In this paper, a polymer electro-optic modulator has been designed and optimized using the full vectorial finite element method. For this purpose, the effects of magnesium oxide (MgO) and down cladding thicknesses, distance between two ground electrodes, hot electrode and modulator widths modulator on the key modulator parameters, such as microwave effective index nm, the characteristic impedance ZC and the microwave losses α are presented. After selecting optimal dimensions of polymer electro-optic modulator, frequency dependent aforementioned parameters and the half-wave voltage-length product (VπL) parameter of polymer electro-optic modulator are extracted and as a consequence, an optimized design is reported. Finally, the optical and electrical modulation responses of polymer electro-optic modulator are calculated. The optimized polymer electro-optic modulator exhibits 3-dB electrical bandwidth of 260 GHz and VπL about 2.8 V∙cm in this frequency.
Kambiz ABEDI, Habib VAHIDI. Design optimization of microwave properties for polymer electro-optic modulator using full vectorial finite element method[J]. Frontiers of Optoelectronics, 2013, 6(3): 290-296. DOI: 10.1007/s12200-013-0335-9
Introduction
Optical modulators play an important role in all optical communication systems. These devices can be categorized into two main groups of electroabsorption and electro-optic modulators. Electro-optic modulators have a variety of advantages over electroabsorption modulators, and therefore they are under investigation more than electroabsorption modulators. To realize high bandwidth in electro-optic modulators, we should match the velocities of microwave and optical wave with each other. The line characteristic impedance should be also matched with the source and load impedances that are usually 50 W. Meanwhile, the microwave loss of dielectrics and conductors should be decreased. In addition to bandwidth, the drive voltage is another important parameter in electro-optic modulators, which must be as low as possible. This voltage is dependent on the electro-optic coefficient of used electro-optic material and to the interaction of microwave and optical wave [1-9]. If we increase the interaction, the dielectric and conductor losses will be dominant, and in spite of speeds and impedances matching, we will have small bandwidth.. Therefore, we must choose another way to increase the bandwidth and the only remaining way is to increase the electro-optic coefficient. The electro-optic material has the electro-optic coefficient around 30 pm/V, which is suitable to build low voltage wide bandwidth modulators. If we are interested in larger bandwidths with the low drive voltages, we have to use materials with larger electro-optic coefficients. The commonly used materials for this reason are electro-optic polymers. The early polymer materials were suffered from low thermal and photo stability such that the high electro-optic coefficient of them reduced over the time. But new materials such as CLD-1/APC polymers show good stability of the electooptic coefficient and very large electro-optic coefficient (92 pm/V) and therefore are appropriate for our purpose [1]. Some works with this material attain to bandwidths about 170 GHz for drive voltage of 3 V∙cm [3]. But since this material with this large electro-optic coefficient has potentials over these results, we have optimized one of structures that proposed in Ref. [1] with some basic variation and utilizing CLD-1/APC as electro-optic material to achieve larger bandwidth and lower voltage.
Modulator layout and cross-section
The cross-section of polymer electro-optic modulator that we have designed and optimized in this work is shown in Fig. 1. The core material is the CLD-1/APC polymer. Ribs are formed on top of the core material with dimensions such that the single mode operation is preserved [3]. Two claddings of UFC-170 and UV-15 materials are placed on top and at bottom of this material as upper and lower cladding, respectively. These claddings, in addition to prepare good optical confinement with having lower refractive indexes than CLD-1/APC, didnot physically degrade solvent sensitive CLD-1/APC at their common boundaries [1]. Below the down cladding, a layer of magnesium oxide, MgO, has been placed. This material has large relative permittivity (about 9.8) that helps us in good speed matching between microwave and optical wave [2].
Fig.1 Schematic cross section of polymer electro-optic modulator
Two splitted ground electrodes placed below this MgO material that altogether placed on top of the silicon substrate via one SiO2 layer. One hot electrode also placed on the top of these layers. The refractive index and relative permittivity of all utilized materials have been listed in Table 1.
Tab.1 Relative permittivity and refractive indexes of utilized materials
material
refractive index
permittivity
UFC-170
1.488
2.5
CLD-1/APC
1.612
3.0
UV-15
1.504
2.5
SiO2
1.460
3.9
MgO
1.730
9.8
silicon
3.880
11.7
Some differences of our structure with Ref. [3] are extending ground electrode and it is above layer from sides, using MgO material to achieve good matching, taking different width of hot electrode with respect to modulator width and decreasing distance between hot and ground electrodes to lowering VpL. Furthermore, we have used more dimension in optimization and achieved to better matching and lower attenuation.
Numerical technique
In determining frequency response of one modulator, it is necessary to calculate some parameters. These parameters include microwave effective index, nm, characteristic impedance, ZC, microwave attenuations, a, and optical effective index, neff. From these parameters, the response of the modulator can be calculated using Eq. (1) in which l is the length of the modulator that for our modulator is 1 cm, c is the velocity of the light in vacuum, Z1 and Z2 are source and load impedances and f is the frequency [3,10]. The dB electrical response can be determined by whilst the dB optical response can be determined by .whereand
The only remaining parameter is the drive voltage of the modulator, VpL, which can be defined as the voltage needed to produce 180° phase difference in two arms of Mach-Zehnder (MZ) modulator [7]. This voltage can be evaluated using relationwhere, V0 is the applied voltage, and b1 and b0 are the propagation constants of the light with and without the modulating electric field. From all required parameters in determining response and voltage of the modulator, nm, a, ZC and the modulating electric field must be obtained by microwave analysis, and neff, b0 and b1 by optical analysis. Also one quasi- transverse electromagnetic (TEM) is required to calculate modulating electric field that causes Dn and so Db.
We used the full vectorial finite element method in microwave analysis and quasi-transverse magnetic finite element method (TM FEM) analysis in order to do optical analysis.
The vectorial FEM is based on the edge element formulation [10-14]. The most important advantage of using edge element finite element formulation is to prevent from the appearance of spurious modes [12]. This avoidance will be achieved along with some problems such as loosing sparsity of matrices [14] that prevent us from taking advantage of sparse matrix solver algorithms [15] in solving eigenvalue equation. If we are going to use these algorithms in edge-element formulations, we must use full wave in such a way that the size of our matrices increase from e × e, in which e is total number of edges, to (e + n) × (e + n), in which n is total number of nodes, as done in Refs, [10,11]. Although spurious modes can be removed by some technique in nodal FEM, such as introducing penalty parameter [13], however this cause some error and dependency of our results to proper choice of penalty parameter [13].
The edge element formulation can be extracted from discretization of vector wave equation of(4)
By discretization and applying FEM procedure, the following eigenvalue equation extractedwith
{Et} is the transverse value of electric field at all edges and is complex propagation constant. The shape functions {N}, {U} and {V} are illustrated in Ref. [12].
The longitudinal component can be computed from transverse components by
By obtaining the eigenvalue, the microwave effective index can be obtained from
And the attenuation constant in Neper per meter fromwhere αc and αd are the conductor and dielectric loss parameters, respectively. The characteristic impedance of the transmission line in this case should be evaluated using the power current definition in the form of [10]where P is modal power and I is the total current in z direction carried by center electrode. P and I in the FEM formulations can be evaluated by [10]where for P, the integration carried out over entire waveguide cross section, whereas for I is done only on hot electrode cross section with conductivity of s.
After this microwave analysis, the optical analysis is necessary to obtain neff and VpL. To utilize the large electro-optic coefficient (92 pm/V) of CLD-1/APC which is r33, transverse magnetic (TM) polarization is assumed throughout this paper [3].
The optical analysis also could be from full-vectorial type. But in most of cases, the quasi-TE and quasi-TM analysis work properly. In our paper that TM polarization assumed, the quasi-TM formulation can be used that is based on the following abstraction of wave equation [10].
By applying the FEM to Eq. (12), the following eigenvalue equation can be obtained
In which, the vector {Hx } is the magnetic field in all element nodes and b is the propagation constant and matrixes [K] and [M] are given with below relations [10].
From calculated eigenvalue b, optical effective index will be neff = b/k0, where k0 is wavenumber of the optical wave. This b is the propagation constant of the light without the modulating electric field. . Furthermore, since ny and nz components influence from electro-optic property of core material through microwave modulating electric field component of Ey. Then, the b1 can be evaluated from another quasi-TM analysis with the modulating electric field. Consequently,VpL can be determined from Eq. (3).
Results and discussion
In this section, we use the full vectorial finite element method to calculate the microwave parameters. For this purpose, the dimensions that affect the optical parameter of neff is kept unchanged and constant neff is calculated. Then other parameters are designed to match nm with the constant neff and characteristic impedance with 50 Ω. We are free in this design except that we should not reduce interaction strength by extra increasing of layers thickness that cause additional VpL enlargement. The parameters that affect nm are type of core and claddings material that not changed at all. Dimensional affecting parameters are H1, H2, H3, H4, Wg and G2. The width of modulator, W, has negligible effect on neff and changed during optimization. Among other parameters, only H4 changed. H4 is the thickness of down cladding and will influence neff when its thickness is very small. Therefore, this thickness always is larger than 1.5 mm and thus changing of H4 didnot affect neff severely to be needed updating optical analysis. Inadequate parameters restrict our freedom in playing with nm, ZC and a values. We must change such amount of dimensions that lead to independent alteration of each of these three values. We chose five parameters of G1, H4, H5, W and We.
Figures 2(a)-2(e) show the results of varying these parameters on three major microwave characteristics. It can be observed that as G1 is increased, the effective index and characteristic impedance values are increased, whereas microwave attenuation values are approximately constant (see Fig. 2(a)). It is also shown that as H4 is reduced; the effective index and microwave attenuation values are increased, whereas the characteristic impedance values are reduced (Fig. 2(b)). Figure 2(c) also shows the variations of the nm and ZC and a with H5. It can be noted that nm and ZC decreased as H5 was decreased. It can also be noted that a increased as H5 was decreased. With increasing of the W and We values, nm values are increased, but ZC values are decreasd and microwave attenuation values are approximately constant (Figs. 2(d) and 2(e)). The purpose of these figures is not to precisely study dimensional effects and they only intended to show the behavior of characteristics with respect to these dimensions change. After perceiving these behaviors, we can easily start with one initial dimension values and go ahead through these figures to obtain desirable decrease/increase in nm, ZC and a together with or separately.
Fig.2 Results of varying (a) G1; (b) H4; (c) H5; (d) W and (e) We on microwave important parameters of nm, ZC and a
Note that all other parameters in these figure is the same dimensions that listed in Table 2. Table 2 shows these final optimized dimension values. All these figures were obtained by the full vectorial FEM in the frequency of 250 GHz. The final dimensions are such that two ribs were exactly between top and ground electrodes in expose of maximum static electric field and therefore VpL be the minimum. After final selecting of dimensions, through one microwave quasi-TEM and two optical quasi-TE analyses, the dc value of VpL can be calculated [5].
For these parameters, the value of neff computed from quasi-TM analysis is about 1.59. As seen in Fig. 3, the nm is completely matched with neff value for large frequencies. We have done our full vectorial analysis in 26 4 points from 10 to 260 GHz with steps of 10 GHz. Utilizing MgO brings this good nm increasing to establish matching with neff.
Tab.2 Various dimensions of optimized modulator
modulator structure parameter
value/μm
w
38
we
20
wg
3.5
t
2
te
2
G1
6
G2
8
H1
1.5
H2
1
H3
2
H4
1.7
H5
2.6
Fig.3 Microwave effective index vs frequency along with constant optical effective index
The other parameter calculated from microwave analysis is ZC, which is shown in Fig. 4. As we can see from this figure, ZC is nearly matched with 50 Ω in large frequencies. About 0.5 Ω difference of it with 50 Ω has very negligible effect on modulator bandwidth; for this reason, we didnot make more efforts on its full matching.
Fig.4 Characteristic impedance of modulator vs frequency
The microwave attenuation vs frequency that resulted for optimized modulator is displayed in Fig. 5. One rule of thumb that is common for estimating electrical bandwidth of modulators is that when velocities and impedances were matched, the cut-off frequency is wherever the microwave attenuation reaches to the value of 6.5 dB/cm. Then before we begin choosing and optimizing layers materials and dimensions, we should take a look that is it possible to have below 6.5 dB/cm losses for one desired VπL or not.
The final modulator response is shown in Fig. 6. It is evident from this figure that the electrical bandwidth of modulator is 260 GHz. From Ref. [3], the dc value of VpL was computed about 2 V∙cm. The frequency dependent VpL can be obtained from Ref. [3].In which, m(f) is dB optical response of the modulator. As we can see from Fig. 7, VpL for 260 GHz frequency is about 2.8 V cm that is lower than similar works [3].
Fig.6 Electrical and optical responses of modulator vs frequency
In this paper, we designed and optimized an ultra wideband and low drive voltage polymer electro-optic modulator. For this purpose, frequency dependencies of all important parameters, such as microwave effective index, microwave characteristic impedance and microwave loss, were extracted. Calculated modulator responses showed an electrical bandwidth of 260 GHz and drive voltage of about 2.8 V∙cm in this frequency that is lower than similar works that achieved to lower bandwidths.
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