Electro-optic modulator based on a photonic crystal ... - OSA Publishing

Electro-optic modulator based on a photonic crystal slab with electro-optic polymer Yonghao Gao, Xinnan Huang, and Xingsheng Xu* State Key Laboratory of Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083 China * [email protected]

Abstract: 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 timedomain method. The band structure and the field distribution of the guided mode resonance are simulated and analyzed. The modulation voltage and bandwidth of the hybrid modulator are calculated. 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 OCIS codes: (160.5298) Photonic crystals; (250.4110) Modulator.

References and links 1.

H. C. Nguyen, Y. Sakai, M. Shinkawa, N. Ishikura, and T. Baba, “10 Gb/s operation of photonic crystal silicon optical modulators,” Opt. Express 19(14), 13000–13007 (2011). 2. Y. Hu, X. Xiao, H. Xu, X. Li, K. Xiong, Z. Li, T. Chu, Y. Yu, and J. Yu, “High-speed silicon modulator based on cascaded microring resonators,” Opt. Express 20(14), 15079–15085 (2012). 3. X. Xiao, H. Xu, X. Li, Z. Li, T. Chu, Y. Yu, and J. Yu, “High-speed, low-loss silicon Mach-Zehnder modulators with doping optimization,” Opt. Express 21(4), 4116–4125 (2013). 4. M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004). 5. J. M. Brosi, C. Koos, L. C. Andreani, M. Waldow, J. Leuthold, and W. Freude, “High-speed low-voltage electrooptic modulator with a polymer-infiltrated silicon photonic crystal waveguide,” Opt. Express 16(6), 4177–4191 (2008). 6. J. H. Wülbern, J. Hampe, A. Petrov, M. Eich, J. D. Luo, A. K. Y. Jen, A. Di Falco, T. F. Krauss, and J. Bruns, “Electro-optic modulation in slotted resonant photonic crystal heterostructures,” Appl. Phys. Lett. 94(24), 241107 (2009). 7. C. Y. Lin, X. L. Wang, S. Chakravarty, B. S. Lee, W. C. Lai, J. D. Luo, A. K. Y. Jen, and R. T. Chen, “Electrooptic polymer infiltrated silicon photonic crystal slot waveguide modulator with 23 dB slow light enhancement,” Appl. Phys. Lett. 97(9), 093304 (2010). 8. M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, “Broadband modulation of light by using an electro-optic polymer,” Science 298(5597), 1401–1403 (2002). 9. X. Q. Piao, X. M. Zhang, Y. Mori, M. Koishi, A. Nakaya, S. Inoue, I. Aoki, A. Otomo, and S. Yokoyama, “Nonlinear Optical Side-Chain Polymers Post-Functionalized with High-beta Chromophores Exhibiting Large Electro-Optic Property,” J. Polym. Sci. Pol. Chem. 49(1), 47–54 (2011). 10. S. H. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). 11. C. Y. Qiu, J. B. Chen, Y. Xia, and Q. F. Xu, “Active dielectric antenna on chip for spatial light modulation,” Sci. Rep-Uk 2, 855 (2012). 12. W. X. Liu, Z. Q. Lai, H. Guo, and Y. Liu, “Guided-mode resonance filters with shallow grating,” Opt. Lett. 35(6), 865–867 (2010).

1. Introduction 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 [1]. For micro-ring resonance modulators, the speed can be as high as 40 Gb/s [2], 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 [3]. However, the MZI modulators are

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Received 28 Jan 2014; revised 16 Mar 2014; accepted 16 Mar 2014; published 4 Apr 2014 7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.008765 | OPTICS EXPRESS 8765

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 [4]. Recently, Hong C. Nguyen et al. experimentally demonstrated a 10 Gb/s modulation in a PC waveguide-based MZI modulator with 200 μ m arms [1]. 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 speed ( ≥ 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 [5]. EO modulation in slotted PC waveguides based on SOI substrates covered and infiltrated with EO polymer has also been experimentally demonstrated [6]. Che-Yun Lin et al. reported a similar structure with 23 dB EO coefficient enhancement using the PC slow light effect [7]. Up to THz modulation effects in EO modulators have been reported [8]. The value of the second-order nonlinearity of a recently reported EO polymer is several times higher than that of the commercial modulator material LiNbO3 [9]. 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. [10]. 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 usually 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 [11]. 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 original 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 [10]. 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 of the PC slab 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.

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1.0

Transmission

0.9

0.8

0.7

220 nm 260 nm 340 nm

0.6

0.5 1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Wavelength (μm)

Fig. 1. Normal-incidence transmission spectra for various slab thicknesses. The index of the slab is 3.4, and the index of the background is 1.5.

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. [10] 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. Ordinary PhC slab without cladding layer Ordinary PhC slab with polymer cladding

1.0 0.9 0.8

Transmission

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

Frequency (2πc/a) Fig. 2. Normal-incidence transmission spectra of an ordinary PC slab with and without a cladding layer. The PC period is a, the hole radius is 0.25a, the slab thickness is h = 0.34a , the index of the slab is 3.4, the cladding layer and the material filling in the holes is a polymer of index 1.5, the substrate layer is SiO2 , and the index is assumed to be 1.5.

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S. H. Fan studied the radius dependence of the resonance [10]. 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. Ordinary PhC slab Shallow etch , 0.4h Changed-radius, 0.16a

1.0 0.9

Transmission

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

Frequency (2πc/a) Fig. 3. Normal-incidence transmission spectra of an ordinary PC slab, a shallow-etched PC slab and a changed-radius ordinary PC slab. The ordinary PC slab has a radius of 0.25a and is etched to 1.0h ( h = 0.34a ), as in Fig. 2. The shallow-etched PC slab has a radius of 0.25a and is shallowly etched to 0.4h. The changed-radius PC slab has a radius of 0.16a and is etched to 1.0h. The etch ratios of the shallow-etched PC slab and the changed radius PC slab are the same. Other conditions are the same as those in Fig. 2.

0.50

2000 1800

0.48

Q value

SE-low Q SE-high Q CR-low Q CR-high Q

1200 1000 800

0.46

SE-low SE-high CR-low CR-high

1400

0.44 0.42 0.40

600 400

Frequency (a/λ)

1600

0.38

200 0.36

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Etch ratio Fig. 4. Frequency and Q value for modes in PC slabs as a function of etch ratio. 'SE-low' corresponds to the mode at a frequency around 0.40 a / λ of the shallow-etched PC slab. 'SEhigh' corresponds to the mode at a frequency around 0.47 a / λ of the shallow-etched PC slab. 'CR-low' corresponds to the mode at a frequency around 0.40 a / λ of the changed-radius PC slab. 'CR-high' corresponds to the mode at a frequency around 0.47 a / λ of the changedradius slab.

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 #205668 - $15.00 USD (C) 2014 OSA


through the slab completely. In structures such as waveguide grating and DFB lasers, shallowly etched gratings [12], 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 adding 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 symmetric 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 ( h = 0.34a ), 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 a / λ 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 a / λ , and the changed-radius PC slab mode (CR-low mode) shifts to 0.39 a / λ . The mode at the normalized frequency 0.475 a / λ 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 a / λ , and the mode in the changed-radius PC slab (CR-high mode) shifts to 0.464 a / λ . 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 SEhigh mode is smaller than that for the CR-high mode. In addition, the mode at the lowest frequency (around 0.40 a / λ ) is more sensitive to the reduction of the etching ratio than the mode at a higher frequency (around 0.47 a / λ ). 0.7 0.6

Frequency (a/λ)

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

kx (2π/a) Fig. 5. Band structure of even symmetry with respect to the Y = 0 plane, which refers to TMlike modes. This band structure is for the PC slab with the same structure parameters used in the ordinary PC slab with polymer cladding in Fig. 2. The modes at Γ point ( k x = 0 ) are marked by red squares.

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0.7 0.6

Frequency (a/λ)

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

kx (2π/a) Fig. 6. Band structure of odd symmetry with respect to the Y = 0 plane, which refers to TE-like modes. This band structure is for the PC slab with the same structure parameters used in the ordinary PC slab with polymer cladding in Fig. 2. The modes at Γ points ( k x = 0 ) are marked by red squares.

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 Ez source, TM-like modes will be stimulated as shown in Fig. 5. When setting the plane as odd symmetry and using an H z source, TE-like modes will be stimulated as shown in Fig. 6. The modes in the band structure at the k x = 0 ( Γ point) correspond to the modes in transmission spectrum of PC slab under normal incidence. Therefore, we can see that the mode of normalized frequency around 0.40 a / λ (lowest mode in Fig. 2) belongs to TE-like modes with odd symmetry and the mode of normalized frequency around 0.475 a / λ (second lowest mode in Fig. 2) 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, TElike 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.

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Fig. 7. Structure of the EO polymer modulator with a shallow-etched PC Si slab. (A) Side view; (B) top view.

3.2 Transmission of modulator change with index component As mentioned above, to obtain modulation effect, 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 molecules. 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 driving 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 driving 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 nz (the index component along the poling orientation) will be changed; nx and n y will keep their original values.

1.0

no nz shift nz shift 2% nz shift 4% nz shift 6%

0.9

Transmission

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.63

0.64

0.65

0.66

Frequency (a/λ) Fig. 8. Transmission spectra with nz variation of 0%, 2%, 4% and 6%. The PC period is a, the hole radius is 0.2a, the slab thickness is h = 0.34a and the etching depth is h = 0.34a . The cladding layer and the material filling in the holes is EO polymer of index 1.5. The other refractive index parameters are the same as those used in Fig. 2.

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0.650

60000

0.645

50000

0.640

Frequency

40000

0.635

30000

0.630

20000

0.625

Frequency (a/λ)

Q value

70000

0.620

10000

Q value 0.615

0 0

1

2

3

4

5

6

7

8

9

nz shift (%) Fig. 9. Frequency and Q value as functions of

nz

variation of the structures in Fig. 8.

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 h = 0.34a and the etching depth is 0.3h = 0.1a . We calculate transmission spectra at normal incidence (from + Z to -Z directions) with the EO polymer nz 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 display the variation tendency. The actual 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 nz , especially a dip at frequency 0.65 a / λ . Some other peaks almost do not shift with a change in nz , e.g. the dip at frequency 0.66 a / λ . Only the Z component of the refractive index of the polymer changes with the applied voltage. Therefore, only if the resonance modes in the shallow-etched PC slab are TM-like with a primary E z electric field component and a considerable amount of energy distributed in the polymer layer, will the modes significantly shift with a change in nz . The center frequency and the Q values of the resonances around 0.65 a / λ as function of nz variation are collected and plotted in Fig. 9. It can be seen that as nz variations increase, the Q value of the mode decreases exponentially, while the center frequency decreases almost linearly. As nz variation value changes from 0% to 9%, the Q value decreases from 65000 to 120, while the center frequency changes from 0.654 a / λ to 0.613 a / λ .

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0.70 0.69

Frequency (a/λ)

0.68 0.67 0.66 0.65 0.64 0.63 0.62 0.61 0.60 0.0

0.1

0.2

0.3

0.4

0.5

kx (2π/a) Fig. 10. Band structure of even symmetry with respect to the Y = 0 plane, which refers to TMlike modes. This band structure is for the PC slab with the same structure parameters used in the shallow-etched PC slab with the 0%

nz

variation of polymer cladding in Fig. 8. Mode at

Γ points ( k x = 0 ) is marked by red squares.

0.70 0.69

Frequency (a/λ)

0.68 0.67 0.66 0.65 0.64 0.63 0.62 0.61 0.60 0.0

0.1

0.2

0.3

0.4

0.5

kx (2π/a) Fig. 11. Band structure of odd symmetry with respect to the Y = 0 plane, which refers to TElike modes. This band structure is for the PC slab with the same structure parameters used in the shallow-etched PC slab with the 0%

nz

variation of polymer cladding in Fig. 8. Mode at

Γ points ( k x = 0 ) is marked by red squares.

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 Γ point ( k x = 0 ) correspond to the modes of PC slab under normal incidence. From Fig. 10 and Fig. 11, we can see that the mode of normalized frequency around 0.65 a / λ belongs to TM-like modes with Y = 0 plane even symmetry and the mode of normalized frequency around 0.66 a / λ 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.

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Fig. 12. Energy and field distributions for the corresponding modes in Fig. 8. (A) Side view of the energy distribution of frequency 0.65 a / λ mode. (B) Side view of the electric field distribution of frequency 0.65 a / λ mode. (C) Side view of the energy distribution of frequency 0.66 a / λ mode. (D) Side view of the electric field distribution of frequency 0.66 a / λ mode.

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 a / λ are confined in the slab, where the primary electric field component is E x , and the mode is a TE-like mode. The energy of the mode at frequency 0.65 a / λ is distributed by a considerable amount outside the slab and in the area of the polymer layer, where the primary electric field is Ez ; this mode is a TM-like mode. The field distributions match the band structures well. This is the reason why, with changing polymer index nz , the mode at frequency around 0.65 a / λ shifts significantly, while the mode at frequency around 0.66 a / λ 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 μ m period, 200 nm hole radius and 100 nm 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 μ m 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 nz change is used for modulation. The center wavelength and Q value of the mode are 1.54 μ m and 65000, respectively. The center wavelength of the mode shifts to red by 6 nm with a 1% change of nz . According to previous calculations, the wavelength shifts proportionally to the nz change within a 10% nz variation range. In modulation applications, as the transmission changes from minimum to half-maximum, the spectrum requires a shift of 0.012 nm ( λ / 2Q ). That is, nz needs a change of

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0.012nm × 1% = 0.002% . The polymer index is 1.5, so nz needs a change of 3 × 10−5 . 6nm Recently, the EO coefficients that are applicable in the modulator have been achieved of around 150 pm / V [9]. Here, a coefficient of 150 pm / V is adopted. Therefore, to make an

3 × 10−5 = 0.2V / μ m . In our structure, 150 pm / V the polymer layer thickness is 1 μ m . Therefore, to obtain the modulation effect with transmission changing from minimum to half-maximum, a voltage of 1μ m × 0.2V / μ m = 0.2V should be applied. nz change of 3 × 10 −5 , we should apply a voltage of

1.0 0.9 0.8

Transmission

0.7 0.6 0.5 0.4

nz shift 0% nz shift 0.02% nz shift 1%

0.3 0.2 0.1 0.0 1.51

1.52

1.53

1.54

Wavelength (μm)

Fig. 13. Transmission spectra of the high-Q modulator. The lattice constant of the PC slab is 1 μ m , the hole radius is 200 nm , the slab thickness is 340 nm , the etching depth is 100 nm and the index of the slab is 3.4. The cladding layer and the material filling in the holes is the EO polymer whose index is 1.5. The substrate layer is SiO2 , and the index is also assumed to be 1.5.

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 [5]: 1 1 1 = + fM f RC fτ p

(1)

where f M is the total bandwidth of the modulator, f RC is the bandwidth of the capacitive electrode and fτ p is the bandwidth of the optical mode. The bandwidth of the capacitive electrode is [5] f RC =

1 2π RC

(2)

where C is the capacitance value of the electrodes and R is the effective resistance of the electrodes. Here, we set R = 50Ω . In real devices, we can adjust the doping concentration and the electrode structure to achieve this value. The capacitance value is determined by the effective electrode area, the distance between the two electrodes and the dielectric constant of the polymer:

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C=

εS 4π kd

(3)

where ε is the permittivity of the medium between the two electrodes, S is the area of the electrodes, d is the distance between the two electrodes and k is the Coulomb constant 9.0 × 109 Nm 2 C 2 . S is determined by the period value and the number of the PC structures. Generally speaking, a two-dimensional hole array with more than 10 periods can work as a PC structure. Therefore, we use 10 periods here, which is 10 μ m . Substituting all the parameters in the above, we obtain fτ p

f RC = 1.6 × 1012 Hz = 1.6THz. is determined by the photon lifetime of the guided resonance mode: fτ p =

1 2πτ p

(4)

where τ p is the photon lifetime. It depends on the Q value of the mode [11]:

τp =

Q

(5)

ω0

ω0 = 2π

c

(6)

λ

where ω0 is the circular frequency of the resonance mode, λ is the wavelength of the mode and c is the speed of light in vacuum. We therefore obtain fτ p =

For

the

mode

we

just

calculated,

c λQ Q =

(7) 65000,

λ = 1.54 μ m , and then

fτ p = 3 × 10 Hz = 3GHz . Combined with Eq. (1), the total bandwidth of the modulator is: 9

fM =

fτ p f RC fτ p + f RC

=

3 × 1600 ≈ 3GHz 3 + 1600

(8)

For a shallow-etched EO polymer cladding PC slab modulator, the driving voltage is 0.2 V, which can make transmission of the modulated light change from a minimum to a halfmaximum. The bandwidth is 3 GHz. For modulation of the mode with a high Q in Fig. 13, it should be noted that the driving 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 μ m , while the radius is changed to 250 nm and the etching depth is changed to 136 nm . We calculate transmission and modulation characterization with the TM-like mode, which is sensitive to nz variation, as Fig. 14 shows. Because the PC structure parameters change, the mode shifts to around 1.53 μ m compared with that in Fig. 13. The mode's sensitivity to

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nz variation is the same as the previous structure in Fig. 13, a 6 nm shift for a 1% nz 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 change the probe light transmission from a minimum to a halfmaximum. Using Eq. (1), Eq. (2) and Eq. (7), the bandwidth determined by 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. 1.0 0.9 0.8

Transmission

0.7 0.6 0.5 0.4 0.3

nz shift 0% nz shift 0.043% nz shift 0.5% nz shift 1%

0.2 0.1 0.0 1.530

1.535

1.540

1.545

Wavelength (μm)

Fig. 14. Transmission spectra of the low-Q modulator with nz variations. The PC period is 1 μ m , the hole radius is 250 nm , the slab thickness is 340 nm , the etching depth is 136 nm and the index of the slab is 3.4. The cladding layer and the material filling in the holes is EO polymer with index 1.5. The substrate layer is SiO2, and the index is also assumed to be 1.5.

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 of the modulator. The low-Q PC slab can lower the mode lifetime to increase the total bandwidth, but also increase the driving 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 nm 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. 4. Conclusion

In conclusion, a type of high-performance modulator based on a shallow-etched PC slab with 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 driving 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

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parameters of the driving voltage and the bandwidth. In our calculation, the obtained voltages and bandwidths are 0.2 V and 3 GHz for the low-driving-voltage type modulator, 4.4 V and 62 GHz for the high-bandwidth modulator. Acknowledgments

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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