1. INTRODUCTION

Optical comb generation based on electro-optic (EO) modulators [1–4] has an impact on several fields, such as photonic measurements and fiber-optic telecommunications [5–7]. It is well known that deep optical phase modulation on a continuous-wave (CW) light simply and promptly generates modulation sidebands, which can be used as the frequency components (i.e., comb lines) of optical combs [2,3]. Compared with conventional comb generators relying on mode-locked laser technologies [8–10], the EO-modulator-based ones offer several advantages, particularly in terms of stability and flexibility of operation. They do not experience jitter and amplitude noise enhancement due to mode competition or other instabilities related to multimode laser oscillation. The center wavelength and frequency spacing of the generated combs can be flexibly tuned as desired by simply controlling and shifting the wavelength of the seed CW lights and frequency of the radio-frequency (RF) driving signals input to the comb generators. These features originate from the fact that the comb generator simply consists of a CW light source and EO modulator, where no optical cavities contribute to the comb generation and its spectral enhancement, unlike conventional mode-locked lasers.

One issue that remains in the EO-modulator-based comb generation is that the frequency spacing of the generated combs cannot exceed the electrical bandwidth of the EO modulators and their driver amplifiers, which are currently restricted to 10–40 GHz. The frequency offset of the generated comb should also be within the range. The restriction prevents the application of technologies to systems that demand optical combs with higher-frequency spacing. For example, it is challenging to generate an optical comb with a frequency spacing of 50 or 100 GHz, for applying the generated combs to standard fiber-optic wavelength-division multiplexing (WDM) transmission systems. To solve the issue, we need to seek methods that enable generation of multiple-frequency (MF)-spaced and MF-offset optical combs, where an optical comb with a frequency spacing of $N\!{f_m}$ [Hz] and frequency offset of $L\!{f_m}$ [Hz] ($N,L$: integers) is generated, even if the EO modulators are driven at ${f_m}$ [Hz] [11–14]. For example, it would be remarkable if we can generate optical combs with a frequency spacing of 100 GHz by EO modulation driven at 12.5 or 25 GHz. A straightforward way is to apply optical filters to the generated combs. A problem is that optical filters with excellent spectral resolution are required for selectively filtering the optical comb lines; in addition, the frequency spacing and center wavelength of the generated comb are fixed and hardly tunable because the comb lines need to be strictly aligned to either the pass or stopband of the applied optical filters.

In this paper, we investigate MF-spaced optical comb generation by using an EO multiple-parallel phase modulator (MP-PM). In the proposed approach, the frequency spacing of the generated combs is flexibly multiplied at multiples of the driving frequency. In addition to the operational mode for MF-spaced comb generation without frequency offsets (we reported in [13]), in this paper, we add another mode for functionally enabling MF-offset operation, where the center frequency of the generated MF-spaced comb is also shifted at multiples of the driving frequency as desired. Such MF-spaced, MF-offset comb generation is flexibly programable and dynamically reconfigured by controlling optical offsets in MP-PM without modifying the hardware setup. The control speed is potentially as high as MHz or GHz, thanks to the response speed of the EO modulation. We can build agile multiple-wavelength lasers that can flexibly switch laser wavelengths for reconfigurable WDM applications, for example. In addition, in this paper, a technique called “re-partitioning” of modulator arms in MP-PM is proposed for effectively maximizing the conversion efficiency. This is an effective solution to compete against the depression of conversion efficiency encountered when the modulator arms are naturally arranged according to the analytically derived driving conditions for MF-spaced and MF-offset operations [Eq. (9), described later]. In this paper, all the driving conditions of MP-PM for the operations are analytically clarified and summarized as simple formulas. For all the cases, we specify the driving conditions for spectrally flattening the generated combs, by extending the two-light pairing method [3].

The paper is structured as follows. First, the principles of MF-spaced and MF-offset optical comb generation using MP-PM are described in the next section. The driving conditions for MF spacing and offset are also clarified there. Section 3 describes the basic characteristics of the comb generation. In Section 4, numerical examples are provided for proving the concept and verifying the formulations. Then, the conclusion is drawn in the last section after discussing the perspective of the technology in Section 5.

2. DRIVING CONDITIONS OF MP-PM FOR MF-SPACED, MF-OFFSET COMB GENERATION

An MP-PM [13] is utilized for MF-spaced, MF-offset comb generation. Figure 1 shows its principles. The modulator in Fig. 1(a) shows a $2n$-arm MP-PM consisting of a $1 \times 2n$ input optical splitter, $2n$ sets of optical phase modulators, and a $2n \times 1$ output optical combiner. In the MP-PM, a CW light incident on the splitter is first split into $2n$ arms. Then, the CW light is phase modulated in each arm by using the phase modulator driven with a large-amplitude sinusoidal signal. The phase-modulated lights are combined with the $2n \times 1$ output combiner after their optical offsets are individually adjusted with optical offset controllers (OCs). The OCs may either consist of (1) amplitude and phase shifters (modulators) or simply of (2) phase shifters.

 

Fig. 1. (a) $2n$-arm multiple-parallel phase modulator (MP-PM) for MF-spaced, MF-offset comb generation. MF spacing and offset are achieved by using the $n$ arms in dotted box A. For flattening the comb spectra, part B is also used paired with part A ($N\!{f_{\rm{m}}}$, frequency spacing; $L\!{f_{\rm{m}}}$, frequency offset of the generated comb). (b) Offset controllers (OCs) for parallel superposition of phase-modulated lights, consisting of (1) amplitude and (2) phase shifters. (c) Interferometric blocking of unused lights (useful when we only use phase shifters in OCs). (d) Repartitioning of modulator arms in MP-PM for higher conversion efficiency.

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Here, in the subsections below, we describe the driving conditions of the modulator for MF-spaced and MF-offset optical comb generation. First, in Subsection 2.A, we focus on comb generation using $n$-arm MP-PM employing amplitude and phase OCs. In the analysis, we concentrate on the multiplication of frequency spacing and offset; we do not consider the spectral flatness of the generated comb. In Subsection 2.B, we discuss the configurations that omit amplitude shifters from the optical OC components, which allow us to use an MP-PM by purely employing an optical phase modulator in each arm. Subsection 2.C clarifies the driving conditions for spectrally flattening the generated comb, where a $2n$-arm configuration is used instead of the $n$-arm one. Repartitioning of the modulator arms in the MP-PM for higher conversion-efficiency operation is also described in Subsection 2.D.

A. Conditions for Multiplication of Frequency Spacing and Offset

Here, we clarify the driving conditions for MF-spaced and MF-offset comb generation with the $n$-arm MP-PM, where we only use the modulator arms enclosed in the dotted box A in Fig. 1.

The driving signal fed to the phase modulator at the $i$th arm is set as

The optical field of the light output from the $n$-arm MP-PM is expressed as

From this equation, it is found that the amplitudes of the comb lines can be controlled through the coefficients $\Xi$. In order to design the offset coefficients $\Xi$, the desired spectral coefficients, $\hat \Xi$, should be specified first; then, the values of coefficients for the OCs are determined according to the inverse discrete Fourier transform of $\hat \Xi$. Here, we define the important integer parameters, $N$ and $L$, as $N$: MF-spacing order; $L$: MF-offset order. For generating optical combs with a frequency spacing of $N\!{f_{\rm{m}}}$ and a frequency offset of $L\!{f_{\rm{m}}}$, the spectral coefficients, $\hat \Xi$, should be assigned as

B. Offset Control with Pure Phase Shifters

With the amplitude and phase OCs in the modulator arms, $N$ and $L$ are flexibly configured as desired; however, Eq. (9) suggests that amplitude and phase shifters should be structured in the OC sections, as shown in Fig. 1(b-1), which increases the complexity of the modulator structure. In this subsection, we describe a technique for offset control only by using optical phase shifters without relying on amplitude control in the OCs [Fig. 1(b-2)].

 

Fig. 2. Numerically calculated optical spectra: (a) generated from modulators #0, #1, #2, and #3, (b) from #4, #5, #6, and #7; (c) combining the optical combs of #0, #1, #2, and #3; (d) combining #4, #5, #6, and #7; (e) combining all combs generated from #0, #1, … #7.

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First, it can be seen from Eq. (9) that all coefficients ${\xi _i}$ become pure phase shifts in the case when $n = N$ is satisfied because only a single choice for $l$ exists in the range of $0 \le L + Nl \lt n$. Under this condition, $|{\xi _i}| = 1$ is always satisfied, which means that all the OCs may consist of optical phase shifters without structuring of the amplitude shifters.

In the case $n \ne N$, multiple $l$ exists in the given range. $|{\xi _i}|$ becomes in general not equal to one because of coherent summation of multiple lights, as expressed in Eq. (9), which means that we need to control the optical amplitude in each arm. However, we also notice that $|{\xi _i}|$ only takes a binary value, 0 or 1, if $n$ is divisible by $N$. This means that we should block the light’s output from the arms where the coefficients of $|{\xi _i}| = 0$ are assigned. This can be simply achieved in the following way, as shown in Fig. 1(c). First, we should turn off the RF signals fed to the arms to be blocked (i.e., ${A_i} = 0$ should be assigned). All the modulation sidebands at the arms disappear except the 0th order component at the input carrier frequency ($k = 0$ component). Then, the 0th order components from the arms are combined, giving the following phase offsets to the target arms:

To summarize, for MF-spaced and MF-offset optical comb generation by using $n$-arm MP-PM that employs only phase shifters in OCs, instead of using Eq. (9), we should set the amplitude of the RF signals and the optical offset coefficients for the OCs as follows:

C. Equalization of Comb-Line Amplitudes

An unresolved issue that we should address is that the generated MF-spaced and MF-offset optical comb does not always have a flat spectrum. To spectrally flatten the generated comb, we extend the technique of two-light pairing [3] and apply it to the $2n$-arm MP-PM used here. In the original technique, two phase modulators are driven with signals that have slightly different amplitudes. Here, the driving method is adopted in the MF-spaced and MF-offset comb generation using all the arms in blocks A and B of the $2n$-arm MP-PM shown in Fig. 1(a). The amplitudes of the generated comb lines are equalized independently of $k$ when the paired phase modulators are driven under the following condition:

D. Repartitioning of Modulator Arms in MP-PM for Higher Conversion Efficiency

The lightwave blocking in the unused modulator arms increases the optical loss and decreases conversion efficiency in comb generation if we choose the natural phase-modulator arrangement. This issue is solved if the unused arms are reused instead of blocking the lights. This can be accomplished by repartitioning the arrangement of the modulator arms in the MP-PM as shown in Fig. 1(d) and driving the unused modulator arms exactly the same way as performed for the originally nonblocked arms in the natural arrangement. Multiple optical combs with the same spectral profile are coherently summed up in the modulator by using the unused arms, which increases the optical intensity of the generated comb; consequently, the conversion efficiency becomes higher. This approach is worthwhile if the number of blocked arms, ${n_{{\rm{blocked}}}}$, is greater than that of the nonblocked arms, ${n_{{\rm{non}} - {\rm{blocked}}}}$, in the natural modulator-arm arrangement. Because ${n_{{\rm{non}} - {\rm{blocked}}}} = N$ and ${n_{{\rm{blocked}}}} = n - N$, $N \le n/2$ should be obviously satisfied for the repartitioning. If $n$ is divisible by $N$, none of the arms are blocked, and all the arms can be reused; therefore, the conversion efficiency is maximized. If not, unused arms remain (${n_{{\rm{blocked}}}}$ becomes ${n_{{\rm{blocked}}}} = [n\;\;{\rm{mod}}\;\;N]$ after the repartitioning), and the light’s output from the remaining arms should be interferometrically blocked.

After the repartitioning, by applying the two-light pairing as well, the offset coefficients, ${\hat \Xi _{{\rm{p,re}}}}$, assigned to the OCs should have the elements of

The control of RF phases is easily achieved by using microwave phase shifters.

3. BASIC CHARACTERISTICS

A. Bandwidth

Bandwidth of the generated combs is clarified by following the method developed for characterizing the optical comb generation using the Mach–Zehnder modulator (MZM) [3]. We assume in the analysis that the generated combs have rectangular spectra and that the optical energy is equally distributed to all the frequency components (comb lines). The bandwidth of the comb generated from the MP-PM should be the same as that from the MZM if the modulation depth is set at the same value [13]. This is because the coherent superpositions of the generated combs in the MP-PM structure do not change the overall spectral envelope of the generated comb. That is, the bandwidth is described as

In the equation, $a$ denotes a spectral correction factor for reflecting the actual spectral profiles, which are different from the ideal rectangular shape. It is applied for more precisely describing the bandwidth of the generated combs. Through the analysis of optical comb generation using the MZM, $a = 0.67$ is obtained as the optimum value for the correction [3].

Frequency spacing of the generated comb is multiplied $N$ times; therefore, the number of comb lines within the bandwidth can be estimated as

B. Conversion Efficiency

Conversion efficiency of the comb generation, ${\eta _k}$, is also derived here, which is defined as the intensity ratio from the input CW light to the $k$th generated comb line. When we focus on lightwaves from two modulator arms combined for the two-light pairing operated under Eq. (12), the power ratio between the input and output lights from the modulators is expressed as ${\eta _k} \equiv {P_{{\rm{out}}}}/{P_{{\rm{in}}}} = \frac{{1 - \cos 4\Delta \theta}}{4}$ [3]. Blocked arms in MF-spaced [13] and MF-offset comb generation using the MP-PM introduce additional optical loss. Because the energy of the CW light input to the modulator is equally distributed to the generated comb lines under the flat spectrum condition [Eq. (12)], the conversion efficiency can be described as

From the equation, it is found that the conversion efficiency varies as a function of the optical phase difference between the paired lights, $\Delta \theta$; the highest efficiency is achieved when $\Delta \theta = \frac{\pi}{4}$ is satisfied. We also notice that the conversion efficiency is inversely proportional to the number of comb lines and, consequently the modulation depth. Another important fact is that the net conversion efficiency, ${\eta _{{\rm{net}}}}$, reaches $\frac{1}{2}$ (i.e., the net conversion loss becomes ${-}{{3}}\;{\rm{dB}}$) when we set $\Delta \theta = \Delta A = \pi /4$ and $N/n = 1$ in the natural phase modulator arrangement. The highest net conversion efficiency is also achieved when $n$ is divisible by $N$ in the repartitioned case.

4. NUMERICAL PROOF

Here, we conduct numerical analysis in order to prove the concept. In the simulation, the optical fields output from the MP-PM are numerically calculated in the time domain; then, their spectra are calculated by FFT processing. For all calculations in this paper, the temporal resolution and data points for the FFT are set at 0.3125 [ps] and 4096, respectively. The following parameters are commonly used (except wherever notified): $A = 15.71$ [rad], $\Delta A = 0.79$ [rad], ${f_{\rm{m}}} = 25$ [GHz].

In Subsection 4.A, the analysis is focused on quadruple-frequency-spaced optical comb generation, which helps in understanding the mechanism of the proposed comb generation scheme. Subsection 4.B describes the multiplication of frequency offsets of the generated combs; then, in Subsection 4.C, the analysis is extended to the cases for MF-spaced and MF-offset comb generation with the application of modulator-arm repartitioning. The overall mapping of the driving conditions by scanning specific parameters, which is described in Subsection 4.D, clearly shows the definitions and roles of the parameters for the comb generation. Through numerical analysis in Subsection 4.E, we confirm the bandwidth and conversion efficiency of the comb generation that we derived above.

A. Quadruple-Frequency-Spaced Comb Generation

For the quadruple-frequency-spaced optical comb generation, $N$ and $n$ are set to 4, where the arm number of the parallel phase modulator is four if we ignore spectral flatness; whereas the arm number should be eight for flat-spectrum comb generation; $L = 3$ is chosen in the analysis. Under the above condition, the spectral coefficients are set as

From Eq. (9), the offset coefficients given to the modulation arms become

For the flat spectrum condition, the coefficients become

This driving condition is summarized in Line #1 in Table 1 (note that other driving conditions described in other parts in the paper are also summarized in the table. All conditions are for flat-spectrum comb generation using pure phase shifters for offset control).

Table 1. Driving Conditions of MP-PM for MF-Spaced, MF-Offset Comb Generation

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Figure 2 shows the calculated optical spectra generated from the MP-PM. To clearly show the roles of the arms in the MP-PM, we also calculated the optical spectra obtained when only the subsets of the arms are activated. These are plotted in Figs. 2(a)–2(d). Figure 2(a) is the optical spectra generated from arms #0, #1, …, or #3. These should have identical optical intensity spectral profiles because they are driven at the same modulation depth. Other common spectral outputs from arms #4, #5, …, or #7 are plotted in Fig. 2(b), where the spectral profiles are different from those in Fig. 2(a). Figures 2(c) and 2(d) depict the optical spectra obtained by combining the outputs from the arms #0, #1, … and #3 and those from #4, #5 …, and #7, respectively. These spectra have frequency spacing of $4f$ ($= \;{{100}}$[GHz]), but they inherit the spectral intensity envelopes of Figs. 2(a) and 2(b), respectively. Figure 2(e) is the optical spectrum obtained by combining Figs. 2(c) and 2(d), i.e., by combining all the outputs from the arms #0, #1, … and #7. As we expected, the generated comb becomes spectrally flattened with the frequency spacing of $4f$.

We can understand the mechanism for the multiplication of frequency spacing by watching the phase relationships of the generated comb in each arm. Figure 3 shows the phase relationships among the comb lines. In Figs. 3(a)–3(d), the optical phases of the comb generated from the arms #0, … #3 are plotted against the offset frequencies of the comb lines, where they obviously have different phase patterns. This is because the driving signals with different delays are fed to the arms. Figure 3(e) shows the phasor diagrams at the offset frequencies of A: 75, B: 100, C: 125, D: 150, and E: 175 GHz, where the squares, circles, triangles, and inverted triangles denote the components generated from the arms #0, #1, #2, and #3, respectively. It can be seen that all components generated from the arms have identical phases at the offset frequencies of A: 75, E: 175 GHz; therefore, they are constructively added when they are combined at the output-side optical combiner. On the other hand, the comb lines generated from different arms have phase differences of multiples of 90º or 180º when we analyze the frequencies around B: 100, C: 125, and D: 150 GHz, where their intensities are destructively suppressed to be zero at the output combiner. Therefore, the generated comb should have a frequency spacing of 100 GHz and a frequency offset of 75 GHz, as we expected.

 

Fig. 3. Optical phases of comb lines generated from (a) modulator #0, (b) modulator #1, (c) modulator #2, (d) modulator #3; (e) phasor diagrams at the frequencies A: 75, B: 100, C: 125, D: 150, E: 175 GHz. The symbols for the phasors are the same as those used in (a)–(d).

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B. Multiplication of Frequency Offsets

Next, it is shown that the frequency offsets of the optical combs generated from the MP-PM can be flexibly set at multiples of the driving frequency.

 

Fig. 4. Quadruple-frequency-spaced optical comb generation with different frequency offsets: (a) 0, (b) ${f_{\rm{m}}}$, (c) $2{f_{\rm{m}}}$, (d) 3 ${f_{\rm{m}}}$ frequency offsets.

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Fig. 5. Numerically calculated optical spectra: (a) 2x, (b) 3x, (c) 5x, (d) 8x frequency-spaced combs.

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The offset coefficients are controlled in the natural phase modulator arrangements by employing two-light pairing for spectral flattening. $L$ is changed from 0 to 3; other driving conditions are the same as those used for quadruple-frequency spaced optical comb generation mentioned above (but with $n = 8$). Under the conditions, the spectral offset control coefficients, $\hat \Xi$, are configured as follows: (a) $[{1,0,0,0,1,0,0,0}]$; (b) $[{0,1,0,0,0,1,0,0}]$; (c) $[{0,0,1,0,0,0,1,0}]$; and (d) $[{0,0,0,1,0,0,0,1}]$, which correspond to comb generations with frequency offsets of $0$, ${f_{\rm{m}}}$, $2{f_{\rm{m}}}$, and $3{f_{\rm{m}}}$, respectively. The offset coefficients, $\Xi$, are obtained as (a) $[{1,0,1,0,1,0,1,0}]$, (b) $[{1,0,j,0, - 1,0, - j,0}]$, (c) $[{1,0, - 1,0,1,0, - 1,0}]$, and (d) $[{1,0, - j,0, - 1,0,j,0}]$, translated from the spectral offset control coefficients. The driving conditions, ${\alpha _{\rm{p}}}$ and ${\Xi _{\rm{p}}}$, are summarized in Lines #2, #3, #4, and #5 in Table 1.

Figures 4(a)–4(d) show the optical spectra calculated for the offset coefficients of (a)–(d), respectively. The frequency offsets of the generated combs are 0, 25, 50, and 75 GHz, as we expected.

C. Multiple-Frequency-Spaced Comb Generation

Here, we apply the repartitioning technique for maximizing the conversion efficiency of the comb generation, focusing on the following four cases: (a) $L = 1$, $N = 2$, $n = 8$; (b) $L = 1$, $N = 3$, $n = 8$; (c) $L = 3$, $N = 5$, $n = 8$; and (d) $L = 7$, $N = 8$, $n = 8$. (a) and (b) are the repartitioned cases ($N \le n/2$), whereas (c) and (d) are naturally arranged ones ($N \gt n/2$). The driving conditions are summarized in Lines #6, #7, #8, and #9 in Table 1.

Figure 5 shows the optical spectra of MF-spaced and MF-offset optical combs calculated under the conditions in (a)–(d). Figures 5(a)–5(d) further show that the optical combs have (a) $2f$ (50 GHz), (b) $3f$ (75 GHz), (c) $5f$ (125 GHz), and (d) 8f (200 GHz) frequency spacing, respectively.

All combs in the examples are generated from a common MP-PM with $n = 8$. This means that we can flexibly set the MF-spacing and MF-offset orders without changing the hardware configuration.

D. Overall Mapping of the Driving Conditions

For a more clear understanding, we visualize the driving conditions for the comb generation. Because there are many parameters to control, we discuss the driving conditions by scanning the parameters one by one, where we apply some restrictions on the parameter control.

In Fig. 6, we plot a 2D contour of the optical intensity against the frequency offset ($y$ axis) and a particular scanned parameter ($x$ axis). First, we scan the parameter MF-spacing order denoted by $N$ for visualizing the influence of the parameter on the spectral profiles of the generated combs. In the analysis, the integer $N$ in Eq. (9) is replaced with a continuous value, $\hat N$ (called continuous MF-spacing order), and the offset control coefficients are calculated with the modified equation by scanning $\hat N$. Other parameters follow Line #2 in Table 1 for the analysis in the previous subsection. Figure 6(a) shows the optical spectra of the generated combs calculated for a different value of $\hat N$ scanned in the range of 0 to 10 with a step of 0.1. It can be seen that the frequency spacing becomes multiples of the modulation frequency when $\hat N$ is an integer.

 

Fig. 6. Optical spectra calculated against (a) continuous MF-spacing order ($\hat N$), (b) continuous MF-offset order ($\hat L$), (c) modulation imbalance ($\Delta A$), and (d) modulation depth ($A$).

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In a similar way, the MF-offset order, $L$, is replaced with a continuous one, $\hat L$ (continuous MF-offset order); then, the comb spectra are calculated by scanning $\hat L$. The scanning range and step are 0 to 10 and 0.1, respectively. As shown in Fig. 6(b), the frequency offset shifts proportionally to $\hat L$; strictly speaking, the frequency shift of $L\!{f_{\rm{m}}}$ is achieved when $\hat L$ equals the integers, $L$. We also see that the frequency shifts are periodic along $\hat L$ and its period is $n$ ($= \;{{8}}$ in this example).

The modulation imbalance, $\Delta A$, and modulation depth, $A$, are also important parameters that we can scan. Figure 6(c) shows the optical spectra plotted against $\Delta A$. It clearly shows that the generated comb is spectrally flattened along the dotted line, which corresponds to the flat spectrum condition. Figure 6(d) shows the optical spectral profiles calculated against $A$. As expected in Eq. (17), the number of comb lines increases proportionally to the modulation depth.

 

Fig. 7. Conversion efficiency and number of comb lines calculated against modulation depth; open symbols: number of comb lines; solids: conversion efficiency; red squares: ${\rm{2x}}$, blue triangles: 4x, green inverted triangles: 8x frequency-spaced combs; broken curves: theoretical conversion efficiency [Eq. (18)], dotted: theoretical number of comb lines [Eq. (17)]; calculated under the condition of (a) $n = 8$ and $\Delta A = \Delta \theta = 0.39$ [rad] (naturally arranged); (b) $N/n = 1$, and $\Delta A = \Delta \theta = \pi /4 = 0.79$ [rad] (naturally arranged, highest conversion efficiency); (c) $N = 2$ (red) and 3 (purple) for $n = 8$ and $\Delta A = \Delta \theta = 0.79$ [rad] (repartitioned, highest conversion efficiency).

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E. Bandwidth and Conversion Efficiency

Figure 6(d) gives us information on the bandwidth and conversion efficiency of the generated combs. Here, we extract the information and compare it with the theoretical ones described in Section 3.

Figure 7 shows the conversion efficiency and the number of comb lines plotted against the modulation depth. Figures 7(a) and 7(b) are for the naturally arranged cases and calculated under the conditions: (a) $n = 8,N/n \le 1$ and $\Delta A = \Delta \theta = 0.39$ [rad]; (b) $N/n = 1$ and $\Delta A = \Delta \theta = \pi /4$ ($= \;{0.79}$ [rad]), respectively. Figure 7(c) is for the repartitioned case with $n = 8$ and $\Delta A = \Delta \theta = \pi /4$. In the plots, the solid and open symbols represent the number of comb lines and conversion efficiencies, respectively. The number of comb lines is counted within the 3 dB bandwidth of the generated comb. Red squares, blue triangles, and green inverted triangles correspond to the cases for 2x, 4x, and 8x frequency spacing (i.e., $N = 2,4,8$), respectively. The number of comb lines and conversion efficiency derived with Eqs. (17) and (18) are also plotted as dotted and broken curves in the graphs, and the colors of the curves are the same as those for the numerical symbols ($N = 2,4,8$). We set the spectral correction factor as $a = 0.67$. The numerical values agree well with the theoretical ones.

In Fig. 7(a), $n$ is fixed at 8 for different $N$ ($= 2,4,8$). The conversion efficiency is maximized when $N = 8$, which satisfies $N/n = 1$; however, the efficiency is reduced when $N = 2,4$ because $N/n$ cannot reach one under the conditions. In the region $N/n \lt 1$, optical loss is increased because some light output from the unused arms is blocked, as we expected.

In Fig. 7(b), on the other hand, the conversion efficiency is maximized by setting the condition at $\Delta A = \Delta \theta = \pi /4$. The highest conversion efficiency is achieved regardless of the frequency multiplication factor, $N$ because $N/n = 1$ is always satisfied, and no lightwave output from the arms is blocked, in this situation.

For $N = 2,4,8$, we can obtain the same highest conversion efficiency even if we keep $n = 8$, when we apply the repartitioning of the modulator arms. The case for $N = 2$ is representatively plotted in Fig. 7(c), where the notations for symbols and curves are the same as those for Fig. 7(a). In these cases, $n$ is divisible by $N$; thus, all of the arms are used as a result of the repartitioning. We also plot the results for $N = 3$, $n = 8$, and $\Delta A = \Delta \theta = \pi /4$ ($= \;{0.79}$ [rad]) in Fig. 7(c) for examining the case where $n/N$ is not an integer. Numerical and theoretical values are plotted as purple circular symbols and broken-dotted curves, respectively. It can be seen that the conversion efficiency is reduced slightly if $n(= 8)$ is not divisible by $N(= 3)$. In this situation, there remain some blocked arms even after repartitioning the modulator arms.

5. PERSPECTIVE

As described above, the MF-spacing order, $N$, and MF-offset order, $L$, can be flexibly reconfigured. The flexibility will be increasingly enhanced for larger $n$. Currently, modulators with $n = 2,4$ have already been realized and recognized as dual- [14,15] or quad-parallel MZMs [16]. These kinds of modulators based on ${\rm{LiNb}}{{\rm{O}}_3}$ waveguide technologies are matured well; they also have high-bandwidth electrodes covering 25 GHz. This means that optical comb generation with 50 or 100 GHz frequency spacing can be practically achieved, which fits well in WDM systems. For higher-order multiplication in $N$ and $L$, we need to seek large-scale photonic integration technologies. InP-/silicon-photonic platforms [17,18] will be a solution for the integration in the near future. Bias-control techniques developed for integrated waveguide modulators are also useful for precisely adjusting the coefficients in the OCs; we need to upgrade them suitable for the multiple-parallel modulator configuration.

This paper focuses on discretely controlling the frequency spacing and offset of the generated combs. Wavelength of the generated comb lines needs to be continuously tuned for some purposes. We can flexibly control and configure the wavelengths of the generated comb lines by tuning the frequency of the driving signal, ${f_{\rm{m}}}$, and the center wavelength of the CW light input to the comb generator. It is known that optical single-sideband (SSB) modulation technologies [15] are useful for high-speed and precise wavelength control. We can build comb generators with the capability of agile and flexible wavelength control by applying them to our comb generator.

It is also important to increase the number of comb lines with deeper phase modulation. At this moment, several 10 comb lines with a single frequency spacing of ${f_m}$ can be generated with commercially available EO modulators [2,4]. Waveguide modulators with lower driving voltage [18–20] will accelerate deeper phase modulation, enhancing the generated comb spectra. Cascaded modulation techniques will also be helpful for spectral enhancements [21].

6. CONCLUSION

We have investigated MF-spaced and MF-offset optical comb generation by using an MP-PM. It has been shown that optical combs are generated with great spectral flatness, where the frequency spacing and offset of the generated combs are flexibly set at any multiple frequencies of the driving signal fed to the modulator. We have clarified the driving conditions of the modulator for comb generation and investigated the basic characteristics of the generated combs, which are numerically verified through analysis by focusing on optical comb generation with frequency spacings of $n \times 25$ [GHz].