## Abstract

We propose multiple-frequency-spaced flat optical comb generation using an electro-optic (EO) multiple-parallel phase modulator. We formulate and clarify the operating conditions in which we can generate optical combs adding the two important functionalities of spectral shaping: (1) multiplication of frequency spacing and (2) spectral flattening. The frequency spacing of the generated comb is enhanced much higher than the EO modulation bandwidth. The spectral shaping is achieved fully through the EO modulation process without relying on any optical filters. This filter-less configuration is advantageous for flexible tuning of wavelength and frequency spacing of the generated combs. The concept is numerically verified, focusing on $N\times 25$-GHz-spaced comb generation.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Optical comb generators based on electro-optic (EO) modulation [1–5] contribute a lot in several fields, such as photonic measurements and fiber-optic telecommunications. The technologies are advantageous for stably and flexibly generating optical combs with high frequency spacing, useful especially in ultra-high-speed multi-carrier transmission systems. One issue is that the frequency spacing of generated combs is basically the same as the frequency of microwave signals fed to EO modulators; thus, it cannot exceed the electrical bandwidth of the EO modulators and the driving amplifiers [1–3]. Currently, the frequency spacing is restricted up to 10–40 GHz and hardly reaches 50 or 100 GHz, which is recognized as a spacing of a wavelength grid standardized in fiber-optic wavelength-division multiplexing systems. To solve the issue, we need to develop any techniques that can multiply frequency spacing of generated combs. A straightforward way is to apply optical filters to a generated comb; however, very high spectral resolution is inevitable for slicing optical comb lines; in addition, the frequency spacing and center wavelength of the generated comb are fixed and hardly tunable if any optical filters are applied. There are some reports focusing on double frequency-spaced optical comb generation using serial or parallel EO modulator configurations [4,5].

In this Letter, we propose optical flat comb generation with high-order multiplication of its frequency spacing by using a multiple-parallel phase modulator. Parallel modulator structures themselves have been demonstrated so far mainly in the context of coherent data modulation for telecom applications, fabricated as ${\mathrm{LiNbO}}_{3}$-, InP-, or silicon-photonic-based integrated devices [6,7]. In this Letter, we formulate and clarify the driving conditions of the multiple-parallel phase modulator for optical comb generation with the spectral shaping functionalities of (1) multiplication of frequency spacing and (2) spectral flattening. The comb generation with these spectral shaping functionalities is achieved totally through the EO modulation process, without relying on any optical filters. This means that there are no wavelength sensitive factors in the comb generation, allowing us to flexibly control wavelength and frequency spacing of the generated combs. Another feature is that the multiplication orders of frequency spacing can be flexibly reconfigured by changing only optical offsets (i.e., bias voltages) applied to the modulator. The concept and analytically derived driving conditions are verified by numerical analysis on optical comb generation with $N\times 25\text{-}\mathrm{GHz}$ frequency spacing. Bandwidth and conversion efficiency of the generated comb are also characterized.

For generating a multiple-frequency-spaced comb, our proposal is to use a multiple-parallel optical phase modulator driven with a single-tone sinusoidal signal. Here, the structure and configuration of the modulator are described first; next, we clarify the driving conditions for spectrally flatly generating the optical comb in the modulator; then, we derive the conditions for multiplying the frequency spacing of the generated comb.

Figure 1(a) shows the basic structure of a $2n$-parallel optical phase modulator [6]. The modulator consists of a $1\times 2n$ optical splitter, $2n$ sets of optical phase modulators, and a $2n\times 1$ optical combiner. In the modulator, a continuous-wave (CW) light input to the splitter is split into $2n$ first. Then, the CW light is deeply phase modulated in each arm by using the phase modulator driven with a large-amplitude single-tone sinusoidal signal. Optical offset of the phase-modulated light is controlled with an offset controller placed in each arm. The offset controller comprises the sections for phase and amplitude control, which serve for spectral flattening and multiplication of frequency spacing, respectively. Finally, the lights output from the arms are combined with the optical combiner.

The phase modulator at the $i$-th arm is driven with the signal

*one*($\u03f5=1$), for simplicity. We also assume that the driving signals assigned to the arms have two different amplitudes; that is, ${A}_{i}={A}_{+}$ for $0\le i<n$ and ${A}_{i}={A}_{-}$ for $n\le i<2n$.

In each arm, an optical comb with a frequency spacing of ${f}_{\mathrm{m}}$ is generated, creating numbers of comb lines. However, as shown in subplots (b) and (c) in Fig. 1, the amplitudes of the comb lines are in general not equal, and the generated comb has a non-flat spectrum, because the amplitudes of comb lines generated by optical phase modulation obey Bessel functions in different orders. The non-flat spectral profile becomes an obstacle to practical use of EO-modulator-based comb sources.

To make the generated comb spectrally flattened, the technique based on two-light paring [3] is used, in which slightly different amplitudes, ${A}_{+}$, ${A}_{-}$, are given to the driving signals for the two paired arms of the modulator. Assuming that the $i$-th and $i+n$-th modulators are paired (for $i=\mathrm{0,1},\dots ,n-1$), the driving condition is described as

where $\mathrm{\Delta}\theta \equiv {\theta}_{\mathrm{b},i}-{\theta}_{\mathrm{b},i+n}$; $\mathrm{\Delta}A\equiv |{A}_{i}-{A}_{i+n}|=|{A}_{+}-{A}_{-}|$; and ${\theta}_{\mathrm{b},i}$ is the optical phase shift at the $i$-th arm, set by the phase control section of the offset controller shown in Fig. 1. Under the condition, a combination of the two phase-modulated lights makes the intensities of the generated comb lines equal, as shown in the subplot (d) in Fig. 1. The phase offset coefficients, $\mathrm{\Theta}=[{\theta}_{\mathrm{b},0},{\theta}_{\mathrm{b},1},\cdots ,{\theta}_{\mathrm{b},2n-1}]$ satisfying Eq. (2), are set to the offset controller to make all the paired lights spectrally flattened.This operation for flat-comb synthesis is quite important for multiple-frequency-spaced optical comb generation because the spectral flatness is inhered, even if the frequency spacing of the generated comb is multiplied, as described below.

Next, we clarify the driving conditions for the multiple-frequency-spaced optical comb generation by using the $2n$-parallel phase modulator. It is shown that we can flexibly multiply the frequency spacing in different multiplication orders by appropriately controlling the optical offsets among the arms. The offset coefficients set to the amplitude control sections of the offset controllers are clarified.

The optical field of the light output from the $2n$-parallel phase modulator is expressed as

For proof of concept, we conduct numerical analysis here. In the simulation, optical field output from the $2n$-parallel modulator is numerically calculated and its spectrum is obtained by fast Fourier transform (FFT). Following are the common parameters for the calculation: ${f}_{\mathrm{m}}\equiv {\omega}_{\mathrm{m}}/(2\pi )=25$ [GHz], $n=8$, $A=15.71$ [rad], and $\mathrm{\Delta}\theta =\mathrm{\Delta}A=0.39$ [rad], where Eq. (2) is always satisfied. Temporal resolution and data points for the FFT calculation are set at 0.3125 [ps] and 4096, respectively.

Here, three examples are provided to demonstrate flat optical comb generation with different frequency spacings: (a) $N=2$, (b) $N=4$, and (c) $N=8$. The spectral offset coefficients for comb-line control should be (a) $\widehat{\mathrm{\Xi}}=[\mathrm{1,0},\mathrm{1,0},\mathrm{1,0},\mathrm{1,0}]$, (b) $[\mathrm{1,0},\mathrm{0,0},\mathrm{1,0},\mathrm{0,0}]$, and (c) $[\mathrm{1,0},\mathrm{0,0},\mathrm{0,0},\mathrm{0,0}]$, respectively. The offset coefficients we should assign to the offset controller are derived from Eq. (5), simply yielding (a) $\mathrm{\Xi}=[\mathrm{1,0},\mathrm{0,0},\mathrm{1,0},\mathrm{0,0}]$, (b) $[\mathrm{1,0},\mathrm{1,0},\mathrm{1,0},\mathrm{1,0}]$, and (c) $[\mathrm{1,1},\mathrm{1,1},\mathrm{1,1},\mathrm{1,1}]$, respectively.

Figure 2 shows optical spectra of multiple-frequency-spaced flat optical combs calculated under the conditions, (a)-(c). In plots (a)-(c) in the figure, it is found that the optical combs have frequency spacings of (a) $2{f}_{\mathrm{m}}$ (50 GHz), (b) $4{f}_{\mathrm{m}}$ (100 GHz), and (c) $8{f}_{\mathrm{m}}$ (200 GHz), respectively, where undesired frequency components are suppressed well, as we designed. It is also shown that the spectra are always flat regardless of the multiplication order of frequency spacing. It has been confirmed that we can flexibly generate flat optical combs with different frequency spacings, such as $N=2$, 4, or 8, without changing hardware configuration, keeping $n=8$ in the example.

In the remaining part of this Letter, we investigate bandwidth and conversion efficiency of the generated multiple-frequency-spaced flat combs. We can apply the analysis method developed for characterizing the optical combs generated by a Mach–Zehnder modulator [3], since the spectral envelope will not change even if multiple spectra are superposed in the multiple-parallel modulator structure. That is, the bandwidth is described as

where $\overline{A}\equiv \frac{{A}_{+}+{A}_{-}}{2}$; and $a$ is a spectral correction factor for reflecting actual spectral profiles (different from ideal rectangular shape), applied for more precisely describing the bandwidth of the generated combs. (As discussed in Ref. [3], $a=0.67$ is assigned for describing 3-dB spectral bandwidth.)This equation suggests that the bandwidth of the generated comb is proportional to averaged modulation depth of the phase modulation applied in the modulator arms. Since frequency spacing of the generated comb is multiplied in $N$ times, the number of comb lines is simply estimated as

Figure 3 shows the number of comb lines appearing within the 3-dB bandwidth calculated against the modulation depth. Red dots, blue triangles, and green inverted triangles indicate comb generation for $2\times $, $4\times $, and $8\times $ frequency spacing, respectively. They are in good agreement with the theoretically derived number of comb lines [expressed in Eq. (7)], which are plotted as broken curves in the corresponding colors in the same graph.We also derive conversion efficiency of the generated combs, which is defined here as intensity ratio of each comb line output from the modulator against the input CW light. Under the flat spectrum condition of Eq. (2), the conversion efficiency becomes

*one*($N/n<1$), the conversion efficiency is reduced, where ${\xi}_{k}=0$ is assigned to some arms, and the lights coupled to the arms are blocked out. If $N/n=1$ is satisfied, all arms are set as all-pass; thus, the conversion efficiency is maximized. In the situation, the conversion efficiency becomes independent of the number of branches, $n$, though the optical splitting loss seems to be an issue at first glance. This is because the target comb lines surviving are constructively superposed, compensating for the splitting loss, while others are destructively suppressed. In fact, the net conversion efficiency, the ratio of the total power of the generated comb against that of the input CW light, is ${\eta}_{\mathrm{net}}=\frac{N}{4n}(1-\mathrm{cos}\text{\hspace{0.17em}}4\mathrm{\Delta}\theta )$, which means the energy loss can be minimized as low as $-3\text{\hspace{0.17em}}\mathrm{dB}$ (i.e., ${\eta}_{\mathrm{net}}=\frac{1}{2}$) when we set the phase offset parameter as $\mathrm{\Delta}\theta =\mathrm{\Delta}A=\pi /4$, keeping $N/n=1$. The 3-dB loss comes from the process for spectral flattening by means of the two-light pairing.

Figure 4 shows conversion efficiency calculated against the modulation depth under two different conditions. Notations in the plot are the same as those in Fig. 3. The conditions for plot (a) are $n=8$ and $\mathrm{\Delta}A=\mathrm{\Delta}\theta =0.39$ [rad]; $N=2$, 4, or 8 is taken for the calculation. The numerically calculated conversion efficiencies also agree well with analytical ones [Eq. (8)]. If we configure the modulator as $N/n=1$ and set the phase offset as $\mathrm{\Delta}A=\mathrm{\Delta}\theta =\pi /4$ (= 0.79 [rad]), the conversion efficiency is maximized independent of $N$, as shown in Fig. 4(b).

To discuss practicality and future perspective of the proposed comb generation scheme, we can once again look in the parameters, $N$ and $n$. Parallel modulator structures with a scale order of $n=2$ or 4 (i.e., 4 or 8 arms) have already been realized as dual- or quad-parallel Mach–Zehnder modulators [6], which are used for single- or dual-polarization quadrature-amplitude modulation in realistic optical fiber communication systems. By using the technologies, frequency spacing can be practically multiplied 2 or 4 times ($N=2$ or 4). With a driving frequency of 25 GHz, frequency spacing can reach 50 or 100 GHz, respectively. To pursue greater $N$ for higher frequency spacing and/or for lower driving frequency, larger-scale parallel modulator structures will be required. In addition to ${\mathrm{LiNbO}}_{3}$ waveguide technologies, photonic integration based on InP-/silicon-photonic platforms will be good solutions to such large-scale parallel modulator structures [7].

In conclusion, we have proposed and investigated optical comb generation with a multiple-parallel phase modulator. Operating conditions for multiple-frequency-spaced optical comb generation with great spectral flatness have been analytically clarified and numerically verified. Bandwidth and conversion efficiency of the generated comb have also been characterized.

## Funding

Precursory Research for Embryonic Science and Technology (PRESTO) (JPMJPR15P9); Japan Society for the Promotion of Science (JSPS) (B, 15H04001, C, 15K06050).

## Acknowledgment

The authors thank Dr. T. Kawanishi at Waseda Univ., and Drs. I. Morohashi and N. Yamamoto at NICT for discussion and support.

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