Electrooptic Modulators with Controlled Frequency Responses by Using Nonperiodically Polarization-Reversed Structure

Abstract

We discuss a new method to design traveling-wave electrooptic modulators with controlled frequency responses using nonperiodically polarization-reversed structure. Using our method, the frequency responses of both magnitude and phase of modulation index are controllable. Several electrooptic modulators for advanced modulation formats such as duobinary modulation and wideband single-sideband modulation are proposed.

1. Introduction

High-speed traveling-wave electrooptic modulators are very important devices because of their applications in many optoelectronic fields, such as telecommunication, optical signal processing, and optical measurement. With the increasing demand for broadband communication, many studies on the high-speed electrooptic modulators have been carried out for optical fiber communication system [1]. In addition, the precise control of the frequency responses of high-speed electrooptic modulators has been required for the new generation optical fiber communication systems [2–4]. In a standard traveling-wave electrooptic modulator, the frequency response of the modulation index is restricted by two effects: the velocity mismatching between the lightwave and modulation microwave and the loss of modulation microwave in the traveling-wave electrodes.

The polarization reversal technology of ferroelectric optical crystals, such as LiNbO₃ and LiTaO₃, is attractive for realizing high-performance electrooptic modulators. We have developed traveling-wave electrooptic modulators utilizing the periodically polarization-reversed structure for quasi-velocity-matching (QVM) between the lightwave propagating in the optical waveguide and the modulation microwave traveling along the electrodes. This technique is attractive because it compensates for the velocity mismatch in the high frequency ranges without the requirement of specific and fine structures of the optical waveguide and the electrodes for velocity matching. Several advanced electrooptic modulators using periodically polarization-reversed structure have been proposed and demonstrated, such as a QVM electrooptic phase modulator [5], an electrooptic single-sideband (SSB) modulator [6], and an optical frequency shifter [7]. The bandwidth of the modulation frequency response in these electrooptic modulators with the simple periodically polarization-reversed structure is inversely proportional to its electrode length and becomes relatively narrow for a long-electrode device.

We have proposed a design to realize the traveling-wave electrooptic modulators utilizing nonperiodically polarization-reversed structures [8]. Using the proposed design, it is possible to obtain electrooptic modulators with the arbitrary frequency responses of the magnitude of the modulation index over a specified frequency range. The flat broadband electrooptic modulator with the compensation for the degradation by both velocity mismatch and the microwave loss is obtained. However, the control of not only the magnitude but also the phase of the modulation index is very important in order to design the electrooptic modulators for advanced modulation formats.

In this paper, we extend our design method to control frequency responses of both the magnitude and the phase of modulation index in electrooptic modulators with nonperiodically polarization-reversed structures utilizing new flatness parameters. This new approach is applicable to the electrooptic modulators for advanced modulation formats, such as duobinary modulation or broadband single-sideband modulation. This paper is organized as follows. In Section 2, the basic structure of the electrooptic phase modulator using the nonperiodic polarization reversal and the frequency responses of modulation index are presented. In Section 3, the calculation method to control the frequency responses of both the magnitude and the phase of the modulation index is described. Several calculation results are shown in Section 4. In Section 5, the designs of the duobinary modulator and broadband single-sideband modulator with nonperiodically polarization-reversed structure are discussed.

2. Traveling-Wave Electrooptic Modulators with Polarization-Reversed Structure

A schematic of the basic structure of the traveling-wave electrooptic modulator is shown in Figure 1. It consists of a single-mode channel waveguide and simple coplanar traveling-wave electrodes fabricated on a ferroelectric substrate. A Mach-Zehnder waveguide for intensity modulation is also applicable. The velocity mismatch between the group velocity of the lightwave propagating in the optical waveguide and the phase velocity of a modulation microwave traveling along the electrodes is compensated for by adopting the periodic polarization reversal scheme [9]. The domain length 𝐿 of a standard QVM with the periodic polarization reversal scheme is determined as follows: 1𝐿=2𝑓_𝑚(1/𝑣_𝑚−1/𝑣_𝑔), (1) where 𝑓_𝑚 is the peak modulation frequency, 𝑣_𝑚 is the phase velocity of the modulation microwave traveling along the electrodes, and 𝑣_𝑔 is the group velocity of the lightwave propagating in the waveguide. The parameter 𝑣_𝑚 can be calculated from the effective dielectric constant of the coplanar traveling-wave electrodes taking account of its structure. The parameter 𝑣_𝑔 can be derived from both the material dispersion of the refractive index of the ferroelectric material substrate and the mode dispersion of the optical waveguide.

Figure 1 

Basic structure of traveling-wave electrooptic phase modulator with nonperiodically polarization-reversed structure.

The nonperiodic polarization reversal scheme is used to control the frequency response of the electrooptic modulation. Figure 2 shows the structure of the nonperiodic polarization reversal. The total active length for modulation is 𝐿_𝑡. It is divided into 𝑀 sections of successive polarization-reversed and nonreversed region. Each domain length is 𝐿_𝑖(𝑖=1,2,…,𝑀).

Figure 2 

Nonperiodically polarization-reversed structure.

The modulation index Δ𝜙(𝑓), which is a function of the modulation microwave frequency 𝑓, is calculated using the following equation: Δ𝜙(𝑓)=−^𝑦1₀𝑘Δ𝑛𝑑𝑦+^𝑦2_𝑦1𝑘Δ𝑛𝑑𝑦−⋯+(−1)^𝑀𝑦𝑀_𝑦𝑀−1𝑘Δ𝑛𝑑𝑦, (2)where 𝑘 is the wave vector of the lightwave in vacuum and Δ𝑛 is the refractive index change by the Pockels effect. When the light polarization is along the 𝑧-axis and the modulation electrical field is also along the 𝑧-axis, the refractive index change induced by the electrical field of the microwave at the position 𝑦(0≤𝑦≤𝐿_𝑡) is derived as follows, taking account of the velocity mismatch between the lightwave and the modulation wave, 1Δ𝑛(𝑦)=2𝑛³_𝑒𝑟₃₃𝑉𝑑Γexp(−𝛼𝑦)×exp[𝑗2𝜋𝑓{𝑡₀1−(𝑣_𝑚−1𝑣_𝑔)𝑦}], (3)where 𝑛_𝑒 is the extraordinary ray refractive index of the substrate, 𝑟₃₃ is the Pockels coefficient of the substrate. 𝑉 is the voltage applied to the electrodes, 𝑑 is the gap between the electrodes, Γ is the overlap factor between the optical field and the electrical field, 𝛼 is the attenuation coefficient of the electrical field of the modulation microwave, and 𝑡₀ is the time when the lightwave arrived at the position 𝑦=0.

3. Control of Frequency Response

When tuning the length of each domain, the frequency responses of the modulation index magnitude |Δ𝜙(𝑓)| and phase arg[Δ𝜙(𝑓)] change. In order to design a nonperiodic polarization reversal structure for an arbitrary frequency response of the modulation index magnitude, we have proposed the modified flatness parameter MF [8], which expresses the difference between the magnitude of the modulation index |Δ𝜙(𝑓)| and the target function TF(𝑓) in the defined frequency range from 𝑓₁ to 𝑓₂MF=(𝑓₂−𝑓₁)∫^𝑓2_𝑓1|Δ𝜙(𝑓)/TF(𝑓)|²𝑑𝑓[∫^𝑓2_𝑓1|Δ𝜙(𝑓)/TF(𝑓)|𝑑𝑓]²−1. (4) By tuning the length of each domain, the value of the modified flatness parameter changes. If the modified flatness parameter becomes zero, the magnitude of the modulation index and the target function completely coincide in the frequency range.

We extended this method for dealing with both the modulation index magnitude and the phase. We define two flatness parameters, MF_𝑀 and MF_𝑃, for the magnitude and the phase of the modulation index by using two target functions: TF_𝑀(𝑓) for the magnitude and TF_𝑃(𝑓) for the phase in the defined frequency range from 𝑓₁ to 𝑓₂, as follows: MF_𝑀=(𝑓₂−𝑓₁)∫^𝑓2_𝑓1(|Δ𝜙(𝑓)|²/|TF_𝑀(𝑓)|²)𝑑𝑓[∫^𝑓2_𝑓1(|Δ𝜙(𝑓)|/|TF_𝑀(𝑓)|)𝑑𝑓]²−1,MF_𝑃=(𝑓₂−𝑓₁)∫^𝑓2_𝑓1(|arg[Δ𝜙(𝑓)]|²/|TF_𝑃(𝑓)|²)𝑑𝑓{∫^𝑓2_𝑓1(|arg[Δ𝜙(𝑓)]|/|TF_𝑃(𝑓)|)𝑑𝑓}²−1, (5)where MF_𝑀 is the difference between the magnitude of the modulation index |Δ𝜙(𝑓)| and TF_𝑀(𝑓), and MF_𝑃 is the difference between the phase of the modulation index arg[Δ𝜙(𝑓)] and TF_𝑃(𝑓).

If the modified flatness parameter MF_𝑀 becomes zero, the magnitude of the modulation index |Δ𝜙(𝑓)| and the target function TF_𝑀(𝑓) completely coincide in the defined frequency range. And if the modified flatness parameter MF_𝑃 becomes zero, the phase of the modulation index arg[Δ𝜙(𝑓)] and the target function TF_𝑃(𝑓) completely coincide in that defined frequency range. We tried to find the condition for minimizing the two flatness parameters MF_𝑀 and MF_𝑃 at the same time by tuning the domain lengths 𝐿_𝑖(𝑖=1,2,…,𝑀) successively and repeatedly. In this calculation, the initial value for each domain length is set as the same value in the periodic case 𝐿_𝑖=𝐿(𝑖=1,2,…,𝑀). At the first step of calculation, only the length of the first domain 𝐿₁ is changed, and the corresponding modulation index value is calculated. By repeating the calculation with tuning the length 𝐿₁, the smallest values of MF_𝑀 and MF_𝑃 which are called the local minimizing values in this calculation step are obtained. After that, only the next domain length 𝐿₂ is tuned and the next local minimal values of MF_𝑀 and MF_𝑃 are obtained. These local minimal values are smaller than the previous ones, respectively. This calculation step with tuning the domain length is repeated by changing the domain to be tuned from the first domain to the last domain successively. When finishing the calculation step with tuning the last domain length, the domain to be tuned is set to the first domain again and these calculation steps are repeated. In these repeating calculations, the minimal values of the two flatness parameters MF_𝑀 and MF_𝑃 are obtained when the local minimal values become the same with the values in the previous calculation step, respectively. As a result, the length of each domain is optimal and the difference between the frequency responses of modulation index and the target functions becomes minimum. Therefore, it is possible to control the frequency responses of the magnitude and the phase of modulation index.

4. Calculation Results

We tried to calculate the frequency response using our method shown in the previous section. In these calculations, the attenuation coefficient 𝛼 was set as negligible for simplicity. It is also possible to calculate and design the frequency response of the modulation index taking into account the attenuation coefficient [8].

Figure 3 shows examples of calculated frequency responses of the magnitude and the phase of the modulation index in the traveling-wave electrooptic modulators with nonperiodic polarization reversals for the flat, parabolic magnitude responses and linear phase responses. The number of domains was set as 𝑀=6 and the frequency range was set from 𝑓₁=0.7𝑓_𝑚 to 𝑓₂=1.3𝑓_𝑚. The target function for the flat magnitude response was defined as TF_𝑀1(𝑓)=𝑐₁ (𝑐₁ is constant, and its value is proportional to the applied modulation voltage to the electrodes), and the target function for the linear phase response was defined as TF_𝑃1(𝑓)=𝑎₁[(𝑓/𝑓_𝑚)−1] (𝑎₁ is constant, and it is set as 6.5 rad in calculation). With these conditions, the calculation for minimizing the flatness parameters was carried out. The obtained frequency responses of the magnitude and the phase are shown by the curves (1) in Figures 3(a) and 3(b), respectively.

(a)

(b)

Figure 3 

Frequency dependence of modulation index in traveling-wave electrooptic modulators for flat, parabolic magnitude responses with linear phase responses. (a) Magnitude responses, (b) phase responses.

The target function for the parabolic magnitude response was also defined as TF_𝑀2(𝑓)=𝑐₂[(𝑓/𝑓_𝑚)²−1]+𝑐₁ (with 𝑐₂/𝑐₁ is set as 0.6) and the target function for the linear phase response was defined as TF_𝑃2(𝑓)=𝑎₂[(𝑓/𝑓_𝑚)−1] (with 𝑎₂ is set as 10 rad). The calculation was also carried out. The obtained frequency responses of the magnitude and the phase are also shown as the curves (2) in Figures 3(a) and 3(b), respectively. For comparison, the frequency responses of the magnitude and the phase in traveling-wave electrooptic modulator with the periodic polarization reversal are also shown in the figures.

The obtained frequency responses almost coincide with the target functions in the designed frequency ranges. Table 1 shows the values of the obtained domain lengths normalized by the length 𝐿. The length 𝐿 is the domain length in the periodic polarization reversal for QVM at the peak modulation frequency 𝑓_𝑚. For example, if we set the peak modulation frequency as 𝑓_𝑚=15GHz and the lightwave wavelength as λ = 1.55 μm, then the corresponding length of 𝐿 becomes 𝐿=4.33mm for a LiTaO₃ guided-wave modulator (microwave phase velocity 𝑣_𝑚=6.47×10⁷m/s and lightwave group velocity 𝑣_𝑔=1.29×10⁸m/s).

Table 1 

(1) Obtained normalized domain lengths for the flat magnitude response and the linear phase response, and (2) the parabolic magnitude response and the linear phase response.

We confirmed the validity of the proposed method. It is possible to obtain the traveling-wave electrooptic modulators with the required frequency responses of both the magnitude and the phase of the modulation index by using our new approach.

5. Applications

Recently, there are many reports related to the advanced modulation formats such as duobinary, SSB, QPSK modulation to increase the capacity of the data transmission for the new generation optical fiber communication systems. Using the control technique of the frequency responses in the electrooptic modulators with the nonperiodic polarization reversal, it is possible to design the new electrooptic modulators for the advanced modulation formats.

5.1. Electrooptic Intensity Modulator for Duobinary Modulation

The optical duobinary modulation format offers a larger capacity of data transmission and a large tolerance for fiber chromatic dispersion in optical fiber communication system [3, 10]. Several experimental studies on the generation of the optical duobinary modulation signals have been reported. In these reports, the electrical/optical filters are required to limit the spectrum bandwidth of the generated signals. By using the nonperiodically polarization-reversed structure, it is possible to obtain the electrooptic modulators with specifically tuned filter-like modulation frequency responses with any passband (low-pass, high-pass, band-pass, Gaussian like, etc.). It can be used for the generation of the duobinary modulation signals without the external electric/optic filters.

Figure 4 shows the structure of the electrooptic intensity modulator for duobinary modulation using nonperiodic polarization reversal. It consists of the Mach-Zehnder waveguide, coplanar traveling-wave electrodes, and nonperiodically polarization-reversed structures. The lengths of each domain are the same in the two arms of the Mach-Zehnder waveguide, however, the polarity is opposite in the two arms. Therefore, by applying the electrical fields with the same direction, the push-pull intensity modulation with the same modulation index magnitude for the duobinary operation is obtained with a single modulation signal.

(a)

(b)

(c)

Figure 4 

Basic structure of duobinary modulator with nonperiodic polarization reversal. (a) Whole view, (b) cross-section view, (c) polarization-reversed patterns.

Using the proposed method for the frequency response control, we tried to obtain a Gaussian filter-like frequency response in the electrooptic modulation with a cutoff frequency 𝑓_𝑐 for generating the optical duobinary modulation signals. The target function for the Gaussian filter-like magnitude response was defined as TF_MG(𝑓)=Δ𝜙₀exp[−𝑔(𝑓/𝑓_𝑚)²] (with 𝑔 is set as 2) and the target function for the linear phase response was defined as TF_PG(𝑓)=𝑎(𝑓/𝑓_𝑚) (with 𝑎 is set as 5.5 rad). The number of domains was set as 𝑀=8 and the frequency range was set as the baseband from 𝑓₁=0 to 𝑓₂=𝑓_𝑚. The obtained results are shown in Figure 5. The targets functions are shown as the dashed lines. The obtained frequency responses of the magnitude and the phase of electrooptic modulation are shown as the solid lines. Table 2 shows the obtained domain lengths normalized by the periodic case length 𝐿. By using the Gaussian filter-like magnitude and the linear phase responses in electrooptic modulation, it is possible to generate the duobinary modulation signals by simple digital electrical driving signal without any filter.

Table 2 

Obtained normalized domain lengths for the Gaussian-like magnitude response and the linear phase response.

Figure 5 

Frequency dependence of modulation index in traveling-wave electrooptic modulator for the Gaussian-like filter magnitude response and the linear phase response.

We also applied the calculated polarization reversal pattern into the two arms of the Mach-Zehnder waveguide in the proposed duobinary modulator and calculated its performance. For the design of a 10 Gbps duobinary modulator, the cutoff frequency 𝑓_𝑐 was set as 5 GHz and the lightwave wavelength as 𝜆=1.55𝜇m. Then, the total electrode length 𝐿_𝑡 became 𝐿_𝑡=24mm for a LiTaO₃-based device.

We also checked the effect caused by the attenuation of the modulation microwave in the electrodes. We assumed a typical attenuation coefficient of the modulation microwave (𝛼=0.25dB/cm/GHz^0.5) in coplanar electrodes. As a result, the frequency response of the 10 Gbps duobinary modulator with the microwave attenuation became almost the same with the calculated one with the negligible attenuation coefficient in the frequency range from 0 to 10 GHz. Therefore, an ∼10Gbps duobinary modulator is expected using our new approach.

5.2. Broadband Single-Sideband Modulator

By using optical single-sideband modulation formats, it is possible to reduce the optical power in the signal transmission. This is attractive in long-haul optical fiber communication systems with higher density wavelength multiplexing [4]. We proposed and demonstrated the SSB modulators using the periodic polarization reversal [6]. By using our new approach, the SSB modulator with the flat and wider operational frequency range can be designed.

Figure 6 shows the structure of this electrooptic modulator using nonperiodic polarization reversal. It consists of the Mach-Zehnder waveguide, coplanar traveling-wave electrodes, and two nonperiodically polarization-reversed patterns set in the two arms of the Mach-Zehnder waveguide. In order to have the SSB modulation characteristics, it is required that the magnitudes of the modulation index are the same in the two arms of the Mach-Zehnder waveguide and the phases of the modulation index in the two arms are different as 𝜋/2 in the designed frequency range. For realizing these two modulation characteristics, we applied the control method of the frequency responses.

(a)

(b)

(c)

Figure 6 

Basic structure of SSB modulator with nonperiodic polarization reversal. (a) Whole view, (b) cross-section view, (c) polarization-reversed patterns.

Figure 7 shows the calculated frequency responses for the SSB modulator with nonperiodic polarization reversal for the same target function TF_MA(𝑓)=TF_MB(𝑓)=TF_𝑀(𝑓)=𝑐 (constant) and linear target functions TF_PA(𝑓)=𝑎(𝑓/𝑓_𝑚)+𝑏 and TF_PB(𝑓)=𝑎(𝑓/𝑓_𝑚)+𝑏−𝜋/2 (with 𝑎=4rad, 𝑏 is a common phase offset value). In this example, the number of domains was set as 𝑀=6 and the frequency range was set as 𝑓₁=0.7𝑓_𝑚 and 𝑓₂=1.3𝑓_𝑚. The calculated frequency responses of the magnitude and the phase of modulation index are shown by curves (A) and (B) in Figure 7. The obtained frequency responses of the modulation index magnitudes are flat and almost the same. The obtained frequency responses of the modulation index phase are linear and have a phase difference of about 𝜋/2 in the designed frequency range. Table 3 shows the obtained domain lengths normalized by the periodic case length 𝐿.

Table 3 

Obtained normalized domain lengths for the designed SSB modulator.

(a)

(b)

Figure 7 

Frequency dependence of modulation index in designed SSB modulator for flat magnitude responses and linear phase responses with 𝜋/2 difference. (a) Magnitude responses, (b) phase responses.

We applied the calculated polarization reversal patterns (A) and (B) into the two arms of the Mach-Zehnder waveguide in the SSB modulator and calculated the performance of the SSB modulation. The obtained frequency dependence of the upper sideband (USB) and the lower sideband (LSB) is shown in Figure 8, where we assumed the optical delay of 𝜆/4 in the two arms of the Mach-Zehnder waveguide by supplying an appropriate DC bias voltage to the electrodes. It is shown that the USB is generated around the designed frequency range, while the LSB is almost suppressed. The sideband suppression is expected over 20 dB around the designed frequency range. The characteristics of the USB and the LSB can be switched by changing the DC bias voltage. Therefore, the broadband SSB modulation characteristics are expected with our new approach.

Figure 8 

Frequency responses of USB and LSB components of designed SSB modulator.

6. Conclusion

We proposed the novel design of the traveling-wave electrooptic modulators using nonperiodically polarization-reversed structures to control frequency responses of both the magnitude and the phase of the modulation index. It is possible to design advanced electrooptic modulators such as duobinary modulators or broadband SSB modulators. This approach is also applicable to design other devices such as broadband optical frequency-shift-keying (FSK) modulator, and broadband optical frequency shifter (OFS).

Acknowledgments

This work was supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. This research was also partially support by a grant from the Global COE Program, “Center for Electronic Devices Innovation,” from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.