Abstract

We propose and demonstrate a frequency multiplying optoelectronic oscillator with nonlinearly-coupled double loops based on two cascaded Mach–Zehnder modulators, to generate high frequency microwave signals using only low-frequency devices. We find the final oscillation modes are only determined by the length of the master oscillation loop. Frequency multiplying signals are generated via nonlinearly-coupled double loops, the output of one loop being used to modulate the other. In the experiments, microwave signals at 10 GHz with −121 dBc/Hz phase noise at 10 kHz offset and 20 GHz with −112.8 dBc/Hz phase noise at 10 kHz offset are generated. Meanwhile, their side-mode suppression ratios are also evaluated and the maximum ratio of 70 dB is obtained.

©2013 Optical Society of America

1. Introduction

Optoelectronic oscillators (OEOs) with pure emitted microwave signals has attracted great attention recently due to its numerous potential applications and advantages in wireless communication system, radar systems, modern electronic warfare [1,2], optical signal processing [36], high sensitivity sensors [7], microwave signals detection [8], etc. Due to the electronic bottleneck of the electrical and electro-optical devices used, the generated frequency of conventional OEOs is limited to only few 10s of GHz [9]. To extend the operational frequency range, some frequency multiplying OEOs (FM-OEOs) concepts have been suggested [1016]. For instance, a frequency-doubling OEO using a double-drive Mach–Zehnder modulator (MZM) and a frequency shifter was proposed in [10], where the MZM was biased at the minimum transmission point to realize double sideband suppressed carrier (DSB-SC) modulation. In order to generate higher frequency band of the microwave signals, a polarization modulator (PolM) aided by two polarizers was applied [11]. Moreover, a frequency-doubling OEO can also be realized using a phase modulator and a phase-shifted fiber Bragg grating (PS-FBG), albeit with the use of an optical notch filter [12]. However, a frequency-doubling OEO without an optical notch filter has been reported in [13], employing a dual parallel Mach Zehnder modulator (DP-MZM). Recently, a frequency-quadrupling OEO for multichannel up conversion was proposed and demonstrated based on two cascaded polarization modulators [14].

In this paper, a FM-OEO with nonlinearly-coupled double loops is established based on two cascaded MZMs. Two MZMs are applied to compose the nonlinearly-coupled double loops. The master loop, Loop1, is designed to be relatively short in order to generate a single-mode microwave signal at the output, which the slave loop, Loop2, is significantly longer in order to enhance the Q of the FM-OEO signal. A theoretical analysis is performed, which is validated with an experiment by adjusting the optical variable delay lines (OVDL) in Loop1 and Loop2. The generation of frequency-multiplying microwave signals at 10GHz and 20GHz are demonstrated. To the best of our knowledge, this is the first FM-OEO scheme without optical filter, dual-parallel Mach-Zehnder modulator or polarization modulator.

2. Operation principle

The implementation of FM-OEO incorporates nonlinearly-coupled double loops using two MZMs as shown in Fig. 1. The output light from first MZM (MZM1) is equally split into two parts via an optical coupler (OC). One part of the output signal is sent through to the second MZM (MZM2) while the other part is directed towards Loop2 through a longer length single mode fibre (SMF) (several hundred meters) and an OVDL, after which it is converted into an electrical signal using a photodetector (PD), PD2, before being fed back into the modulator part of MZM2.

 

Fig. 1 The experimental setup of FM-OEO scheme. (MZM, Mach–Zehnder modulator; OC, optical coupler; EC, electrical coupler; PC, polarization controller; EDFA, Erbium doped fiber amplifier; OVDL, optical variable delay line; PD, photodetector; EA, electrical amplifier; EBPF, electrical bandpass filter; ESA, electrical spectrum analyzer).

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Suppose the electrical input signal V1(t) to MZM1 is a sinusoidal wave with an angular frequency of ωm, an amplitude of VRF1, and a DC bias of Vb1, then

V1(t)=Vb1+VRF1sin(ωmt)
Thus, the output electric field of MZM1 can be expressed by
E1(t)=12E0exp(jω0t)+12E0exp(jω0t)exp[jπVπ1V1(t)]
where E0 and ω0 are the amplitude and the angular frequency of the input electric field, respectively.Vπ1 is the half-wave voltage of MZM1. Then expand the left-hand side of Eq. (2) with Bessel functions:
E1(t)E02{J0(β1)exp(jω0t)+J1(β1)exp[j(ω0+ωm)t]+J1(β1)exp[j(ω0ωm)t]}
where β1=πVRF1Vπ1. J0(β1) and J1(β1) is the zero-order and the first-order Bessel functions of the first kind, respectively. Higher-order harmonic terms are theoretically suppressed due to small signal modulation.

The electrical signal V2 (t), proportional to|E1(t+τ2)|2, is delayed by a time τ2 and fed back into MZM2. According to Eq. (3), V2(t) can be expressed as follows:

V2(t)={R2GA22E02J0(β1)J1(β1)exp[j(ωmt+ωmτ2)]+R2GA24E02J1(β1)J1(β1)exp[j(2ωmt+2ωmτ2)]}
where R2is the photo-responsivity of PD2, and GA2 is the voltage gain of EA2.The output electric field of MZM2 can be written as:
 E2(t)=14E1(t)+14E1(t)exp{jπVπ2Vb2+jπ2Vπ2R2GA2E02J0(β1)J1(β1)exp[j(ωmt+ωmτ2)]+jπ4Vπ2R2GA2E02J1(β1)J1(β1)exp[j(2ωmt+2ωmτ2)]}
where Vπ2 is the half-wave voltage of MZM2. Thus the output of PD1 E3(t)can be simply described as follows:
E3(t)R1E024{[J0(β1)J0(β2)+J0(β1)J0(β3)]exp(jωmt)+[J0(β1)J1(β2)+J0(β1)J1(β3)]exp[jωm(t+τ2)]+[J1(β1)J1(β2)+J1(β1)J1(β3)]exp[j2ωm(t+τ2)]+[J0(β1)J0(β2)+J0(β1)J1(β2)]exp[j3ωm(t+τ2)]+[J0(β1)J0(β3)+J0(β1)J1(β3)]exp[j4ωm(t+τ2)]}
whereβ2=π2Vπ2R2GA2E02J0(β1)J1(β1), β3=π4Vπ2R2GA2E02J1(β1)J1(β1).R1 is the photo-responsivity of PD1.

It is clear from Eq. (6) that there are many harmonic components of ωm, such as frequency-doubling signal and -quadrupling signals. A particular signal can be isolated using a suitable RF filter with a sufficiently narrow bandwidth to block all other harmonic components. Using such filter, Eq. (6) can be re-written as:

E3(t)R1E024{[J0(β1)J0(β2)+J0(β1)J0(β3)]exp(jωmt)+[J0(β1)J1(β2)+J0(β1)J1(β3)]exp[jωm(t+τ2)]}=R1E024{[J0(β1)J0(β2)+J0(β1)J0(β3)]+[J0(β1)J1(β2)+J0(β1)J1(β3)]exp(jωmτ2)}exp(jωmt)
If the total gain in the system is larger than the total loss, once the nonlinearly-couple double loops are closed, the FM-OEO will start to oscillate. The total output of PD1 at any instant time is the summation of all circulating filed in the system. When the FM-OEO oscillates stably, the total output of PD1 can be expressed as:
E3(t)exp[jωm(t+nτ1)]n=0{GA1R1E024[J0(β1)J0(β2)+J0(β1)J0(β3)+J0(β1)J1(β2)exp(jωmτ2)+J0(β1)J1(β3)exp(jωmτ2)]}n=exp(jωmt)n=0[Ge(ωm)]nexp(jωmnτ1)
where GA1is the voltage gain of EA1, n is the number of times the field has circulated around Loop1 and Ge(ωm) is the effective open-loop gain, given by
Ge(ωm)={GA1R1E024[J0(β1)J0(β2)+J0(β1)J0(β3)+J0(β1)J1(β2)exp(jωmτ2)+J0(β1)J1(β3)exp(jωmτ2)]}
After the FM-OEO starts oscillation, the effective open-loop gain Ge(ωm) is a little less than unity, Eq. (8) can be simplified as
E3(t)exp(jωmt)1Ge(ωm)exp(jωmτ1)
The corresponding microwave power P(ωm,t) with stable oscillation is then given:
P(ωm,t)|E3(t)|22R=12R[1+Ge2(ωm)2Ge(ωm)cos(ωmτ1)]
where R is the load impedance of PD1.

Once the FM-OEO oscillates stably, only the oscillation frequency components with the minimum loss and those total loop phase shift being a multiple of 2π can oscillate. Therefore, the oscillation frequency must meet the phase matching condition as:

ωmτ=2kπ,k=0,1,2,3,...,
where the oscillation frequency peaks will locate at

ωm=2kπτ1

Therefore, the relationship between frequency stability and loop delay can be described by:

Δωω=Δff=Δττ
where Δω is the variation of angular frequency, Δfis the variation of frequency and Δτ is the variation of delay.

Obviously, the final oscillation frequencies are determined by the delay of loop1 τ1, but not the delay of loop2 τ2. From Eq. (6), it is apparent that τ2 only influences the initial phase of the oscillation signals, which is starkly different to the principle of the second loop in conventional dual-loop-OEO systems [17], whereby the longer loop presents low spurious peaks due to unwanted interferences. Only those modes that are closest to the filter center frequency and meet both the phase matching conditions of long loop and short loop will oscillate since they have constructive interferences. In other words, the double loops are linearly-coupled and determine the final oscillation modes altogether. However, the double loops are nonlinearly-coupled in our FM-OEO scheme, in which the output of one loop is used to modulate the other. This is quite an asymmetrical configuration because the two loops play different roles to guarantee the stable operation of the system. Loop1 acts as the master oscillation loop and determines the oscillation modes, while Loop2 plays a role as a slave oscillation loop to reduce the detrimental effect of the multiplicative phase noise and dump delay-induced spurious peaks [18].

3. Experiment result and discussion

An experiment based on the scheme shown in Fig. 1 is carried out. The DFB laser has a fixed wavelength of 1550.6 nm and the EDFA has a saturated output optical power of 23 dBm. The MZMs from JDSU have a 3 dB bandwidth of 20 GHz, whose half-wave voltages of 5.7 V. The PDs have a 3 dB bandwidth of 30 GHz with a responsivity of 0.9 A/W. The EAs with a bandwidth of 2~33 GHz and a maximum gain of 48 dB is applied to compensate the loop loss. The EBPF has a 3 dB bandwidth of 30 MHz with the center frequency of 10 GHz. The length of Loop1 is nearly 10 m with a time delay τ1 of 50ns, while the length of Loop2 is nearly 300 m and its delay τ2 is 1.5μs. An OVDL with the maximum delay of 330ps is used to adjust the feedback phase and tune the oscillation mode when needed. An ESA Agilent 8593E with input frequency ranging from 9 kHz to 23 GHz is used to observe the output microwave signals.

When the loop is closed, the single frequency oscillation of OEO is realized, as is shown in Fig. 2. By appending OVDL1 in Loop1, the initial oscillating signal is 10.00383 GHz withτOVDL1=0ps, as is shown in Fig. 2(a). By adjusting OVDL1, the output spectrum of the generated oscillation signal is changed to 10.00360 GHz with τOVDL1=10ps, as is shown in Fig. 2(b). It is apparent that the relationship between frequency stability and the loop delay is in agreement with Eq. (6). However, Fig. 2(c) and Fig. 2(d) show that there is little change in the frequency of the oscillation signal when adjusting OVDL2. This strongly supports the case that the final oscillation modes are solely determined by the delay of Loop1, and not that of Loop2.

 

Fig. 2 Electrical spectrum of the FM-OEO. Span = 30MHz. (a) OVDL1withτOVDL1=0ps; (b) OVDL1 withτOVDL1=10ps; (c) OVDL2 withτOVDL2=0ps; (d) OVDL2 withτOVDL2=10ps.

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To evaluate the side-mode suppression ratio (SMSR), the comparative analysis between FM-OEO and the conventional double-loop-OEO [17] with the same length of loops is made in Fig. 3(a), (b). The SMSR of FM-OEO is nearly 70 dB, which is more than 30 dB higher than that of the conventional double-loop OEO. The side-modes are sufficiently suppressed due to the function of the nonlinearly-coupled double loops.

 

Fig. 3 (a) Electrical spectrum of the FM-OEO. (b) Electrical spectrum of the conventional dual-loop OEO.

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Figure 4 shows the microwave signals at 10 GHz and 20 GHz. Due to the limit on the bandwidth of the employed ESA, only the frequency-doubled microwave signal at 20 GHz can be measured. As is seen in Fig. 4, the spectrum of the fundamental-frequency component is 21 dB higher than that of the frequency-doubled microwave signal. To investigate the spectral quality of the generated microwave signal, the single-sideband (SSB) phase noises of the signals are measured by the frequency discriminator method [19]. Figure 5 shows the results. As a comparison, the phase noise spectrums for different operating conditions are also shown in Fig. 5. The phase noises of four different signals at a 10-kHz offset frequency are −94.6 dBc/Hz,−106.9 dBc/Hz, −121 dBc/Hz and −112.8 dBc/Hz,respectively. Compared with the single-loop OEO, the SSB phase noise is reduced remarkably in the conventional double-loop OEO, as is shown in Fig. 5(a), (b). The FM-OEO is also superior to the conventional double-loop OEO in its less SSB phase noise due to the elimination of multiplicative phase noise, as is shown in Fig. 5(b), (c). The frequency-doubled microwave signal has 8.2 dB phase noise degradation, compared with that of the fundamental-frequency microwave signal. Theoretically, the phase noise of a frequency-doubled signal should have a phase noise degradation of about 10log10226dB.

 

Fig. 4 Electrical spectrum of fundamental-frequency and frequency-doubled microwave signals.

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Fig. 5 Phase noise spectrum of the (a) single-loop OEO measured at 10 GHz, (b) double-loop OEO measured at 10 GHz, (c) FM-OEO measured at 10 GHz and, (d) FM-OEO measured at 20 GHz.

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4. Conclusion

A FM-OEO with nonlinearly-coupled double loops based on two cascaded MZMs is proposed and its mathematical model is demonstrated. In a conventional double-loop OEO, both loops are coupled linearly and play a similar role, while in our system the loops play different roles and the coupling is nonlinear to form the master-slave oscillation loops. The final oscillation modes are only determined by the length of the master loop, which accords with the theoretical analysis. The SMSR of FM-OEO is nearly 70 dB, which is more than 30 dB higher than that of the conventional double-loop OEO. The SSB phase noise of 10 GHz microwave signal is −121 dBc/Hz at a 10-kHz offset frequency. We believe that the proposed FM-OEO can find applications in wireless communication, radars devices, electronic warfare and optical signal processing.

Acknowledgments

We thank the reviewers for their comments that substantially improved this work by the National Basic Research Program of China (973 Program 2012CB315703) and the National Natural Science Foundation of China (No. 61275027).

References and links

1. X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32(7), 1141–1149 (1996). [CrossRef]  

2. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” IEEE J. Opt. Soc. 13(8), 1725–1734 (1996). [CrossRef]  

3. I. Ozdur, D. Mandridis, N. Hoghooghi, and P. J. Delfyett, “Low noise optically tunable opto-electronic oscillator with Fabry–Pérot Etalon,” J. Lightwave Technol. 28(21), 3100–3106 (2010).

4. S. L. Pan and J. P. Yao, “Multichannel optical signal processing in NRZ systems based on a frequency-doubling optoelectronic oscillator,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1460–1468 (2010). [CrossRef]  

5. L. Huo, Y. Dong, C. Lou, and Y. Gao, “Clock extraction using an opto-electronic oscillator from high-speed NRZ signal and NRZ-to-RZ format transformation,” IEEE Photon. Technol. Lett. 15(7), 981–983 (2003). [CrossRef]  

6. Y. C. Chi, P. C. Peng, and G. R. Lin, “Clock-free RZ-BPSK data generation using self-starting optoelectronic oscillator,” J. Lightwave Technol. 29(11), 1702–1707 (2011). [CrossRef]  

7. L. D. Nguyen, K. Nakatani, and B. Journet, “Refractive index measurement by using an optoelectronic oscillator,” IEEE Photon. Technol. Lett. 22(12), 857–859 (2010). [CrossRef]  

8. P. S. Devgan, M. W. Pruessner, V. J. Urick, and K. J. Williams, “Detecting low-power RF signals using a multimode optoelectronic oscillator and integrated optical filter,” IEEE Photon. Technol. Lett. 22(3), 152–154 (2010). [CrossRef]  

9. D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002). [CrossRef]  

10. T. Sakamoto, T. Kawanishi, and M. Izutsu, “Optoelectronic oscillator using push-pull Mach–Zehnder modulator biased at point for optical two-tone signal generation,” in Conf. Lasers Electro-Optics (CLEO2005). (Tokyo, Japan., 2005), pp. 877–879.

11. S. L. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009). [CrossRef]  

12. W. Z. Li and J. P. Yao, “An optically tunable frequency-doubling optoelectronic oscillator incorporating a phase shifted fiber bragg grating based frequency-tunable photonic microwave filter,” Proc. MWP 2011, (Singapore, 2011) pp. 429–432. [CrossRef]  

13. L. X. Wang, N. H. Zhu, W. Li, and J. G. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber bragg grating,” IEEE Photon. Technol. Lett. 23(22), 1688–1690 (2011). [CrossRef]  

14. D. Zhu, S. F. Liu, D. Ben, and S. Pan, “Frequency-Quadrupling optoelectronic oscillator for multichannel upconversion,” IEEE Photon. Technol. Lett. 25(5), 426–429 (2013). [CrossRef]  

15. D. Zhu, S. L. Pan, and S. H. Cai, “High-Performance Photonic Microwave Downconverter Based on a Frequency-Doubling Optoelectronic Oscillator,” J. Lightwave Technol. 30(18), 3036–3042 (2012). [CrossRef]  

16. M. Haji, L. P. Hou, A. E. Kelly, J. Akbar, J. H. Marsh, J. M. Arnold, and C. N. Ironside, “High frequency optoelectronic oscillators based on the optical feedback of semiconductor mode-locked laser diodes,” Opt. Express 20(3), 3268–3274 (2012). [CrossRef]   [PubMed]  

17. D. Eliyahu and L. Maleki, “Low phase noise and spurious level in multi-loop opto-electronic oscillators,” in proceeding of IEEE Conference on International Frequency Control Symposium and PDA Exhibition Jointly with the 17th European Frequency and Time Forum(Tempa, Florida, U.S.A, 2003), pp. 405–410. [CrossRef]  

18. R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48(11), 1415–1423 (2012). [CrossRef]  

19. Hewlett-Packard, “Phase noise characterization of microwave oscillators—frequency discriminator method,” product note 11729C–2 (Hewlett-Packard, Santa Clara, Calif.).

References

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  • |
  • |

  1. X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32(7), 1141–1149 (1996).
    [Crossref]
  2. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” IEEE J. Opt. Soc. 13(8), 1725–1734 (1996).
    [Crossref]
  3. I. Ozdur, D. Mandridis, N. Hoghooghi, and P. J. Delfyett, “Low noise optically tunable opto-electronic oscillator with Fabry–Pérot Etalon,” J. Lightwave Technol. 28(21), 3100–3106 (2010).
  4. S. L. Pan and J. P. Yao, “Multichannel optical signal processing in NRZ systems based on a frequency-doubling optoelectronic oscillator,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1460–1468 (2010).
    [Crossref]
  5. L. Huo, Y. Dong, C. Lou, and Y. Gao, “Clock extraction using an opto-electronic oscillator from high-speed NRZ signal and NRZ-to-RZ format transformation,” IEEE Photon. Technol. Lett. 15(7), 981–983 (2003).
    [Crossref]
  6. Y. C. Chi, P. C. Peng, and G. R. Lin, “Clock-free RZ-BPSK data generation using self-starting optoelectronic oscillator,” J. Lightwave Technol. 29(11), 1702–1707 (2011).
    [Crossref]
  7. L. D. Nguyen, K. Nakatani, and B. Journet, “Refractive index measurement by using an optoelectronic oscillator,” IEEE Photon. Technol. Lett. 22(12), 857–859 (2010).
    [Crossref]
  8. P. S. Devgan, M. W. Pruessner, V. J. Urick, and K. J. Williams, “Detecting low-power RF signals using a multimode optoelectronic oscillator and integrated optical filter,” IEEE Photon. Technol. Lett. 22(3), 152–154 (2010).
    [Crossref]
  9. D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
    [Crossref]
  10. T. Sakamoto, T. Kawanishi, and M. Izutsu, “Optoelectronic oscillator using push-pull Mach–Zehnder modulator biased at point for optical two-tone signal generation,” in Conf. Lasers Electro-Optics (CLEO2005). (Tokyo, Japan., 2005), pp. 877–879.
  11. S. L. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
    [Crossref]
  12. W. Z. Li and J. P. Yao, “An optically tunable frequency-doubling optoelectronic oscillator incorporating a phase shifted fiber bragg grating based frequency-tunable photonic microwave filter,” Proc. MWP 2011, (Singapore, 2011) pp. 429–432.
    [Crossref]
  13. L. X. Wang, N. H. Zhu, W. Li, and J. G. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber bragg grating,” IEEE Photon. Technol. Lett. 23(22), 1688–1690 (2011).
    [Crossref]
  14. D. Zhu, S. F. Liu, D. Ben, and S. Pan, “Frequency-Quadrupling optoelectronic oscillator for multichannel upconversion,” IEEE Photon. Technol. Lett. 25(5), 426–429 (2013).
    [Crossref]
  15. D. Zhu, S. L. Pan, and S. H. Cai, “High-Performance Photonic Microwave Downconverter Based on a Frequency-Doubling Optoelectronic Oscillator,” J. Lightwave Technol. 30(18), 3036–3042 (2012).
    [Crossref]
  16. M. Haji, L. P. Hou, A. E. Kelly, J. Akbar, J. H. Marsh, J. M. Arnold, and C. N. Ironside, “High frequency optoelectronic oscillators based on the optical feedback of semiconductor mode-locked laser diodes,” Opt. Express 20(3), 3268–3274 (2012).
    [Crossref] [PubMed]
  17. D. Eliyahu and L. Maleki, “Low phase noise and spurious level in multi-loop opto-electronic oscillators,” in proceeding of IEEE Conference on International Frequency Control Symposium and PDA Exhibition Jointly with the 17th European Frequency and Time Forum(Tempa, Florida, U.S.A, 2003), pp. 405–410.
    [Crossref]
  18. R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48(11), 1415–1423 (2012).
    [Crossref]
  19. Hewlett-Packard, “Phase noise characterization of microwave oscillators—frequency discriminator method,” product note 11729C–2 (Hewlett-Packard, Santa Clara, Calif.).

2013 (1)

D. Zhu, S. F. Liu, D. Ben, and S. Pan, “Frequency-Quadrupling optoelectronic oscillator for multichannel upconversion,” IEEE Photon. Technol. Lett. 25(5), 426–429 (2013).
[Crossref]

2012 (3)

2011 (2)

L. X. Wang, N. H. Zhu, W. Li, and J. G. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber bragg grating,” IEEE Photon. Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

Y. C. Chi, P. C. Peng, and G. R. Lin, “Clock-free RZ-BPSK data generation using self-starting optoelectronic oscillator,” J. Lightwave Technol. 29(11), 1702–1707 (2011).
[Crossref]

2010 (4)

L. D. Nguyen, K. Nakatani, and B. Journet, “Refractive index measurement by using an optoelectronic oscillator,” IEEE Photon. Technol. Lett. 22(12), 857–859 (2010).
[Crossref]

P. S. Devgan, M. W. Pruessner, V. J. Urick, and K. J. Williams, “Detecting low-power RF signals using a multimode optoelectronic oscillator and integrated optical filter,” IEEE Photon. Technol. Lett. 22(3), 152–154 (2010).
[Crossref]

I. Ozdur, D. Mandridis, N. Hoghooghi, and P. J. Delfyett, “Low noise optically tunable opto-electronic oscillator with Fabry–Pérot Etalon,” J. Lightwave Technol. 28(21), 3100–3106 (2010).

S. L. Pan and J. P. Yao, “Multichannel optical signal processing in NRZ systems based on a frequency-doubling optoelectronic oscillator,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1460–1468 (2010).
[Crossref]

2009 (1)

S. L. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
[Crossref]

2003 (1)

L. Huo, Y. Dong, C. Lou, and Y. Gao, “Clock extraction using an opto-electronic oscillator from high-speed NRZ signal and NRZ-to-RZ format transformation,” IEEE Photon. Technol. Lett. 15(7), 981–983 (2003).
[Crossref]

2002 (1)

D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
[Crossref]

1996 (2)

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32(7), 1141–1149 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” IEEE J. Opt. Soc. 13(8), 1725–1734 (1996).
[Crossref]

Akbar, J.

Arnold, J. M.

Ben, D.

D. Zhu, S. F. Liu, D. Ben, and S. Pan, “Frequency-Quadrupling optoelectronic oscillator for multichannel upconversion,” IEEE Photon. Technol. Lett. 25(5), 426–429 (2013).
[Crossref]

Cai, S. H.

Chang, D.

D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
[Crossref]

Chembo, Y. K.

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48(11), 1415–1423 (2012).
[Crossref]

Chi, Y. C.

Y. C. Chi, P. C. Peng, and G. R. Lin, “Clock-free RZ-BPSK data generation using self-starting optoelectronic oscillator,” J. Lightwave Technol. 29(11), 1702–1707 (2011).
[Crossref]

Colet, P.

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48(11), 1415–1423 (2012).
[Crossref]

Delfyett, P. J.

Devgan, P. S.

P. S. Devgan, M. W. Pruessner, V. J. Urick, and K. J. Williams, “Detecting low-power RF signals using a multimode optoelectronic oscillator and integrated optical filter,” IEEE Photon. Technol. Lett. 22(3), 152–154 (2010).
[Crossref]

Dong, Y.

L. Huo, Y. Dong, C. Lou, and Y. Gao, “Clock extraction using an opto-electronic oscillator from high-speed NRZ signal and NRZ-to-RZ format transformation,” IEEE Photon. Technol. Lett. 15(7), 981–983 (2003).
[Crossref]

Erlig, H.

D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
[Crossref]

Fetterman, H. R.

D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
[Crossref]

Gao, Y.

L. Huo, Y. Dong, C. Lou, and Y. Gao, “Clock extraction using an opto-electronic oscillator from high-speed NRZ signal and NRZ-to-RZ format transformation,” IEEE Photon. Technol. Lett. 15(7), 981–983 (2003).
[Crossref]

Haji, M.

Hoghooghi, N.

Hou, L. P.

Huo, L.

L. Huo, Y. Dong, C. Lou, and Y. Gao, “Clock extraction using an opto-electronic oscillator from high-speed NRZ signal and NRZ-to-RZ format transformation,” IEEE Photon. Technol. Lett. 15(7), 981–983 (2003).
[Crossref]

Ironside, C. N.

Journet, B.

L. D. Nguyen, K. Nakatani, and B. Journet, “Refractive index measurement by using an optoelectronic oscillator,” IEEE Photon. Technol. Lett. 22(12), 857–859 (2010).
[Crossref]

Kelly, A. E.

Larger, L.

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48(11), 1415–1423 (2012).
[Crossref]

Li, W.

L. X. Wang, N. H. Zhu, W. Li, and J. G. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber bragg grating,” IEEE Photon. Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

Li, W. Z.

W. Z. Li and J. P. Yao, “An optically tunable frequency-doubling optoelectronic oscillator incorporating a phase shifted fiber bragg grating based frequency-tunable photonic microwave filter,” Proc. MWP 2011, (Singapore, 2011) pp. 429–432.
[Crossref]

Lin, G. R.

Y. C. Chi, P. C. Peng, and G. R. Lin, “Clock-free RZ-BPSK data generation using self-starting optoelectronic oscillator,” J. Lightwave Technol. 29(11), 1702–1707 (2011).
[Crossref]

Liu, J. G.

L. X. Wang, N. H. Zhu, W. Li, and J. G. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber bragg grating,” IEEE Photon. Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

Liu, S. F.

D. Zhu, S. F. Liu, D. Ben, and S. Pan, “Frequency-Quadrupling optoelectronic oscillator for multichannel upconversion,” IEEE Photon. Technol. Lett. 25(5), 426–429 (2013).
[Crossref]

Lou, C.

L. Huo, Y. Dong, C. Lou, and Y. Gao, “Clock extraction using an opto-electronic oscillator from high-speed NRZ signal and NRZ-to-RZ format transformation,” IEEE Photon. Technol. Lett. 15(7), 981–983 (2003).
[Crossref]

Maleki, L.

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32(7), 1141–1149 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” IEEE J. Opt. Soc. 13(8), 1725–1734 (1996).
[Crossref]

Mandridis, D.

Marsh, J. H.

Nakatani, K.

L. D. Nguyen, K. Nakatani, and B. Journet, “Refractive index measurement by using an optoelectronic oscillator,” IEEE Photon. Technol. Lett. 22(12), 857–859 (2010).
[Crossref]

Nguimdo, R. M.

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48(11), 1415–1423 (2012).
[Crossref]

Nguyen, L. D.

L. D. Nguyen, K. Nakatani, and B. Journet, “Refractive index measurement by using an optoelectronic oscillator,” IEEE Photon. Technol. Lett. 22(12), 857–859 (2010).
[Crossref]

Oh, A. C.

D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
[Crossref]

Ozdur, I.

Pan, S.

D. Zhu, S. F. Liu, D. Ben, and S. Pan, “Frequency-Quadrupling optoelectronic oscillator for multichannel upconversion,” IEEE Photon. Technol. Lett. 25(5), 426–429 (2013).
[Crossref]

Pan, S. L.

D. Zhu, S. L. Pan, and S. H. Cai, “High-Performance Photonic Microwave Downconverter Based on a Frequency-Doubling Optoelectronic Oscillator,” J. Lightwave Technol. 30(18), 3036–3042 (2012).
[Crossref]

S. L. Pan and J. P. Yao, “Multichannel optical signal processing in NRZ systems based on a frequency-doubling optoelectronic oscillator,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1460–1468 (2010).
[Crossref]

S. L. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
[Crossref]

Peng, P. C.

Y. C. Chi, P. C. Peng, and G. R. Lin, “Clock-free RZ-BPSK data generation using self-starting optoelectronic oscillator,” J. Lightwave Technol. 29(11), 1702–1707 (2011).
[Crossref]

Pruessner, M. W.

P. S. Devgan, M. W. Pruessner, V. J. Urick, and K. J. Williams, “Detecting low-power RF signals using a multimode optoelectronic oscillator and integrated optical filter,” IEEE Photon. Technol. Lett. 22(3), 152–154 (2010).
[Crossref]

Steier, W. H.

D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
[Crossref]

Urick, V. J.

P. S. Devgan, M. W. Pruessner, V. J. Urick, and K. J. Williams, “Detecting low-power RF signals using a multimode optoelectronic oscillator and integrated optical filter,” IEEE Photon. Technol. Lett. 22(3), 152–154 (2010).
[Crossref]

Wang, L. X.

L. X. Wang, N. H. Zhu, W. Li, and J. G. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber bragg grating,” IEEE Photon. Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

Williams, K. J.

P. S. Devgan, M. W. Pruessner, V. J. Urick, and K. J. Williams, “Detecting low-power RF signals using a multimode optoelectronic oscillator and integrated optical filter,” IEEE Photon. Technol. Lett. 22(3), 152–154 (2010).
[Crossref]

Yao, J. P.

S. L. Pan and J. P. Yao, “Multichannel optical signal processing in NRZ systems based on a frequency-doubling optoelectronic oscillator,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1460–1468 (2010).
[Crossref]

S. L. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
[Crossref]

W. Z. Li and J. P. Yao, “An optically tunable frequency-doubling optoelectronic oscillator incorporating a phase shifted fiber bragg grating based frequency-tunable photonic microwave filter,” Proc. MWP 2011, (Singapore, 2011) pp. 429–432.
[Crossref]

Yao, X. S.

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32(7), 1141–1149 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” IEEE J. Opt. Soc. 13(8), 1725–1734 (1996).
[Crossref]

Zhang, C.

D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
[Crossref]

Zhang, H.

D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
[Crossref]

Zhu, D.

D. Zhu, S. F. Liu, D. Ben, and S. Pan, “Frequency-Quadrupling optoelectronic oscillator for multichannel upconversion,” IEEE Photon. Technol. Lett. 25(5), 426–429 (2013).
[Crossref]

D. Zhu, S. L. Pan, and S. H. Cai, “High-Performance Photonic Microwave Downconverter Based on a Frequency-Doubling Optoelectronic Oscillator,” J. Lightwave Technol. 30(18), 3036–3042 (2012).
[Crossref]

Zhu, N. H.

L. X. Wang, N. H. Zhu, W. Li, and J. G. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber bragg grating,” IEEE Photon. Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

IEEE J. Opt. Soc. (1)

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” IEEE J. Opt. Soc. 13(8), 1725–1734 (1996).
[Crossref]

IEEE J. Quantum Electron. (2)

R. M. Nguimdo, Y. K. Chembo, P. Colet, and L. Larger, “On the phase noise performance of nonlinear double-loop optoelectronic microwave oscillators,” IEEE J. Quantum Electron. 48(11), 1415–1423 (2012).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32(7), 1141–1149 (1996).
[Crossref]

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

S. L. Pan and J. P. Yao, “Multichannel optical signal processing in NRZ systems based on a frequency-doubling optoelectronic oscillator,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1460–1468 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (7)

L. Huo, Y. Dong, C. Lou, and Y. Gao, “Clock extraction using an opto-electronic oscillator from high-speed NRZ signal and NRZ-to-RZ format transformation,” IEEE Photon. Technol. Lett. 15(7), 981–983 (2003).
[Crossref]

L. D. Nguyen, K. Nakatani, and B. Journet, “Refractive index measurement by using an optoelectronic oscillator,” IEEE Photon. Technol. Lett. 22(12), 857–859 (2010).
[Crossref]

P. S. Devgan, M. W. Pruessner, V. J. Urick, and K. J. Williams, “Detecting low-power RF signals using a multimode optoelectronic oscillator and integrated optical filter,” IEEE Photon. Technol. Lett. 22(3), 152–154 (2010).
[Crossref]

D. Chang, H. R. Fetterman, H. Erlig, H. Zhang, A. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett. 14(2), 191–193 (2002).
[Crossref]

S. L. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
[Crossref]

L. X. Wang, N. H. Zhu, W. Li, and J. G. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber bragg grating,” IEEE Photon. Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

D. Zhu, S. F. Liu, D. Ben, and S. Pan, “Frequency-Quadrupling optoelectronic oscillator for multichannel upconversion,” IEEE Photon. Technol. Lett. 25(5), 426–429 (2013).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (1)

Other (4)

D. Eliyahu and L. Maleki, “Low phase noise and spurious level in multi-loop opto-electronic oscillators,” in proceeding of IEEE Conference on International Frequency Control Symposium and PDA Exhibition Jointly with the 17th European Frequency and Time Forum(Tempa, Florida, U.S.A, 2003), pp. 405–410.
[Crossref]

W. Z. Li and J. P. Yao, “An optically tunable frequency-doubling optoelectronic oscillator incorporating a phase shifted fiber bragg grating based frequency-tunable photonic microwave filter,” Proc. MWP 2011, (Singapore, 2011) pp. 429–432.
[Crossref]

Hewlett-Packard, “Phase noise characterization of microwave oscillators—frequency discriminator method,” product note 11729C–2 (Hewlett-Packard, Santa Clara, Calif.).

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Optoelectronic oscillator using push-pull Mach–Zehnder modulator biased at point for optical two-tone signal generation,” in Conf. Lasers Electro-Optics (CLEO2005). (Tokyo, Japan., 2005), pp. 877–879.

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Figures (5)

Fig. 1
Fig. 1 The experimental setup of FM-OEO scheme. (MZM, Mach–Zehnder modulator; OC, optical coupler; EC, electrical coupler; PC, polarization controller; EDFA, Erbium doped fiber amplifier; OVDL, optical variable delay line; PD, photodetector; EA, electrical amplifier; EBPF, electrical bandpass filter; ESA, electrical spectrum analyzer).
Fig. 2
Fig. 2 Electrical spectrum of the FM-OEO. Span = 30MHz. (a) OVDL1with τ O V D L 1 =0ps ; (b) OVDL1 with τ O V D L 1 =10ps ; (c) OVDL2 with τ O V D L 2 =0ps ; (d) OVDL2 with τ O V D L 2 =10ps .
Fig. 3
Fig. 3 (a) Electrical spectrum of the FM-OEO. (b) Electrical spectrum of the conventional dual-loop OEO.
Fig. 4
Fig. 4 Electrical spectrum of fundamental-frequency and frequency-doubled microwave signals.
Fig. 5
Fig. 5 Phase noise spectrum of the (a) single-loop OEO measured at 10 GHz, (b) double-loop OEO measured at 10 GHz, (c) FM-OEO measured at 10 GHz and, (d) FM-OEO measured at 20 GHz.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

V 1 (t)= V b1 + V RF1 sin( ω m t )
E 1 ( t )= 1 2 E 0 exp( j ω 0 t )+ 1 2 E 0 exp( j ω 0 t )exp[ j π V π1 V 1 ( t ) ]
E 1 (t) E 0 2 { J 0 ( β 1 )exp(j ω 0 t)+ J 1 ( β 1 )exp[ j( ω 0 + ω m )t ]+ J 1 ( β 1 )exp[ j( ω 0 ω m )t ] }
V 2 ( t )={ R 2 G A2 2 E 0 2 J 0 ( β 1 ) J 1 ( β 1 )exp[ j( ω m t+ ω m τ 2 ) ]+ R 2 G A2 4 E 0 2 J 1 ( β 1 ) J 1 ( β 1 )exp[ j( 2 ω m t+2 ω m τ 2 ) ] }
  E 2 (t)= 1 4 E 1 ( t )+ 1 4 E 1 ( t )exp{ j π V π2 V b2 + j π 2 V π2 R 2 G A2 E 0 2 J 0 ( β 1 ) J 1 ( β 1 )exp[ j( ω m t+ ω m τ 2 ) ]+ j π 4 V π2 R 2 G A2 E 0 2 J 1 ( β 1 ) J 1 ( β 1 )exp[ j( 2 ω m t+2 ω m τ 2 ) ] }
E 3 ( t ) R 1 E 0 2 4 { [ J 0 ( β 1 ) J 0 ( β 2 )+ J 0 ( β 1 ) J 0 ( β 3 ) ]exp(j ω m t) +[ J 0 ( β 1 ) J 1 ( β 2 )+ J 0 ( β 1 ) J 1 ( β 3 ) ]exp[ j ω m ( t+ τ 2 ) ] +[ J 1 ( β 1 ) J 1 ( β 2 )+ J 1 ( β 1 ) J 1 ( β 3 ) ]exp[ j2 ω m ( t+ τ 2 ) ] +[ J 0 ( β 1 ) J 0 ( β 2 )+ J 0 ( β 1 ) J 1 ( β 2 ) ]exp[ j3 ω m ( t+ τ 2 ) ] +[ J 0 ( β 1 ) J 0 ( β 3 )+ J 0 ( β 1 ) J 1 ( β 3 ) ]exp[ j4 ω m ( t+ τ 2 ) ] }
E 3 ( t ) R 1 E 0 2 4 { [ J 0 ( β 1 ) J 0 ( β 2 )+ J 0 ( β 1 ) J 0 ( β 3 ) ]exp(j ω m t) +[ J 0 ( β 1 ) J 1 ( β 2 )+ J 0 ( β 1 ) J 1 ( β 3 ) ]exp[ j ω m ( t+ τ 2 ) ] } = R 1 E 0 2 4 { [ J 0 ( β 1 ) J 0 ( β 2 )+ J 0 ( β 1 ) J 0 ( β 3 ) ] +[ J 0 ( β 1 ) J 1 ( β 2 )+ J 0 ( β 1 ) J 1 ( β 3 ) ]exp( j ω m τ 2 ) }exp(j ω m t)
E 3 ( t )exp[ j ω m ( t+n τ 1 ) ] n=0 { G A1 R 1 E 0 2 4 [ J 0 ( β 1 ) J 0 ( β 2 )+ J 0 ( β 1 ) J 0 ( β 3 ) + J 0 ( β 1 ) J 1 ( β 2 )exp( j ω m τ 2 ) + J 0 ( β 1 ) J 1 ( β 3 )exp( j ω m τ 2 ) ] } n =exp(j ω m t) n=0 [ G e ( ω m ) ] n exp( j ω m n τ 1 )
G e ( ω m )={ G A1 R 1 E 0 2 4 [ J 0 ( β 1 ) J 0 ( β 2 )+ J 0 ( β 1 ) J 0 ( β 3 )+ J 0 ( β 1 ) J 1 ( β 2 )exp( j ω m τ 2 )+ J 0 ( β 1 ) J 1 ( β 3 )exp( j ω m τ 2 ) ] }
E 3 ( t ) exp(j ω m t) 1 G e ( ω m )exp( j ω m τ 1 )
P( ω m ,t ) | E 3 ( t ) | 2 2R = 1 2R[ 1+ G e 2 ( ω m )2 G e ( ω m )cos( ω m τ 1 ) ]
ω m τ=2kπ, k=0,1,2,3,...,
ω m = 2kπ τ 1
Δω ω = Δf f = Δτ τ

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