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×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 [15] 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 [13]. 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 LiNbO3-, 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×25-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 1×2n optical splitter, 2n sets of optical phase modulators, and a 2n×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.

 

Fig. 1. (a) 2n-arm multiple-parallel phase modulator for multiple-frequency-spaced optical comb generation; (b) and (c) typical spectra of modulated lights at the arms, (b) #0, (c) #n; and (d) flattened spectrum combining the lights’ output from the arms, #0 and #n.

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

ai(t)=Aisin(ωmt+2πiln+θs0),
where ωm (2πfm) and θs0 are the common angular frequency and initial phase of the driving signals; Ai is the amplitude of the driving signal at the i-th arm; and l is an arbitrary integer. Modulation efficiencies of all the phase modulators are assumed to be one (ϵ=1), for simplicity. We also assume that the driving signals assigned to the arms have two different amplitudes; that is, Ai=A+ for 0i<n and Ai=A for ni<2n.

In each arm, an optical comb with a frequency spacing of fm 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=0,1,,n1), the driving condition is described as

Δθ=±ΔA,
where Δθθb,iθb,i+n; ΔA|AiAi+n|=|A+A|; and θb,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, Θ=[θb,0,θb,1,,θb,2n1] 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

Eout(t)=i=0n1k=ξis^kej(kωmt+ω0t2πikn+ϕ0),
where s^k is the complex amplitude of the k-th comb line generated by pairing the two lights from the arms at i=0 and n; ξi stands for the offset coefficient given by the optical offset controllers in the i-th and i+n-th arms (the coefficients at the paired arms should be the same); and ω0 and ϕ0 are the angular frequency and initial phase of the input CW light. Taking Fourier transform of Eout, the optical field is expressed in the frequency domain as
E^out(ω)=l=k=0n1ξ^ks^nl+kejϕ0δ(ω(nl+k)ωmω0),
where δ(·) is a delta function; and ξ^k are the elements of spectral offset coefficients, Ξ^[ξ^0,ξ^1,,ξ^n1]. Note that Ξ^ is the discrete Fourier transform of the offset coefficients, Ξ[ξ0,ξ1,,ξn1]. This means that the amplitudes of the comb lines can be controlled through the offset coefficients, Ξ. For generating optical comb with a frequency spacing of Nωm, the spectral offset coefficients, Ξ^, should be assigned as ξ^k=1 for k=Nl and ξ^k=0 for others, where l=0,1,2,. Therefore, taking inverse discrete Fourier transform of Ξ^, the offset coefficients we should assign to the offset controllers result in
Ξ=[1,Nnlej2πNln,Nnlej4πNln,,Nnlej2π(n1)Nln].
This is the driving condition for multiplying the frequency spacing of the generated comb in N times. It can be seen that the elements ξi are always 1 or 0 if n is divisible by N. This suggests that the offset controllers can be simply implemented as optical on–off switches (Fig. 1). The multiplication of frequency spacing is achieved totally without using any optical filters or any other wavelength-sensitive elements. We also notice that the multiplication order of the frequency spacing is flexibly reconfigured by changing only the offset coefficients.

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: fmωm/(2π)=25 [GHz], n=8, A=15.71 [rad], and Δθ=ΔA=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) Ξ^=[1,0,1,0,1,0,1,0], (b) [1,0,0,0,1,0,0,0], and (c) [1,0,0,0,0,0,0,0], respectively. The offset coefficients we should assign to the offset controller are derived from Eq. (5), simply yielding (a) Ξ=[1,0,0,0,1,0,0,0], (b) [1,0,1,0,1,0,1,0], and (c) [1,1,1,1,1,1,1,1], 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) 2fm (50 GHz), (b) 4fm (100 GHz), and (c) 8fm (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.

 

Fig. 2. Calculated optical spectra; (a) 2×, (b) 4×, and (c) 8× frequency spacing; offset coefficients, Ξ, are (a) [1,0,0,0,1,0,0,0], (b) [1,0,1,0,1,0,1,0], and (c) [1,1,1,1,1,1,1,1].

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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

Δω=πaA¯ωm,
where A¯A++A2; 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

M=πaA¯N.
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×, 4×, and 8× 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.

 

Fig. 3. Number of comb lines versus modulation depth; red dots: 2×, blue triangles: 4×, green inverted triangles: 8× frequency-spaced combs, and broken curves: theoretical [Eq. (7)].

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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

ηk=1cos4Δθ4πaA¯(Nn)2.
From the equation, it is found that the conversion efficiency varies as a function of the optical phase offset between the paired lights, Δθ; the highest efficiency is achieved when Δθ=π4 is satisfied. We also notice that the conversion efficiency is inversely proportional to the modulation depth, according to the number of comb lines. This is reasonable because the energy of the CW light input to the modulator is equally distributed to the generated comb lines. The ratio between the multiplication order of the frequency spacing and number of arms, N/n(1), is another important factor for the conversion efficiency. If N/n is smaller than one (N/n<1), the conversion efficiency is reduced, where ξk=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 ηnet=N4n(1cos4Δθ), which means the energy loss can be minimized as low as 3dB (i.e., ηnet=12) when we set the phase offset parameter as Δθ=ΔA=π/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 ΔA=Δθ=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 ΔA=Δθ=π/4 (= 0.79 [rad]), the conversion efficiency is maximized independent of N, as shown in Fig. 4(b).

 

Fig. 4. Conversion efficiency versus modulation depth; red dots: 2×, blue triangles: 4×, green inverted triangles: 8× frequency-spaced combs, and broken curves: theoretical [Eq. (8)]; (a) under the condition of n=8 and ΔA=Δθ=0.39 [rad], and (b) optimal conversion efficiency under the condition of N/n=1, and ΔA=Δθ=π/4=0.79 [rad].

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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 LiNbO3 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.

REFERENCES

1. M. Kourogi, T. Enami, and M. Ohtsu, IEEE Photon. Technol. Lett. 6, 214 (1994). [CrossRef]  

2. M. Fujiwara, J. Kani, H. Suzuki, K. Araya, and M. Teshima, Electron. Lett. 37, 967 (2001). [CrossRef]  

3. T. Sakamoto, T. Kawanishi, and M. Izutsu, Opt. Lett. 32, 1515 (2007). [CrossRef]  

4. T. Hoshi, T. Shioda, Y. Tanaka, and T. Kurokawa, in Conference on Laser and Electro Optics/Pacific Rim (2007), paper ThD2-3.

5. T. Sakamoto, in IEEE Photonics Conference (IPC) (2016), paper WF3.6.

6. T. Sakamoto and A. Chiba, IEEE J. Sel. Top. Quantum Electron. 16, 1140 (2010). [CrossRef]  

7. P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

References

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  1. M. Kourogi, T. Enami, and M. Ohtsu, IEEE Photon. Technol. Lett. 6, 214 (1994).
    [Crossref]
  2. M. Fujiwara, J. Kani, H. Suzuki, K. Araya, and M. Teshima, Electron. Lett. 37, 967 (2001).
    [Crossref]
  3. T. Sakamoto, T. Kawanishi, and M. Izutsu, Opt. Lett. 32, 1515 (2007).
    [Crossref]
  4. T. Hoshi, T. Shioda, Y. Tanaka, and T. Kurokawa, in Conference on Laser and Electro Optics/Pacific Rim (2007), paper ThD2-3.
  5. T. Sakamoto, in IEEE Photonics Conference (IPC) (2016), paper WF3.6.
  6. T. Sakamoto and A. Chiba, IEEE J. Sel. Top. Quantum Electron. 16, 1140 (2010).
    [Crossref]
  7. P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

2010 (1)

T. Sakamoto and A. Chiba, IEEE J. Sel. Top. Quantum Electron. 16, 1140 (2010).
[Crossref]

2007 (1)

2001 (1)

M. Fujiwara, J. Kani, H. Suzuki, K. Araya, and M. Teshima, Electron. Lett. 37, 967 (2001).
[Crossref]

1994 (1)

M. Kourogi, T. Enami, and M. Ohtsu, IEEE Photon. Technol. Lett. 6, 214 (1994).
[Crossref]

Araya, K.

M. Fujiwara, J. Kani, H. Suzuki, K. Araya, and M. Teshima, Electron. Lett. 37, 967 (2001).
[Crossref]

Aroca, R.

P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

Baeyens, Y.

P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

Buhl, L.

P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

Chandrasekhar, S.

P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

Chen, Y.

P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

Chiba, A.

T. Sakamoto and A. Chiba, IEEE J. Sel. Top. Quantum Electron. 16, 1140 (2010).
[Crossref]

Dong, P.

P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

Enami, T.

M. Kourogi, T. Enami, and M. Ohtsu, IEEE Photon. Technol. Lett. 6, 214 (1994).
[Crossref]

Fujiwara, M.

M. Fujiwara, J. Kani, H. Suzuki, K. Araya, and M. Teshima, Electron. Lett. 37, 967 (2001).
[Crossref]

Hoshi, T.

T. Hoshi, T. Shioda, Y. Tanaka, and T. Kurokawa, in Conference on Laser and Electro Optics/Pacific Rim (2007), paper ThD2-3.

Izutsu, M.

Kani, J.

M. Fujiwara, J. Kani, H. Suzuki, K. Araya, and M. Teshima, Electron. Lett. 37, 967 (2001).
[Crossref]

Kawanishi, T.

Kourogi, M.

M. Kourogi, T. Enami, and M. Ohtsu, IEEE Photon. Technol. Lett. 6, 214 (1994).
[Crossref]

Kurokawa, T.

T. Hoshi, T. Shioda, Y. Tanaka, and T. Kurokawa, in Conference on Laser and Electro Optics/Pacific Rim (2007), paper ThD2-3.

Liu, X.

P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

Ohtsu, M.

M. Kourogi, T. Enami, and M. Ohtsu, IEEE Photon. Technol. Lett. 6, 214 (1994).
[Crossref]

Sakamoto, T.

T. Sakamoto and A. Chiba, IEEE J. Sel. Top. Quantum Electron. 16, 1140 (2010).
[Crossref]

T. Sakamoto, T. Kawanishi, and M. Izutsu, Opt. Lett. 32, 1515 (2007).
[Crossref]

T. Sakamoto, in IEEE Photonics Conference (IPC) (2016), paper WF3.6.

Shioda, T.

T. Hoshi, T. Shioda, Y. Tanaka, and T. Kurokawa, in Conference on Laser and Electro Optics/Pacific Rim (2007), paper ThD2-3.

Suzuki, H.

M. Fujiwara, J. Kani, H. Suzuki, K. Araya, and M. Teshima, Electron. Lett. 37, 967 (2001).
[Crossref]

Tanaka, Y.

T. Hoshi, T. Shioda, Y. Tanaka, and T. Kurokawa, in Conference on Laser and Electro Optics/Pacific Rim (2007), paper ThD2-3.

Teshima, M.

M. Fujiwara, J. Kani, H. Suzuki, K. Araya, and M. Teshima, Electron. Lett. 37, 967 (2001).
[Crossref]

Electron. Lett. (1)

M. Fujiwara, J. Kani, H. Suzuki, K. Araya, and M. Teshima, Electron. Lett. 37, 967 (2001).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Sakamoto and A. Chiba, IEEE J. Sel. Top. Quantum Electron. 16, 1140 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Kourogi, T. Enami, and M. Ohtsu, IEEE Photon. Technol. Lett. 6, 214 (1994).
[Crossref]

Opt. Lett. (1)

Other (3)

T. Hoshi, T. Shioda, Y. Tanaka, and T. Kurokawa, in Conference on Laser and Electro Optics/Pacific Rim (2007), paper ThD2-3.

T. Sakamoto, in IEEE Photonics Conference (IPC) (2016), paper WF3.6.

P. Dong, X. Liu, S. Chandrasekhar, L. Buhl, R. Aroca, Y. Baeyens, and Y. Chen, in Optical Fiber Communication Conference (OFC) (2013), paper PDP5C.6.

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Figures (4)

Fig. 1.
Fig. 1. (a)  2 n -arm multiple-parallel phase modulator for multiple-frequency-spaced optical comb generation; (b) and (c) typical spectra of modulated lights at the arms, (b) #0, (c)  # n ; and (d) flattened spectrum combining the lights’ output from the arms, #0 and # n .
Fig. 2.
Fig. 2. Calculated optical spectra; (a)  2 × , (b)  4 × , and (c)  8 × frequency spacing; offset coefficients, Ξ , are (a)  [ 1,0 , 0,0 , 1,0 , 0,0 ] , (b)  [ 1,0 , 1,0 , 1,0 , 1,0 ] , and (c)  [ 1,1 , 1,1 , 1,1 , 1,1 ] .
Fig. 3.
Fig. 3. Number of comb lines versus modulation depth; red dots: 2 × , blue triangles: 4 × , green inverted triangles: 8 × frequency-spaced combs, and broken curves: theoretical [Eq. (7)].
Fig. 4.
Fig. 4. Conversion efficiency versus modulation depth; red dots: 2 × , blue triangles: 4 × , green inverted triangles: 8 × frequency-spaced combs, and broken curves: theoretical [Eq. (8)]; (a) under the condition of n = 8 and Δ A = Δ θ = 0.39 [rad], and (b) optimal conversion efficiency under the condition of N / n = 1 , and Δ A = Δ θ = π / 4 = 0.79 [rad].

Equations (8)

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a i ( t ) = A i sin ( ω m t + 2 π i l n + θ s 0 ) ,
Δ θ = ± Δ A ,
E out ( t ) = i = 0 n 1 k = ξ i s ^ k e j ( k ω m t + ω 0 t 2 π i k n + ϕ 0 ) ,
E ^ out ( ω ) = l = k = 0 n 1 ξ ^ k s ^ n l + k e j ϕ 0 δ ( ω ( n l + k ) ω m ω 0 ) ,
Ξ = [ 1 , N n l e j 2 π N l n , N n l e j 4 π N l n , , N n l e j 2 π ( n 1 ) N l n ] .
Δ ω = π a A ¯ ω m ,
M = π a A ¯ N .
η k = 1 cos 4 Δ θ 4 π a A ¯ ( N n ) 2 .

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