Versatile photonic microwave waveforms generation using a dual-parallel Mach–Zehnder modulator without other dispersive elements

Optics Communications 396 (2017) 134–140

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Versatile photonic microwave waveforms generation using a dual-parallel Mach–Zehnder modulator without other dispersive elements

MARK

⁎

Guang-Fu Baia,b, , Lin Hub, Yang Jiangb, Jing Tianb, Yue-Jiao Zib, Ting-Wei Wub, FengQin Huangb a b

College of Big Data and Information Engineering, Guizhou University, Guiyang 550025, China College of Physics, Guizhou University, Guiyang 550025, Guizhou, China

A R T I C L E I N F O

A BS T RAC T

Keywords: Radio frequency photonics Analog optical signal processing Fiber optics

In this paper, a photonic microwave waveform generator based on a dual-parallel Mach–Zehnder modulator is proposed and experimentally demonstrated. In this reported scheme, only one radio frequency signal is used to drive the dual-parallel Mach–Zehnder modulator. Meanwhile, dispersive elements or filters are not required in the proposed scheme, which make the scheme simpler and more stable. In this way, six variables can be adjusted. Through the different combinations of these variables, basic waveforms with full duty and small duty cycle can be generated. Tunability of the generator can be achieved by adjusting the frequency of the RF signal and the optical carrier. The corresponding theoretical analysis and simulation have been conducted. With guidance of theory and simulation, proof-of-concept experiments are carried out. The basic waveforms, including Gaussian, saw-up, and saw-down waveforms, with full duty and small duty cycle are generated at the repetition rate of 2 GHz. The theoretical and simulation results agree with the experimental results very well.

1. Introduction In recent years, photonic generation of optical microwave waveforms has attracted much attention because of its important applications in all-optical signal processing and future all-optical networks [1– 3]. For examples, the triangular pulse is used in all-optical conversion of time-division multiplexed (TDM) to wavelength-division multiplexed (WDM) signals [4]; Geisler et al. [5] have invented a high bandwidth scalable, coherent transmitter based on the parallel synthesis of multiple spectral slices using optical arbitrary waveform generation. To generate microwave waveforms in optical domain, many approaches have been proposed. A method based on space-to-time (STT) pulse shaping was reported by several groups [6–8]. In this method, the arbitrary optical pulse sequence, which was generated by using free-space optical components or an arrayed waveguide grating (AWG) in the optical domain, was applied to a high-speed optical-to-electrical converter to generate a microwave waveform. Obviously, those additional components may increase the power loss, and large dimension of system is needed. Frequency-to-time mapping (FTTM) technique was also reported for arbitrary waveform generation [9,10]. In this method, the spectral envelope of a pulsed laser is modified by a spectral shaper, and these shaped spectra undergo frequency-to-time mapping in a dispersive

⁎

component. Unfortunately, ultrashort pulsed laser source can lead to a high cost, and the generated pulse usually has a small duty cycle ( < 1). In many application fields, photonic waveforms with full duty cycle (=1) is required. For example, saw-tooth pulse train with full-dutycycle is needed in all-optical wavelength-conversion scheme based on a saw-tooth pulse shaper [11]. Another photonic microwave generation scheme is based on photonic microwave delay-line filter [12–15]. However, delay-line filter generally has a linear group delay response. To generate a chirped microwave waveform, one filter with group nonlinear delay response is needed, which means that the delay-line filter should have complex tap coefficients. Unfortunately, it is a challenge to implement, especially for a filter with many taps [2]. Generation of a photonic arbitrary waveform can also be implemented through external modulation of a continuous wave (CW) laser [16]. The fundamental principle of this method is to manipulate the phases and amplitudes of these modulated sidebands, so that the optical intensity of sidebands approximately equals to the corresponding Fourier components of a photonic arbitrary waveforms. In Refs. [17,18], a dual electrode Mach–Zehnder modulator (DE-MZM) followed by a length of dispersion fiber was used to generate arbitrary waveforms, where the dispersion fiber was used to suppress the unwanted harmonic. In Ref. [19], one dual-parallel Mach–Zehnder modulator (DP-MZM) followed by an optical bandpass filter was utilized for the

Corresponding author at: College of Big Data and Information Engineering, Guizhou University, Guiyang 550025, China. E-mail address: [email protected] (G.-F. Bai).

http://dx.doi.org/10.1016/j.optcom.2017.03.050 Received 3 February 2017; Received in revised form 15 March 2017; Accepted 21 March 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.

Optics Communications 396 (2017) 134–140

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balanced （50/50）splitting ratio, the optical field of MZM-a output is

triangular pulse generation. A tunable bandpass filter was used to remove the negative sidebands in the ref. [19]. However, the generated RF waveforms are sensitive to the wavelength fluctuation of the optical carrier because the wavelength of the optical carrier has to match the transmission response of the TBPF in order to efficiently remove the undesired optical sidebands. Meanwhile, there are only two parameters that could be adjusted to generate the desired RF waveform. From the Fourier analysis method, an arbitrary waveform can be synthesized by a fundamental frequency and many high order harmonics. Therefore, in the ref. [19], the harmonics higher than third order can’t be controlled, which leads that the generated waveform are not very accurate. At last, only the triangular and rectangular waveforms were verified by experiment in the ref. [19]. Besides above methods, Jiang et al. [20,21] have recently reported a photonic waveform generation by using time-domain synthesis. Some waveforms can be generated by arranging optical power in time domain. The method is still not so satisfied. Firstly, one initial photonic waveform such as square waveform is required to generate other photonic waveform, but generation of the initial photonic waveform is based on external modulation in Ref. [21]. Secondly, the cascaded MZM modulators, and other dispersive components can increase complexity of the system. Among the methods, external modulation has been significantly developed to simplify the system. In Ref. [22], only one DP-MZM was utilized for the triangular pulse generation, but two radio frequency (RF) signals with different frequencies (f and 3f) are required to drive the DP-MZM. In Ref. [23], the authors proposed and experimentally demonstrated an approach to generate triangular pulses using a DPMZM driven by a single-frequency RF signal. Here, we proposed a new approach to generate photonic waveforms based on a DP-MZM. Different from other schemes in Refs. [19–23], firstly, dispersive elements or filters are not required in the proposed scheme, which make the scheme simpler and more stable. Secondly, the generated RF waveforms has higher precision as there are six parameters could be adjusted in the proposed scheme. Thirdly, the different photonic waveforms, including but not limited to triangular waveform are theoretically analyzed and verified through experiment. Comparing with the works from Refs. [9,10], our approach is capable of generating photonic waveforms with full duty cycle (=1). The proposed scheme can also generate photonic waveforms with small duty cycle ( < 1).

Ea =

1 jΔϕ E0 e jω0 t (e jΔϕuper a + ee down a ), 2 2

(1)

where E0 and ω0 indicate the amplitude and the angular frequency of π optical feild at the laser output, respectively, Δϕuper a = V (Vrf a + Vbias a ) π

π

and Δϕdown a = V (−Vrf a ) indicate the phase shifts of the two arms of the π MZM-a introduced by the voltages Vrf a and Vbias a , Vπ is switching voltage of the MZMs. Following the Euler's formula cos x = as

Ea =

eix + e−ix , 2

Eq. (1) can be written

1 π j π V E0 e jω0 t cos [ (2Vrf a + Vbias a )] e 2Vπ bias a . 2Vπ 2

(2)

Similarly, the optical field at the output of the MZM-b can be expressed as

Eb =

1 π −j π V E0 e jω0 t cos [ (2Vrf b + Vbias b )] e 2Vπ bias b . 2Vπ 2

(3)

The optical field of the DP-MZM output is the superposition of the two waves. It can be written as

Ec = Ea + e

−j π Vbias c Vπ Eb.

(4)

Therefore, the power envelope of the light can be written as

I (t ) = Ec 2 = Ec Ec* π

Vbias a )] 2

+ 2 E02 cos2 { V [Ab cos(ωt + φ)

π

Vbias a )] 2

cos { V [Ab cos(ωt + φ) +

e j 2Vπ (Vbias a+ Vbias b+2Vbias c) π 1 + 2 E02 cos [ V (Aa cos ωt +

Vbias a )] 2

cos { V [Ab cos(ωt + φ) +

1

= 2 E02 cos2 [ V (Aa cos ωt + π

+

π

Vbias b ]} 2 π

1

π

1

+ 2 E02 cos [ V (Aa cos ωt +

π

π

Vbias b ]} 2

π

π

π

π

Vbias b ]} 2

π

e−j 2Vπ (Vbias a+ Vbias b+2Vbias c). (5) Expanding Eq. (5) based on Jacobi–Anger expansions, as detailed in Appendix A, the optic intensity at the output of the MZM can be written as

2. Principle of the proposed scheme Fig. 1 gives the principle of the proposed photonic microwave waveform generation system. A DP-MZM comprising two singleelectrode child MZMs and a third parent MZM is the key part to generate a photonic waveform. The DP-MZM has two RF inputs and three independent DC bias voltages. Vrf a = Aa cos(ωt ) and Vrf b = Ab cos(ωt + φ) represent the RF signals applied to MZM-a and MZM-b, respectively, where Aa and Ab are the amplitudes of the two RF signals, ω is the angular frequency of the modulating RF signals, and φ is the phase difference between the two RF signals. Vbias a ,Vbias b , and Vbias c are the three bias voltages applied to MZM-a, MZM-b, and the parent MZM-c, respectively. Let every Y splitter of the DP-MZM has a

Fig. 1. Schematic diagram of the photonic microwave waveforms generation using a DP-MZM.

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G.-F. Bai et al. 1

1

I (t ) = 2 E02 + 4 E02 J0 ( +

1 2 E 2 0

∞ ∑n =1

1

π 2πAa )cos( V Vbias a ) Vπ π

π 2πAb )cos( V Vbias b ) Vπ π

parts, and the two RF signals are applied to MZM-a and MZM-b, respectively. The attenuator is used to control the power ratio of the two RF signals, and the phase difference φ between the two RF signals is controlled by the phaseshifter. The bias voltages Vbias a ,Vbias b , and Vbias c are adjusted by the DC voltage controllers. After square-law detecting using a PD (XPDV2320R-16120701) with a bandwidth of 40 GHz, the generated waveform is measured by an oscilloscope (DSO91204A), and its electrical spectra are measured by an electrical spectrum analyzer (ESA-E4405B). By adjusting the values of those six variables, we can obtain different waveforms.

2πA π (−1)n J2n ( V a )cos(2nωt )cos( V Vbias a ) π π

∞

+ 2 E02 ∑n =1 (−1)n J2n −1 ( 1

1

+ 4 E02 J0 (

∞

+ 2 E02 ∑n =1 (−1)n J2n ( 1

∞

+ E02 {[J0 (

πAa ) Vπ

2πAa )cos Vπ

2πAb )cos Vπ

+ 2 E02 ∑n =1 (−1)n J2n −1 (

π

2πAb )cos Vπ

∞

∞

π

⋅ cos( V Vbias a )}{[J0 ( π

πAa )sin Vπ

πAb ) Vπ

π

π

[(2n (ωt + φ)] cos( V Vbias b )

+ 2 ∑n =1 (−1)n J2n (

+ 2 ∑n =1 (−1)n J2n −1 (

π

[(2n − 1) ωt ] sin( V Vbias a )

π

[(2n − 1)(ωt + φ)] sin( V Vbias b ) π

πAa )cos(2nωt )] Vπ

π

cos( 2V Vbias a ) π

[(2n − 1) ωt )] ∞

+ 2 ∑n =1 (−1)n J2n (

πAb )] Vπ

cos [2n (ωt + φ)]

4. Experimental results and discussions

π cos( 2V Vbias b ) π ∞

+ 2 ∑n =1 (−1)n J2n −1 (

πAb )cos Vπ

Firstly, we investigate the generation of the triangular waveform based on Eqs. (7) and (8). For a periodically triangular waveform, if we neglect the harmonics higher than third order, the Fourier series can be written as

π

[(2n − 1)(ωt + φ)] sin( V Vbias b )} π

π

cos( 2V (Vbias a + Vbias b + 2Vbias c )), π

(6)

∞

where Jn is the n-th order of the first kind of Bessel function, and n is integral number. Obviously, the modulation can generate spectral comb with frequency spacing of ω . After square-law photo-detection using a PD with responsitivity R , the optical signal is converted to electronical signal such that the detected signal i (t ) equals to RI (t ). The Eq. (6) indicates that the detected signal consists of the DC term iDC and an infinite number of harmonics inω ,

n =1,3,5,...

1 + cos(3(ωt + φ)). 9

∑ inω, n =1

(7)

where iDC and inω are functions of the modulating RF signals ( Aa and Ab ), the phase difference φ between the two RF signals, and the bias voltages Vbias a ,Vbias b and Vbias c . From Eq. (7), we can find that each harmonic is resolved and can be analyzed individually. In another hand, from the Fourier analysis method, an arbitrary waveform can be expressed by the following Fourier series

i (t ) ≈ iDC +

∑ Cn cos(nωt ). n =1

RE02 2

πA

πA

πA

π

π

π

[J1 ( V )cos ωt + J3 ( V )cos ωt + J1 ( V )sin 3ωt

πA

+ J3 ( V )sin 3ωt ] π

π

∝iDC + cos(ωt − 4 ) +

1 9

π

cos(3(ωt − 4 )).

(10)

Comparing Eq. (9) with Eq. (10), the desired triangular waveform may be generated. In our experiment, the null points and full points of the child MZMs and the parent MZM are measured firstly. The values are shown in Table 1. From the table, we can find that the switching voltages (i.e. Vπ = null point−full point ) are different; the switching voltages of the three sub-MZMs are 4.8 V, 4.6 V, and 6.2 V, respectively. The operation values are also shown in Table 1. We set Vbias a = −6.5V ,Vbias b = 1.4V , Vbias c = 0.1V . The values agree well with 3V V theoretical values (i.e. Vbias a = 2π ,Vbias b = 2π , Vbias c = 0 ). Furthermore, π the φ is controlled to be , and the powers of the two RF signals are 2 both 19 dBm. Fig. 3 shows the experimental results of the triangular waveform with repetition frequency of 2 GHz. Fig. 2(a) and (b) shows the measured triangular waveform with time scale of 100 ps/div and 300 ps/div, and the corresponding spectrum is shown in Fig. 2(c). As shown in the Fig. 2(c), the power of the fundamental tone at 2 GHz is 19.8 dB higher than third order harmonic at 6 GHz. it is close to the theoretical value of 19.08 dB. The second-order harmonic is suppressed and is 40 dB lower than the fundamental tone. To generate other waveforms, it is needed to find the optimum values of the variables using the multivariable control method [18]. During the simulation, the switching voltages of the three sub-MZM are

∞

TAr (t ) = C0 +

(9)

Those six variables must be adjusted so that the first and the third harmonics of Eq. (7) are equal to the corresponding harmonics of Eq. (9). 3V V π Aa = Aa = A = 0.48Vπ , φ = 2 , Vbias a = 2π ,Vbias b = 2π , Setting Vbias c = 0 , and neglecting the harmonics larger than third order, the Eq. (7) can be written as

∞

i (t ) = iDC +

1 cos(n (ωt + φ)) ≈ C0 + cos(ωt + φ) n2

∑

Ttri (t ) = C0 +

(8)

Comparing Eq. (7) with Eq. (8), any disired waveform can be obtained by adjusting the variables in Eq. (7), if the corresponding harmonics in Eq. (8) equal to those in Eq. (7). The waveform is determined by the amplitudes of the modulating RF signals ( Aa and Ab ), the phase difference φ between the two RF signals, and the bias voltages Vbias a ,Vbias b , and Vbias c . In application, the Eq. (7) can reach a favorite approximation to the desired waveform, when the main orders of harmonics are kept. For example, the triangular waveform can be realized in a favorite precision, when only the first three harmonics are kept [20]. For other waveforms, the first five harmonics are kept in this proposed scheme. The concrete expressions of these harmonics are derived in Appendix B. Actually, there are more than one possible series of parameters to get some desired waveform. Multivariable control method [18,24] can be used to find the optimum values of the variables in order to get the expected waveform. This method is introduced in details in Ref. [18]. 3. Experimental setup

Table 1 The null point and full point of the three sub-MZMs and operation value of bias voltages for triangular generation.

The experimental setup of the proposed photonic microwave waveform generation is shown in Fig. 1. A tunable laser (Ando AQ4321D) outputs a CW with wavelength of 1558.9 nm, which is modulated by the DP-MZM (Fujitsu FTM7977HQA). The maximum output of the RF signal generator （Anristu MG3690B）with tunable frequency reaches 26 dBm, and the frequency is fixed to 2 GHz in our experiment. After a hybrid coupler, the RF signal is diveded into two 136

Bias voltage (V)

Null point

Full point

Operation value

Vbias a Vbias b Vbias c

0.7 −1 0

−4.1 3.6 −6.2

−6.5 1.4 0.1

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G.-F. Bai et al.

Fig. 2. the measured triangular waveform with time scale of (a) 100 ps/div and (b) 300 ps/div, the corresponding spectrum of (a) and (b).

equal to 6.5 V, and the first five harmonics are considered in the algorithm. Firstly, a certain target waveform with given linearity and power spectrum is proposed. The optimum waveform may be found within the searching range based on Eq. (7). When similarity coefficient of the searched waveform is greater than 0.9, the waveform and corresponding variables are recorded. The similarity coefficient is defined as

Table 2 optimal variables of different waveform.

N

ξ=1−

∑k =1 (Χk − Yk )2 N

∑k =1 (Χk 2 + Yk 2 )

, (11)

where N is the number of sampling points in a period, Χk is the k-th sampled amplitude of the target waveform, and Yk is the k-th sampled amplitude of the searched waveform. After finishing the search, the similarity coefficient close to 1 mostly is regarded as optimum waveform. The optimal variables are reserved. If there is no records, it means that the target waveform can not be generated using this system. The optimal variables of different waveforms are shown in Table 2. Based on the above optimal theoretical values, the output signal is manipulated by adjusting all of the variables. Fig. 3(a) shows the simulated results of the trapezoidal waveform. Fig. 3(b) and (c) show the measured trapezoidal waveform and the corresponding spectrum.

Parameters

Trapezoidal

Gaussian

Saw-down

Saw-up

Vbias a (V) Vbias b (V) Vbias c (V) φ (rad) Aa (V) Ab (V) similarity coefficient

−2.7 9.2 0 0.2 1.7 4.8 0.977

4.3 7.1 −4.7 0 1.5 5.3 0.954

8.5 −2.7 4.5 1.7 0.5 5.0 0.962

4.8 −5.7 5.5 4.7 0.2 5.0 0.948

Both the measured rising time and falling time are 125 ps. Comparing Fig. 3(a) with Fig. 3(b), the experimental and simulation results agree with each other very well. The Gaussian waveform can also be generated by changing the variables. To generate basic waveforms with small duty cycle, we set target Gaussian waveform with small duty cycle of 0.89, and the searched optimal variables are given in Table 2. The simulation and experimental results of Gaussian waveform are shown in Fig. 3(d) and (e), and (f) gives the spectrum of the measured Gaussian waveform. The results show that a basic waveform with small

Fig. 3. trapezoidal and Gaussian waveforms. (a) simulated and (b) measured trapezoidal waveform, (c) spectrum of (b), (d) simulated and (e) measured Gaussian waveform, (f) spectrum of (e).

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Fig. 4. saw-down and saw-up waveforms. (a) simulated and (b) measured saw-down waveform, (c) spectrum of (b), (d) simulated and (e) measured saw-up waveform, (f) spectrum of (e).

by the bandwidths of the DP-MZM and PD.

duty cycle can be generated based on the proposed scheme. The generation of sawtooth waveform is verified as well. Fig. 4(a) and (b) shows the simulated and measured saw-down waveforms. The corresponding spectrum of measured saw-down waveforms is given by Fig. 4(c). The measured rising time and falling time are 125 ps and 375 ps, respectively. Fig. 4(d) and (e) show the simulated and measured saw-up waveforms, The corresponding spectrum of measured saw-up waveform is shown in Fig. 4(f). The measured rising time and falling time are 377 ps and 123 ps, respectively. Both of the measured waveform and electrical spectrum indicate that they are saw-down and saw-up pulse strains with repetition frequency of 2 GHz. All of the above measured waveforms show that they have full duty cycle except for Gaussian waveform. Finally, we should note that the repetition rate of the generated microwave waveforms is fixed to 2 GHz in our experiment to give an effective experimental verification, because the spectrum range of our ESA (Agilent Technologies ESAE4405B) is from 9 kHz to13.2 GHz. Actually, we can generate microwave signal with repetition rate of about 10 GHz by using the experimental equipment which we have now if we don't measure the frequency of the generated microwave waveform. From the above experimental verification, the practicality and the significances of the proposed scheme are well exhibited. Obviously, this is a flexible scheme to generate versatile photonic microwave waveforms. Only one DP-MZM is used to manipulate sidebands without using additional dispersive element or filter. It can also be regarded as an intuitive way to manage spectral lines by adjusting the six variables. Although only the modulation frequency of 2 GHz is demonstrated, the tunability of the proposed scheme is reliable. The operation bandwidth and the repetition frequency of the generated waveform are determined

5. Conclusions In this work, we have proposed a photonic waveform generation method based on a DP-MZM. A mathematical expression has derived, which can help with the prediction of the waveform generation. In this system, there are six variables affecting the waveform formation. Through the different combinations of these variables, we can generate basic waveforms with full duty and small duty cycle. To verify this method, we have measured versatile waveforms such as triangular, trapezoidal, Gaussian waveform, saw-up, and saw-down waveforms at the repetition rate of 2 GHz in the experiment, these waveforms are full duty cycle except for Gaussian waveform. In addition, only one DPMZM is used to manipulate sidebands, and no additional dispersive element or filter is needed. Finally, tunability of the generator is reliable due to the adjustable RF signal and CW light source. This scheme is particularly attractive as a waveform generation source for photonic network applications. Acknowledgements Our research is supported in part by Guizhou Province for Society Development (Grant Numbers: QianKeHe-2013–3125), in part by the Overseas Returning Persons Technology and Innovation Foundation of Guizhou Province of China (Grant Numbers: 2016-23), in part by the National Natural Science Foundation of China (Grant Numbers: 11264006, 61465002). The authors would also like to thank Prof. Chun-Ting Lin of NCTU of Taiwan for his technique support.

Appendix A By substituting the Euler's formula, Eq. (5) can be written as

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Optics Communications 396 (2017) 134–140

G.-F. Bai et al. π

1

Vbias a )] 2

I (t ) = 2 E02 cos2 [ V (Aa cos ωt + π

π

+ E02 cos [ V (Aa cos ωt + π

Vbias a )] 2

π

1

+ 2 E02 cos2 { V [Ab cos(ωt + φ) + π

π

cos { V [Ab cos(ωt + φ) + π

Vbias b ]} 2

Vbias b ]} 2

π

⋅ cos [ 2V (Vbias a + Vbias b + 2Vbias c )] .

(A1)

π

1 + cos 2θ 2

cos2 θ

= Using the trigonometric power reduction formula cos(θ + φ) = cos θ cos φ − sin θ sin φ , Eq. (A1) can be written as 1 2 E [1 4 0

I (t ) =

2π

+ cos( V Aa cos ωt )cos( π

2π

1

Vbias a ) Vπ

2π

− sin( V Aa cos ωt )sin( π

and the addition angle formula,

Vbias a )] Vπ

Vbias b V 2π b ) − sin [ V Ab cos(ωt + φ)] sin( bias )} Vπ Vπ π π Vbias a sin( V Aa cos ωt )sin( 2V )] π π π V π b − sin [ V Ab cos(ωt + φ)] sin( bias )} cos [ 2V (Vbias a 2Vπ π π

+ 4 E02 {1 + cos [ V Ab cos(ωt + φ)] cos( π

π

Vbias a ) − 2Vπ Vbias b cos( 2V ) π

+ E02 [ cos( V Aa cos ωt )cos( π

π

⋅ { cos [ V Ab cos(ωt + φ)] π

+ Vbias b + 2Vbias c )].

(A2)

Expanding Eq. (A2) using Jacobi–Anger expansions, the Eq. (6) can be obtained. Appendix B Because the first five harmonics are considered in our simulation and experiment, we derived the concrete expressions of the DC term and the first five harmonics as following, 1

iω = − 2 RE02 J1 (

πV 2πAa a )sin( bias ) Vπ Vπ

1

cos(ωt ) − 2 RE02 J1 (

2πAb πV b )sin( bias )cos(ωt Vπ Vπ

+

πA πA πV πV a b )cos( 2bias )sin(ωt RE02 {−2J5 ( V a ) J4 ( V b )cos( bias V V

−

πA πA πV πV a b 2J1 ( V a ) J0 ( V b )cos( bias )cos( 2bias )sin(ωt ) Vπ Vπ π π

−

πA πA πV πV a b 2J2 ( V a ) J3 ( V b )cos( 2bias )sin( bias ) Vπ Vπ π π

+

πA πA πV πV a b 2J4 ( V a ) J3 ( V b )cos( 2bias )sin( bias )cos(ωt Vπ Vπ π π

π

π

π

− 4φ ) −

π

−

πA πA πV πV a b 2J1 ( V a ) J2 ( V b )cos( bias )cos( 2bias ) Vπ Vπ π π

sin(ωt + 4φ)

πA πA πV πV a b 2J2 ( V a ) J1 ( V b )cos( 2bias )sin( bias )cos(ωt Vπ Vπ π π

− 3φ ) +

πA πA πV πV a b 2J4 ( V a ) J5 ( V b )cos( 2bias )sin( bias )cos(ωt Vπ Vπ π π

+ φ) −

sin(ωt − 2φ)

sin(ωt + 2φ)

πA πA πV πV a b 2J3 ( V a ) J4 ( V b )cos( bias )cos( 2bias ) Vπ Vπ π π

sin(ωt + 3φ) −

πA πA πV πV a b − 2J0 ( V a ) J1 ( V b )cos( 2bias )sin( bias )cos(ωt Vπ Vπ π π π ⋅ cos( 2V (Vbias a + Vbias b + 2Vbias c )

+ φ)

πA πA πV πV a b 2J3 ( V a ) J2 ( V b )cos( bias )cos( 2bias ) Vπ Vπ π π

− φ)

+ 5φ)} (A3)

π

1

i2ω = −R 2 E02 J2 (

πV 2πAa a )cos( bias ) Vπ Vπ

1

cos(2ωt ) − 2 RE02 J2 (

πV 2πAb b )cos( bias )cos(2ωt Vπ Vπ

+ 2φ)

+

πA πA πV πV a b )sin( bias )sin(2ωt RE02 {−2J3 ( V b ) J5 ( V a )cos( bias V V

− 3φ ) −

+

πA πA πV πV a b 2J1 ( V a ) J1 ( V b )cos( bias )sin( bias ) Vπ Vπ π π

πA πA πV πV a b 2J1 ( V a ) J3 ( V b )cos( bias )sin( 2bias )sin(2ωt Vπ Vπ π π

+

πA πA πV πV a b 2J2 ( V a ) J4 ( V b )cos( 2bias )cos( 2bias ) Vπ Vπ π π

−

πA πA πV πV a b 2J2 ( V b ) J4 ( V a )cos( 2bias )sin( bias )cos(2ωt Vπ Vπ π π

− 2φ ) −

−

πA πA πV πV a b 2J0 ( V a ) J2 ( V b )cos( 2bias )cos( 2bias )cos(2ωt Vπ Vπ π π

+ 2φ)} cos( 2V (Vbias a + Vbias b + 2Vbias c )

π

1

i3ω = 2 RE02 J3 (

π

π

πV 2πAa a )sin( bias ) Vπ Vπ

π

sin(2ωt + φ ) +

sin(2ωt + 4φ) +

1

cos(3ωt ) + 2 RE02 J3 (

+ +

πA πA πV πV a b 2J1 ( V a ) J4 ( V b )cos( bias )cos( 2bias ) Vπ Vπ π π

π

1

i4ω = 2 RE02 J4 (

πV 2πAa a )cos( bias ) Vπ Vπ

+

1

cos(4ωt ) + 2 RE02 J4 (

−

πA πA πV πV a b 2J1 ( V a ) J3 ( V b )cos( bias )sin( bias ) Vπ Vπ π π

+

πA πA πV πV a b 2J4 ( V a ) J0 ( V b )cos( 2bias )cos( 2bias )cos(4ωt ) Vπ Vπ π π

+

πA πA πV πV a b 2J0 ( V a ) J4 ( V b )cos( 2bias )cos( 2bias )cos(4ωt Vπ Vπ π π

π

π

+ 3φ ) − φ)

+ 2φ )

πA πA πV πV a b 2J2 ( V a ) J1 ( V b )cos( 2bias )sin( bias )cos(3ωt Vπ Vπ π π

+ φ)

πA πA πV πV a b 2J2 ( V a ) J5 ( V b )cos( 2bias )sin( bias )cos(3ωt Vπ Vπ π π

+ 5φ)} (A5)

+

π

(A4)

πA πA πV πV a b 2J4 ( V a ) J1 ( V b )cos( 2bias )sin( bias )sin(3ωt Vπ Vπ π π

πA πA πV πV a b 2J1 ( V a ) J2 ( V b )cos( bias )cos( 2bias )sin(3ωt Vπ Vπ π π

+ 3φ ) +

πA πA πV πV a b )sin( bias ) RE02 {2J5 ( V a ) J1 ( V b )cos( bias V V π

− 2φ) −

sin(3ωt + 4φ) +

πA πA πV πV a b + 2J0 ( V a ) J3 ( V b )cos( 2bias )sin( bias )cos(3ωt Vπ Vπ π π π ⋅ cos( 2V (Vbias a + Vbias b + 2Vbias c ) π

+ 5φ )

πA πA πV πV a b 2J0 ( V b ) J2 ( V a )cos( 2bias )sin( 2bias )cos(2ωt ) Vπ Vπ π π

πV 2πAb b )sin( bias )cos(3ωt Vπ Vπ

πA πA πV πV a b 2J3 ( V a ) J0 ( V b )cos( bias )cos( 2bias )sin(3ωt ) Vπ Vπ π π

π

+ 3φ )

πA πA πV πV a b 2J3 ( V a ) J5 ( V b )cos( bias )sin( bias )sin(2ωt Vπ Vπ π π

π

+

π

− φ)

π

πA πA πV πV a b )cos( 2bias )sin(3ωt RE02 {2J5 ( V a ) J2 ( V b )cos( bias V V π

πA πA πV πV a b 2J1 ( V b ) J3 ( V a )cos( bias )sin( bias )sin(2ωt Vπ Vπ π π

πV 2πAb b )cos( bias )cos(4ωt Vπ Vπ

sin(4ωt − φ) −

sin(4ωt + 3φ) − +

+ 4φ)

πA πA πV πV a b 2J3 ( V a ) J1 ( V b )cos( bias )sin( bias ) Vπ Vπ π π

πA πA πV πV a b 2J2 ( V a ) J2 ( V b )cos( 2bias )cos( 2bias )cos(4ωt Vπ Vπ π π

+ 4φ)}

sin(4ωt + φ)

πA πA πV πV a b 2J1 ( V a ) J5 ( V b )cos( bias )sin( bias )sin(4ωt Vπ Vπ π π

π cos( 2V π

(Vbias a + Vbias b + 2Vbias c )

139

+ 5φ )

+ 2φ) (A6)

Optics Communications 396 (2017) 134–140

G.-F. Bai et al. 1

i5ω = − 2 RE02 J5 (

2πAa πV a )sin( bias ) Vπ Vπ

1

cos(5ωt ) − 2 RE02 J5 (

2πAb πV b )sin( bias )cos(5ωt Vπ Vπ

+

πA πA πV πV a b )cos( 2bias )sin(5ωt ) RE02 {−2J5 ( V a ) J0 ( V b )cos( bias V V

−

πA πA πV πV a b 2J2 ( V a ) J2 ( V b )cos( bias )cos( 2bias )sin(5ωt Vπ Vπ π π

π

π

π

πA πA πV πV a b − 2J2 ( V a ) J3 ( V b )cos( 2bias )sin( bias ) Vπ Vπ π π π ⋅ cos( 2V (Vbias a + Vbias b + 2Vbias c ) π

π

−

+ 5φ )

πA πA πV πV a b 2J4 ( V a ) J1 ( V b )cos( 2bias )sin( bias )sin(5ωt Vπ Vπ π π

+ φ)

+ 2φ ) −

πA πA πV πV a b 2J1 ( V a ) J4 ( V b )cos( bias )cos( 2bias )sin(5ωt Vπ Vπ π π

+ 4φ )

cos(5ωt + 3φ) −

πA πA πV πV a b 2J0 ( V a ) J5 ( V b )cos( 2bias )sin( bias )cos(5ωt Vπ Vπ π π

+ 5φ)} (A7)

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