology and remote sensing.
2. Guided resonance transmission of the PC slab
It is found that the guided resonance transmission of the PC slab overlaps the background transmission of the slab without PC. The formation of transmission dips or the Fano line shape of the PC slab result from the interaction between the guided resonance modes and the original transmission. This phenomenon was discussed by S. H. Fan . The original transmission spectra of the SOI slabs with different slab thicknesses are shown in Fig. 1.It can be seen that the transmission oscillates slowly with wavelength, and the maximum and minimum values are located at different wavelengths for different slab thicknesses. If the resonance frequency is located at a high original transmission region with transmission around 1.0, the guided resonance mode will form a symmetric dip at that frequency. If the location of the resonance is at an original transmission of 0.5, the guided resonance mode usually forms an asymmetric peak with the transmission changing from a minimum to a maximum within a very narrow frequency range, which is the Fano mode. Therefore, we need to choose an appropriate thickness of the PC slab to obtain the required line shape of the resonance guide mode.
There are two commonly used SOI materials with top Si layer thicknesses of 220 nm and 340 nm. As demonstrated in Fig. 1, the transmission of the SOI slab with a thickness of 220 nm is 1.0 at a wavelength of 1550 nm, while for a slab thickness of 340 nm, the transmission is 0.55. To obtain a sharp Fano line shape, we choose the 340 nm material.
Fan et al.  have presented an equation for describing the line shape of the Fano mode in the transmission spectra of the PC slab. Actually, the equation presented by those authors is a semi-empirical formula. The parameters of the guided resonance modes and the accurate spectra should be obtained from appropriate numerical simulations like the finite-difference time-domain (FDTD) method and rigorous coupled-wave analysis method. The characteristics of the transmission spectrum of the PC slab result from the interaction between the guided resonance modes and the original transmission. In essence, these phenomena originate from multiple scattering and interference of light by the PC holes, which depend on the complicated structure of the PC slab.
S. H. Fan studied the radius dependence of the resonance . Firstly, reducing the hole radius will increase the effective index of the PC slab structure, which results in red shifts of the resonance peaks. Secondly, as the holes shrink, the Q factors of all resonance peaks increase. Higher Q factors mean narrower resonance modes. This is useful for some applications such as optical switches and modulators.
Here, we use another strategy to increase the Q factor: the shallow-etching method. Shallow etching means that the PC holes are etched shallowly onto the slab without etching through the slab completely. In structures such as waveguide grating and DFB lasers, shallowly etched gratings , which can also be called one-dimensional PCs, are commonly used.
We calculate transmission spectra of ordinary PC slabs with and without the cladding EO polymer layer, as shown in Fig. 2. It can be seen that all peaks in the spectra shift to the red after addition of the EO polymer layer; the highest transmission of the slab with the EO polymer is higher than that without the polymer layer. It shows that the symmetry structure is suitable for high extinction ratio applications such as switches and modulators.
The etching ratio is defined as the ratio of the volume of the etched slab to that of the whole slab. Figure 3 shows the normal incidence transmission spectra for the shallow-etched PC slab with radius 0.25a and depth 0.4h (), the ordinary PC slab with 0.16a radius and the ordinary PC slab with 0.25a radius. The two structures of shallow-etched and changed-radius PC slabs both have red shifts and increase the Q values of the resonance peaks. Transmission spectra of shallow-etched and changed-radius PC slabs were calculated, and the frequencies and Q values of the two modes in the low-frequency region are collected and shown in Fig. 4. It is obvious that the red shifts are different for the same resonance modes in the two different structures. The mode around the normalized frequency 0.40 of the ordinary PC slab with 0.25a radius has a large shift when the etching ratio changes. When the etching ratio reduces to 40%, the shallow-etched PC slab mode (SE-low mode) shifts to the normalized frequency 0.38, and the changed-radius PC slab mode (CR-low mode) shifts to 0.39. The mode at the normalized frequency 0.475 of the ordinary PC slab with 0.25a radius shifts a small amount when the etching ratio is changed. When the etching ratio reduces to 40%, the mode in the shallow-etched PC slab (SE-high mode) shifts to the normalized frequency around 0.466, and the mode in the changed-radius PC slab (CR-high mode) shifts to 0.464. With the same reduced etching ratio, the shift value for the SE-low mode is larger than that for the CR-low mode, while the shift value for the SE-high mode is smaller than that for the CR-high mode. In addition, the mode at the lowest frequency (around 0.40) is more sensitive to the reduction of the etching ratio than the mode at a higher frequency (around 0.47).
To further understand these phenomena, we calculate the band structure of the PC slab with the same structure parameters used in the ordinary PC slab with polymer cladding in Fig. 2. The mirror plane of Y = 0 is used to separate modes with even or odd symmetries. When setting the mirror plane as even symmetry and using an source, TM-like modes will be stimulated as shown in Fig. 5.When setting the plane as odd symmetry and using an source, TE-like modes will be stimulated as shown in Fig. 6.The modes in the band structure at the ( point) correspond to normal incidence light from free space to the slab. Therefore, we can see that the mode of normalized frequency around 0.40 (lowest mode) belongs to TE-like modes with odd symmetry and the mode of normalized frequency around 0.475 (second lowest mode) belongs to TM-like modes with even symmetry. Moreover, comparing the calculated transmission with the band structure, as shown in Figs. 4–6, it can be seen that TE-like modes are more sensitive to a change in etching ratio than TM-like modes; whilst keeping the same etching ratio, TE-like modes are more sensitive to the shallow-etched scheme than to the changed-radius scheme, but the case for TM-like modes is the opposite.
3. Characteristics of the modulator
3.1 Structure of the modulator
Except for raising the Q value, the shallow-etched PC slab is also used as one of the electrodes. A designed optical modulator hybrid of an EO polymer and shallow-etched SOI PC slab is shown in Fig. 7.Firstly, the PC with shallow holes is etched onto the SOI slab, where the silicon is lightly doped; secondly, a layer of the EO polymer is spin-coated on the patterned SOI slab; thirdly, a layer of the transparent metal electrode is deposited on the EO polymer; finally, the EO polymer is poled by applying a voltage between the SOI electrode and the top transparent metal. In this structure, we set the polymer as the cladding layer of the PC slab. Modulations are achieved by changing the cladding polymer layer's index ellipsoid to affect the guided resonance modes in the PC slab. If the PC slab has been etched completely, the polymer in the holes only touches one electrode, in which case the polymer cannot be poled. The shallow-etched PC slab is conductive after lightly doping; the polymer in the shallow holes can be driven by the slab and the top transparent electrode above the polymer layer. The modulator will form an electric capacitance.
3.2 Transmission of modulator change with index component
As mentioned above, to obtain modulation, the EO polymer has to be poled. The general poling process consists of the following. First, we heat the polymer to the glass transition temperature, and then apply a voltage to the polymer to change the orientation of its molecule. The temperature and the voltage are maintained for several minutes, and then the temperature is rapidly decreased to lock the molecular orientation. Finally, the poling voltage is stopped, and poling is completed. After poling, the polymer and the hybrid modulator structure will possess second-order nonlinearity properties. It should be noted that the poled polymer has a fixed orientation. The polymer index ellipsoid can only be changed by applying a driven voltage to the same orientation of the poling process. Only the index component of the same orientation can be changed. For the above modulator structure, after poling, if the driven voltage is applied between the transparent electrode and the light-doped Si (through the metal layer and heavy-doped Si), among the index components of the polymer, only (the index component along the poling orientation) will be changed; and will keep their original values.
The computation model is formulated as follows. The EO polymer is the cladding layer on the shallow-etched PC slab and the filling material in the shallow holes. The substrate material under the Si slab is SiO2. The index values of the EO polymer, Si and SiO2 are 1.5, 3.4 and 1.5, respectively. The PC period is a, the slab thickness is and the etching depth is . We calculate transmission spectra at normal incidence (from + Z to -Z directions) with the EO polymer index changing by 1–9%, that is, the index vector changes from (1.5, 1.5, 1.5) to (1.5, 1.5, 1.515) for a 1% change in degree. It should be noted that in real devices it is impossible to make changes of several percent in the refractive index of the EO polymer. It is a pure optical simulation analysis without considering the polymer index change limitation. These simulations are just to obtain the shift properties. The index changes in the modulation model are far smaller than 1% as the following discussion shows.
The calculated results are shown in Fig. 8. An interesting phenomenon is that some dips existing in the transmission spectra shift with a change in , especially a dip at frequency 0.65. Some other peaks almost do not shift with a change in , e.g. the dip at frequency 0.66. Only the Z component of the refractive index of the polymer changes with applied voltage. Therefore, only if the resonance modes in the shallow-etched PC slab are TM-like with a primary electric field component and a considerable amount of energy distributed in the polymer layer, will the peaks significantly shift with a change in . The center frequency and the Q values of the resonances around 0.65 as function of change are collected and plotted in Fig. 9.It can be seen that as shifts increase, the Q value of the peak decreases exponentially, while the center frequency decreases almost linearly. As changes from 0% to 9%, the Q value decreases from 65000 to 120, while the center frequency changes from 0.654 to 0.613.
To understand this phenomenon, we calculate the band structure using the same structure parameters that are used in Fig. 8. The modes in the band structure at the point () correspond to normal incident light from free space to the slab. From Fig. 10 and Fig. 11, we can see that the mode of normalized frequency around 0.65 belongs to TM-like modes with Y = 0 plane even symmetry and the mode of normalized frequency around 0.66 belongs to TE-like modes with Y = 0 plane odd symmetry. To verify these, we calculate the energy and field distributions of the two modes, as shown in Fig. 12.
From the field and energy distributions in Fig. 12, we can see that the symmetry and the intensity of these two modes are significantly different. Most of the field and the energy of the mode at frequency 0.66 are confined in the slab, where the primary electric field component is , and the mode is a TE-like mode. The energy of the mode at frequency 0.65 is distributed by a considerable amount outside the slab and in the area of the polymer layer, where the primary electric field is ; this mode is a TM-like mode. The field distributions match the band structures well. This is the reason why, with changing polymer index , the mode at frequency around 0.65 shifts significantly, while the mode at frequency around 0.66 almost does not shift.
3.3 Modulation characteristics
After analysis of the transmission spectra of the shallow-etched PC slab with the polymer cladding layer, we begin to design modulators utilizing the guided resonance mode. We use a type of SOI material with a top Si thickness of 340 nm and pattern the PC structure to the top Si layer. A cubic lattice PC is used with a 1 period, 200 hole radius and 100 etching depth. The PC area is lightly doped. The area adjacent to the PC structure is heavily doped with the same width. The metal electrode is deposited on this area. Upon the PC structure is the polymer layer with a 1 thickness, which ensures that most energy outside the slab is within the polymer layer. The polymer on the metal electrode is removed. The transparent electrode is on the polymer layer.
The transmission and the modulation character of the modulator are shown in Fig. 13. The TM-like mode sensitive to the change is used for modulation. The center wavelength and Q value of the mode are 1.54 and 65000, respectively. The center wavelength of the mode shifts to the red by 6 with a 1% change of . According to previous calculations, the wavelength shifts proportionally to the change within a 10% change range. In modulation applications, as the transmission changes from minimum to half-maximum, the spectrum requires a shift of 0.012 (). That is, needs a change of . The polymer index is 1.5, so needs a change of . Recently, the EO coefficients that are applicable in the modulator have been achieved of around 150 . Here, a coefficient of 150 is adopted. Therefore, to make an change of , we should apply a voltage of . In our structure, the polymer layer thickness is 1. Therefore, to obtain the modulation effect with transmission changing from minimum to half-maximum, a voltage of should be applied.
Another important property of the modulator is modulation speed. The modulator speed in this work is limited by two factors. One is the time constant of the capacitive electrode, the other is the photon lifetime of the guided resonance mode. The modulation bandwidth can be expressed as the following :
The bandwidth of the capacitive electrode is 
The capacitance value is determined by the effective electrode area, the distance between the two electrodes and the dielectric constant of the polymer:
Substituting all the parameters in the above, we obtain
is determined by the photon lifetime of the guided resonance mode:11]:
For the modes we just calculated, Q = 65000, , and then . Combined with Eq. (1), the total bandwidth of the modulator is:
For a shallow-etched EO polymer cladding PC slab modulator, the driven voltage is 0.2 V, which can make the probe light transmission change from a minimum to a half-maximum. The bandwidth is 3 GHz.
For modulation of the mode with a high Q in Fig. 13, it should be noted that the driven voltage is as low as 0.2 V, and the bandwidth is not high. It can be seen from Eq. (1) that the bottleneck is the long lifetime of the mode, i.e. the Q value is too high. To obtain a high-speed modulator, we should choose a mode with a low Q value.
According to above analyses, for the shallow-etched PC slab, the lower the depth of the etching is and the smaller the radius is, the higher the Q value is. Therefore, to obtain low-Q modes for high-speed modulation, we should etch deeper and enlarge the hole radius. To achieve that, the lattice constant of the cubic lattice PC is still used with a period of 1 , while the radius is changed to 250 and the etching depth is changed to 136 .
We calculate transmission and modulation characterization with the TM-like mode, which is sensitive to change, as Fig. 14 shows. Because the PC structure parameters change, the mode shifts to around 1.53 compared with that in Fig. 13. The mode's sensitivity to change is the same as the previous structure in Fig. 13, a 6 shift for a 1% change. Now, the Q value of the mode is 3000. Using the parameters from Fig. 14, we find that a voltage of 4.4 V can shift the probe light transmission from a minimum to a half-maximum. Using Eq. (1), Eq. (2) and Eq. (7), the bandwidth of the photon lifetime is calculated as 65 GHz, and the modulator total bandwidth is 62 GHz. This modulation voltage of less than 5 V is reliable in practical operation. If we continue to reduce the Q value to 1500, which corresponds to a structure with 300 nm hole radius and 145 nm etching depth, the modulation bandwidth can increase to 120 GHz, though the modulation voltage will rise to 8.7 V.
From the two design cases shown in Figs. 13 and 14, it is clarified that high-speed modulation and low-voltage modulation are mutual restraints. The high-Q PC slab can lower the modulation voltage, but the mode lifetime is long, which will decrease the total bandwidth. The low-Q PC slab can lower the mode lifetime to increase the total bandwidth, but also increase the driven voltage. In practical applications, we can design the structure for specific requirements. The flexible design is an important advantage of this kind of modulator.
Because we choose the shallow-etched scheme to increase the Q value, the hole radius can be larger than the case of changed-radius of holes to achieve the same mode Q value. This design will decrease the requirement of the fabrication process. The minimum size in our design is a 400 hole diameter. This size is easy to achieve using deep-UV lithography or electron beam lithography. Because the structure is a cubic lattice, it is possible to use holographic lithography for mass production to further reduce the cost. Other processes such as doping and deposition are routine technologies.
In conclusion, a type of high-performance modulator based on a shallow-etched PC slab with an EO polymer is investigated. With a simulation based on the FDTD method, the transmission spectra, the Q factors and the band structures of the structure are obtained. A significant difference between the shifting of the resonant guide modes with different symmetries is found. By researching the modulation principle, the driven voltage as a function of mode shift, and modulation bandwidth as a function of mode Q factor, is deduced and analyzed. Adapted to different applications, two subtype modulators could be designed by balancing the parameters of the driven voltage and the bandwidth. In our calculation, the obtained voltages and bandwidths are 0.2 V and 3 GHz of the low-drive-voltage type modulator, 4.4 V and 62 GHz for the high-bandwidth modulator.
We acknowledge financial support from the National Natural Science Foundation of China under Grant Nos. 91121019, 61275045 and 61021003, and the National Basic Research Program under Grant No. 2013CB632105. All calculations in this work, including the transmission spectra, the band structure and the energy and field distribution, use MEEP software with the FDTD method. We thank the MIT photonic group who developed MEEP.
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