Abstract

Based on optical frequency comb (OFC), a photonic-assisted ultra-broadband microwave signal down-converting method is proposed. In the proposed scheme, microwave signal at 2~20GHz can be down-converted to 0~1GHz intermediate frequency (IF) signals by an OFC of 2GHz frequency space at different order of comb lines. By slightly switching the frequency space of OFC, the frequency of the signal to be measured can be retrieved through the frequency shift of the down-converted IF signal. The validity of this proposed unknown signal detection method is verified by the experiments. The proposed method is proven to be flexible, low-cost and easily implemented, which requires only a low-frequency tunable microwave source while provides ultra-broadband down-converting frequency range.

© 2015 Optical Society of America

1. Introduction

Microwave photonics provide a new perspective to solve the difficulties that we encounter at electrical domain [1] [2]. Limited by the feature of electrical devices, it is quite hard for electrical receiving systems to measure ultra-broadband and large dynamic microwave signals [35]. Optical frequency comb [6] [7] is a considerably important invention and has been applied in many fields. The most important characteristic of OFC is that it has a series of coherent optical comb lines with the same frequency space. Since each comb line has the characteristic of coherence and low phase noise, they can be utilized as the local oscillator (LO) signals which down-convert microwave signals to IF signals that can be directly sampled by electrical devices such as A/D converters. Due to the broad spectrum band of OFC, the LO signals have can broad bandwidth, which provides the designers of ultra-broadband receiver a new idea.

Many works have been implemented on down-converting high frequency and wideband microwave signals to IF signals in optical link [812]. Since the invention of OFC, many methods based on the OFC have been proposed to down-convert high-frequency microwave signals [1322]. Dual-combs technique and channelization method has been reported [1517,20].

Dual OFCs with different mode spacing can generate LOs with more flexible frequencies than those generated by single OFC. As an example in [20], the heterodyne takes place in separated channels with the periodic optical filter. The channelization technology separates every comb and nearby sideband, so the heterodyning only takes place between one comb line and their sideband which suppressed the sideband beats from other comb lines. Therefore, the channelization technology can restrain cross interference, broaden bandwidth and increase the frequency of receiving signals.

Inspired by dual-combs technology, we realize that the frequency space of OFC restrains the frequency range of IF signal and different order comb lines can beat with different frequency signals in wide possible range. Thus we proposed a new and simple scheme that is available for down-converting ultra-wideband microwave signals. As shown in Fig. 1(a) , the modulated optical signal carrying the information of received microwave signal is coupled with an OFC. The IF signals which represent the down-converted signals is measured by the photodetector, and it is produced by the two sidebands of modulated optical signal beating with the nearest comb lines. From Fig. 1(a), we can see that the frequency range of IF signal is restrained by the frequency spacing of OFC. As the frequencies of received signals increase, the orders of nearest comb lines that beat with modulated optical signal become larger. Thus the spectrum width of OFC determines the frequency range of the down-converting system.

 figure: Fig. 1

Fig. 1 (a) Schematic diagram of photonic system of down-converting microwave signals. (b) Basic scheme of the method that is used to derive out the frequency of received microwave signals:

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After down-converting, the amplitude, phase and frequency of IF signal can be easily obtained. However, the frequency of original signal remains unknown. We hereby propose a simple method that can be used to derive out the frequency of unknown-frequency signals. The basic scheme is shown in Fig. 1(b). Based on the characteristic of optical comb, the frequency between every comb line is uniform. If the frequency space of OFC is slightly switched, the frequency of IF signal also switches accordingly. Based on the phenomenon, the frequency of received microwave signal can be figured out through simple calculation.

2. Theoretical analysis of schematic diagram

As illustrated in Fig. 1, light source is split into two paths via an optical coupler. In one path, light is modulated by received microwave signal through the modulator; in the other path, an OFC is shaped through the OFC generator. The two paths are coupled back together via an optical coupler, and then the combined light is feed into a photodiode where the heterodyne process is conducted.

Here the down-converting progress is analyzed to help to understand the influencing factors in the optical link. We assume the angular frequency of the continuous wave light is ωc, and its optical field can be described as

Ea(t)Pcejωct.
where Pc is the optical intensity.

Then the 3dB optical coupler spilt the light into two paths of equal power. One path light passing though the Mach-Zehnder (MZ) modulator is modulated with carrier suppressed. This can be obtained by MZ modulator with adjusted bias voltage. So the output optical field expression of the MZ modulator can be expressed as

EbPc2ejπvsVπcos(ωst)+c.c.
where vs is the voltage amplitude of received microwave signal, Vπ is the half voltage of the MZ modulator, ωs is the frequency of received signal, and c.c. stands for the complex conjugate of the first term on the right hand side [21].

The other path light is turned into optical comb consisting of N lines on single side-band and writes its optical field as

Ecn=0N1Pnej(ωnt+ϕn)+c.c.
where Pn andϕn are the optical intensity and phase of the nth line respectively. For the OFC, the frequency of nth comb line can be expressed as ωn=ωc+nΔω. Combining the output of the modulator and the OFC generator together, we can write the output electric field of the second optical coupler as

Edn=0N1Pnej(ωnt+ϕn)+Pc2ejπvsVπcos(ωst)+c.c.

The current produced by the total optical field in a square-law photodiode is

i(t)R|Ed(t)|2.

By analyzing photocurrenti(t)we can find that the current has frequencies components of nΔω, nΔω±ωs and 2ωRF, where n[0,N]. We assume frequency of received microwave is n1Δωωs(n1+1)Δω and (ωsn1Δω)((n1+1)Δωωs), where n1[0,N]. To get lower frequency microwave signal, we only need the signal with frequency (ωsn1Δω) which is produced by beating between the n1th and n1th comb lines and the two sidebands.

For simplification, we show the current with frequency ωIF=ωsn1Δω as

iIF2RPn1PcJm(πvsVπ)ej(ωIF+ϕn1)

After the optical-line processing, if the increase of noise floor of received signal is not counted, from Eq. (6), we can find that the signal to noise ratio (SNR) of IF signal is mostly determined by the power of light which is finally feed into photodiode. The power of IF signal is determined by power of CW laser, modulation depth of modulator and responsivity of the photodiode, without considering the loss of optical link.

3. Basic equations calculating the frequency of unknown signals

During the heterodyne process, the received signal beats with almost each comb lines of OFC, however only the beat between its nearest comb line is the desired IF signal. Thus the frequency of IF signal is in the range of 0 to Δω/2, where Δω is frequency space of OFC, and can be written as

ωIF[0,Δω2]

If the frequency space is slightly switched to Δω+Δω' where Δω'Δω, the frequency of IF signal is accordingly switched. Thus two IF signals of different frequencies are produced. Depending on the two IF signals, frequency of received signal ωs can be calculated as

ωs=n1Δω+ωIF.
where n1 is the order of comb line which is the nearest to received signal.

When switching the frequency space of OFC, the received signal beat with different comb lines. Considering all the possible beat frequency situations, four cases can be summarized, as illustrated in Fig. 2 .

 figure: Fig. 2

Fig. 2 Schematic diagram of four down-converting cases.

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As shown in Fig. 2(a), the first case is that the received signal beats with the nth line. Increasing the frequency space, accordingly, ωIF1 switches to ωIF2, where ωIF1>ωIF2. Fig. 2(b) shows the second case that received signal beats with the (n+1)th line and the two IF signals are ωIF1<ωIF2. Analyzing the relationship between ωIF and ωs, we can derive out ωs as

ωs=sign(ωIF1ωIF2)(ωIF1+ωIF1ωIF2Δω'Δω)

The third case is shown in Fig. 2(c). First ωIF1 is generated. Then increasing the frequency space, ωIF2 is the result of beating between received signal and the nth comb line of OFC2. We can easily derive out ωs as

ωs=ωIF1+ωIF1+ωIF2Δω'Δω

As show in Fig. 2(d), the fourth case is that the received signal beats with the (n+1) th line. Switching the frequency spacing, we get ωIF2 that represents beating between received signal and nth line. ωs is derived out as

ωs=Δω(ωIF1+ωIF2)+Δω'Δω'ΔωωIF1

4. Experiment results and analyses

We demonstrate in the experimental setup as shown in Fig. 3 . A CW light centered at f = 1550nm with a power of 16dBm is divided into two branches by an optical coupler. The light in the lower branch is modulated to generate the optical comb, the frequency space of which is determined by the microwave driving signal. An intensity modulator (IM) and two cascaded phase modulators (PM) are used to generate a frequency comb [23]. The microwave driving signal is spilt to two paths. One of the two paths signal is further spilt into two paths. Then the phase of each path signal is shifted and the amplitude is amplified. The light in the higher branch is modulated by the received signal which is amplified by microwave amplifier.

 figure: Fig. 3

Fig. 3 Experimental setup of the photonics down-converting system. CW: continuous laswer, IM: intensity modulator, PM: phase modulator, PS: microwave phase shifter, PA: microwave power amplifier, PD: photo detector, EF: low-pass electrical filter

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In our experiment, a sinusoidal wave with frequency f = 2GHz is chosen as driving signal to generate frequency comb. Thus the frequency range of IF signal is 0~1GHz according to Eq. (8). Limited by the resolution of our OSA (optical spectrum analyzer), the OFC with frequency space of 2GHz cannot be observed. Microwave signal (2~20GHz) is down-converted to IF signal, the frequency of which is under 1GHz. For demonstration, we chose different frequencies of microwave signals as 2.5GHz, 7.6GHz, 13.6GHz and 19.5GHz respectively. The resulted OFC that combines with modulated optical signal is then heterodyne detected by a 3GHz photodiode. Then microwave signal is filtered out by a low-pass electrical filter at 0~1GHz. The frequency spectrums of the four IF signals for the corresponding microwave signals are measured by the ESA (electrical spectrum analyzer) and are demonstrated in Fig. 4(a)-(d) respectively.

 figure: Fig. 4

Fig. 4 Measured IF signals that different received RF signals down-convert to. (a) fre=2.5GHz, (b) fre=7.6GHz, (c) fre=13.6GHz, (d) fre=19.5GHz

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From the above measurement results, the high-frequency microwave signals are down-converted to IF signals with frequencies 500MHz, 400MHz, 400MHz and 500MHz respectively, which completely coincides with theoretical analysis. The SNR of IF signals are 39dB, 38dB, 34dB and 22dB respectively, under a resolution bandwidth (RBW) of 15KHz measurement condition. To analyze the performance of SNR in changing with the received microwave signals, we plot the curve of the SNRs of the down-converted IF signals in changing with the received microwave signals (2GHz ~20GHz) in Fig. 5 .

 figure: Fig. 5

Fig. 5 Measured SNRs of IF signals that received microwave signals downconvert to, and the frequency range of received RF signals is 2~20GHz.

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In our experiment, the floor noise of the optical link is around −100dbm. We observe the fact that, when the received unknown microwave signal is introduced into the optical link, the floor noise of the system remains unchanged, or the change is unnoticeable. Limited to the noise level of our ESA, the noise floor of the optical link might be covered by the noise floor of the analyzer. However, the floor noise of the system is no more than −100dBm which can be accepted by many applications. Therefore, according to the mathematical analysis of section 2.1, the SNRs of IF signals depend on optical power of each comb line and sidebands of modulated optical signal under the condition of same link noise. As the frequency of received signal increases, it can be seen from Fig. 5 that SNRs of IF signals decreases. This phenomenon can be explained that the high order comb lines of OFC have low optical power. Therefore we can enlarge the power of high order comb line of OFC through implementing a modulation index that is large enough.

The method calculating the frequency of unknown signals is slightly switching the frequency space of OFC. In the experiment, the switching time from one stable OFC to another is much less than the post-processing time for calculating the frequency, which does not affect the process of instantaneous signal down-conversion. In order to verify the algorithm that is proposed to calculate the frequency of received signal, the frequency of received microwave signals are set as 14.7GHz, 15.6GHz, 14.05GHz and 15.05GHz respectively. Firstly, setting frequency space of OFC as 2GHz and beating between OFC and modulated optical signal, high-frequency microwave signals are down-converted to IF signals and the frequencies become to 0.7GHz, 0.4GHz, 0.05GHz and 0.95GHz. Then FS (frequency space) of OFC is altered to 2.01GHz, frequencies of IF signals turn into 0.63GHz, 0.48GHz, 0.02GHz and 0.98GHz. The results are plot in Fig. 6 .

 figure: Fig. 6

Fig. 6 Measured IF signals that verify the rightness of frequency-calculating method. (a)fre=14.7GHz, FS of OFC1 = 2GHz, FS of OFC2 = 2.01GHz. (b)fre=15.6GHz, FS of OFC1 = 2GHz, FS of OFC2 = 2.01GHz, (c)fre=14.05GHz, FS of OFC1 = 2GHz, FS of OFC2 = 2.01GHz, (d)fre=15.05GHz, FS of OFC1 = 2GHz, FS of OFC2 = 2.01GHz.

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Thus plugging ωIF1=0.7GHz and ωIF2=0.63GHz into Eq. (9), we can easily get ωs=14.7GHz. Similarly, plugging the other three results into Eq. (9), Eq. (10) and Eq. (11) respectively, we get ωs=15.6GHz,14.05GHzand15.05GHz. These experimental results show that our theoretical analysis is sound and the proposed method is quite effective in calculating frequency of unknown microwave signals.

5. Conclusion

In this work, we propose an OFC-assisted microwave photonics system that is available for down-converting unknown-frequency ultra-wideband microwave signals, and also propose a method that is effective for deriving out the frequency of received microwave signals. The scheme of the system is simple, low-cost and easily implemented. And the miniaturization and productivity of system is convenient. Through experiment, we successfully down-covert 2~20GHz microwave signals to 0~1GHz IF signals. Validity of the method used to calculate the frequency of unknown signal is verified in the experiment. The system has excellent potential applications such as microwave reconnaissance receiver. And it provides great flexibility for microwave receiver designing, such as reducing the frequency space of OFC to bring down the frequency of IF signal, increasing the comb line to spread the receiving bandwidth.

Acknowledgments

This work was supported by National 973 Program (2012CB345600).

References and links

1. J. P. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009). [CrossRef]  

2. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007). [CrossRef]  

3. P. H. Gruchala and M. Czyzewski, “The instantaneous frequency measurement receiver in the complex electromagnetic environment,” in Proc. Int. Conf. Microwave, Radar, and Wireless Communications (MIKON2004), May 2004, 1, 155–158. [CrossRef]  

4. L. Po, Z. Yue, and C. Z. Ping, “Design and implementation of a wideband digital reconnaissance receiver, ” in Proc. Int. Conf. Radar (IET2009), April 2009, pp. 1–5.

5. G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht II, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005). [CrossRef]  

6. J. Ye and S. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005).

7. S. Schiller, “Spectrometry with frequency combs,” Opt. Lett. 27(9), 766–768 (2002). [CrossRef]   [PubMed]  

8. X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012). [CrossRef]  

9. G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995). [CrossRef]  

10. A. C. Lindsay, G. A. Knight, and S. T. Winnall, “Photonic mixers for wide bandwidth RF receiver application,” IEEE Trans. Microw. Theory Tech. 43(9), 2311–2317 (1995). [CrossRef]  

11. C. K. Sun, R. J. Orazi, S. A. Pappert, and W. K. Bums, “A photonic-link millimeter-wave mixer using cascaded optical modulators and harmonic carrier generation,” IEEE Photon. Technol. Lett. 8(9), 1166–1168 (1996). [CrossRef]  

12. A. Karim and J. Devenport, “High dynamic range microwave photonic links for RF signal transport and RF-IF conversion,” J. Lightwave Technol. 26(15), 2718–2724 (2008). [CrossRef]  

13. V. R. Pagán, B. M. Haas, and T. E. Murphy, “Linearized electrooptic microwave downconversion using phase modulation and optical filtering,” Opt. Express 19(2), 883–895 (2011). [CrossRef]   [PubMed]  

14. A. K. M. Lam, M. Fairburn, and N. A. F. Jaeger, “Wide-band electrooptic intensity modulator frequency response measurement using an optical heterodyne down-conversion technique,” IEEE Trans. Microw. Theory Tech. 54(1), 240–246 (2006). [CrossRef]  

15. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008). [CrossRef]   [PubMed]  

16. F. Ferdous, D. E. Leaird, C. B. Huang, and A. M. Weiner, “Dual-comb electric-field cross-correlation technique for optical arbitrary waveform characterization,” Opt. Lett. 34(24), 3875–3877 (2009). [PubMed]  

17. J. D. Rodriguez, M. Bagnell, C. Williams, and P. J. Delfyett, “Multiheterodyne detection for spectral compression and downconversion of arbitrary periodic optical signals,” J. Lightwave Technol. 29(20), 3091–3098 (2011). [CrossRef]  

18. Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012). [CrossRef]  

19. A. O. Wiberg, L. Liu, Z. Tong, E. Myslivets, V. Ataie, B. P. Kuo, N. Alic, and S. Radic, “Photonic preprocessor for analog-to-digital-converter using a cavity-less pulse source,” Opt. Express 20(26), B419–B427 (2012). [CrossRef]   [PubMed]  

20. X. Yang, K. Xu, J. Yin, Y. Dai, F. Yin, J. Li, H. Lu, T. Liu, and Y. Ji, “Optical frequency comb based multi-band microwave frequency conversion for satellite applications,” Opt. Express 22(1), 869–877 (2014). [CrossRef]   [PubMed]  

21. E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic filters based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010). [CrossRef]  

22. C. B. Huang, S. G. Park, D. E. Leaird, and A. M. Weiner, “Nonlinearly broadened phase-modulated continuous-wave laser frequency combs characterized using DPSK decoding,” Opt. Express 16(4), 2520–2527 (2008). [CrossRef]   [PubMed]  

References

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  1. J. P. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
    [Crossref]
  2. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
    [Crossref]
  3. P. H. Gruchala and M. Czyzewski, “The instantaneous frequency measurement receiver in the complex electromagnetic environment,” in Proc. Int. Conf. Microwave, Radar, and Wireless Communications (MIKON2004), May 2004, 1, 155–158.
    [Crossref]
  4. L. Po, Z. Yue, and C. Z. Ping, “Design and implementation of a wideband digital reconnaissance receiver, ” in Proc. Int. Conf. Radar (IET2009), April 2009, pp. 1–5.
  5. G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
    [Crossref]
  6. J. Ye and S. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005).
  7. S. Schiller, “Spectrometry with frequency combs,” Opt. Lett. 27(9), 766–768 (2002).
    [Crossref] [PubMed]
  8. X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
    [Crossref]
  9. G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995).
    [Crossref]
  10. A. C. Lindsay, G. A. Knight, and S. T. Winnall, “Photonic mixers for wide bandwidth RF receiver application,” IEEE Trans. Microw. Theory Tech. 43(9), 2311–2317 (1995).
    [Crossref]
  11. C. K. Sun, R. J. Orazi, S. A. Pappert, and W. K. Bums, “A photonic-link millimeter-wave mixer using cascaded optical modulators and harmonic carrier generation,” IEEE Photon. Technol. Lett. 8(9), 1166–1168 (1996).
    [Crossref]
  12. A. Karim and J. Devenport, “High dynamic range microwave photonic links for RF signal transport and RF-IF conversion,” J. Lightwave Technol. 26(15), 2718–2724 (2008).
    [Crossref]
  13. V. R. Pagán, B. M. Haas, and T. E. Murphy, “Linearized electrooptic microwave downconversion using phase modulation and optical filtering,” Opt. Express 19(2), 883–895 (2011).
    [Crossref] [PubMed]
  14. A. K. M. Lam, M. Fairburn, and N. A. F. Jaeger, “Wide-band electrooptic intensity modulator frequency response measurement using an optical heterodyne down-conversion technique,” IEEE Trans. Microw. Theory Tech. 54(1), 240–246 (2006).
    [Crossref]
  15. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
    [Crossref] [PubMed]
  16. F. Ferdous, D. E. Leaird, C. B. Huang, and A. M. Weiner, “Dual-comb electric-field cross-correlation technique for optical arbitrary waveform characterization,” Opt. Lett. 34(24), 3875–3877 (2009).
    [PubMed]
  17. J. D. Rodriguez, M. Bagnell, C. Williams, and P. J. Delfyett, “Multiheterodyne detection for spectral compression and downconversion of arbitrary periodic optical signals,” J. Lightwave Technol. 29(20), 3091–3098 (2011).
    [Crossref]
  18. Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
    [Crossref]
  19. A. O. Wiberg, L. Liu, Z. Tong, E. Myslivets, V. Ataie, B. P. Kuo, N. Alic, and S. Radic, “Photonic preprocessor for analog-to-digital-converter using a cavity-less pulse source,” Opt. Express 20(26), B419–B427 (2012).
    [Crossref] [PubMed]
  20. X. Yang, K. Xu, J. Yin, Y. Dai, F. Yin, J. Li, H. Lu, T. Liu, and Y. Ji, “Optical frequency comb based multi-band microwave frequency conversion for satellite applications,” Opt. Express 22(1), 869–877 (2014).
    [Crossref] [PubMed]
  21. E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic filters based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010).
    [Crossref]
  22. C. B. Huang, S. G. Park, D. E. Leaird, and A. M. Weiner, “Nonlinearly broadened phase-modulated continuous-wave laser frequency combs characterized using DPSK decoding,” Opt. Express 16(4), 2520–2527 (2008).
    [Crossref] [PubMed]

2014 (1)

2012 (3)

Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

A. O. Wiberg, L. Liu, Z. Tong, E. Myslivets, V. Ataie, B. P. Kuo, N. Alic, and S. Radic, “Photonic preprocessor for analog-to-digital-converter using a cavity-less pulse source,” Opt. Express 20(26), B419–B427 (2012).
[Crossref] [PubMed]

X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
[Crossref]

2011 (2)

2010 (1)

E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic filters based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010).
[Crossref]

2009 (2)

2008 (3)

2007 (1)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

2006 (1)

A. K. M. Lam, M. Fairburn, and N. A. F. Jaeger, “Wide-band electrooptic intensity modulator frequency response measurement using an optical heterodyne down-conversion technique,” IEEE Trans. Microw. Theory Tech. 54(1), 240–246 (2006).
[Crossref]

2005 (1)

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

2002 (1)

1996 (1)

C. K. Sun, R. J. Orazi, S. A. Pappert, and W. K. Bums, “A photonic-link millimeter-wave mixer using cascaded optical modulators and harmonic carrier generation,” IEEE Photon. Technol. Lett. 8(9), 1166–1168 (1996).
[Crossref]

1995 (2)

G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995).
[Crossref]

A. C. Lindsay, G. A. Knight, and S. T. Winnall, “Photonic mixers for wide bandwidth RF receiver application,” IEEE Trans. Microw. Theory Tech. 43(9), 2311–2317 (1995).
[Crossref]

Alic, N.

Alter, J. J.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Ataie, V.

Bagnell, M.

Bums, W. K.

C. K. Sun, R. J. Orazi, S. A. Pappert, and W. K. Bums, “A photonic-link millimeter-wave mixer using cascaded optical modulators and harmonic carrier generation,” IEEE Photon. Technol. Lett. 8(9), 1166–1168 (1996).
[Crossref]

Burns, W. K.

G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995).
[Crossref]

Capmany, J.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Coddington, I.

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
[Crossref] [PubMed]

Crnkovich, J. G.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Dai, Y.

Dai, Y. T.

X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
[Crossref]

de Graaf, J. J. W.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Delfyett, P. J.

Deng, L.

Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Devenport, J.

Esman, R. D.

G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995).
[Crossref]

Evins, J. B.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Fairburn, M.

A. K. M. Lam, M. Fairburn, and N. A. F. Jaeger, “Wide-band electrooptic intensity modulator frequency response measurement using an optical heterodyne down-conversion technique,” IEEE Trans. Microw. Theory Tech. 54(1), 240–246 (2006).
[Crossref]

Ferdous, F.

Gopalakrishnan, G. K.

G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995).
[Crossref]

Haas, B. M.

Habicht, W.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Hagewood, S. M.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Hamidi, E.

E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic filters based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010).
[Crossref]

Hilterbrick, C. L.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Howerton, M. M.

G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995).
[Crossref]

Hrin, G. P.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Huang, C. B.

Jaeger, N. A. F.

A. K. M. Lam, M. Fairburn, and N. A. F. Jaeger, “Wide-band electrooptic intensity modulator frequency response measurement using an optical heterodyne down-conversion technique,” IEEE Trans. Microw. Theory Tech. 54(1), 240–246 (2006).
[Crossref]

Ji, Y.

Karim, A.

Knight, G. A.

A. C. Lindsay, G. A. Knight, and S. T. Winnall, “Photonic mixers for wide bandwidth RF receiver application,” IEEE Trans. Microw. Theory Tech. 43(9), 2311–2317 (1995).
[Crossref]

Kuo, B. P.

Lam, A. K. M.

A. K. M. Lam, M. Fairburn, and N. A. F. Jaeger, “Wide-band electrooptic intensity modulator frequency response measurement using an optical heterodyne down-conversion technique,” IEEE Trans. Microw. Theory Tech. 54(1), 240–246 (2006).
[Crossref]

Leaird, D. E.

Lessin, S. A.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Li, J.

Lin, J. T.

X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
[Crossref]

Lindsay, A. C.

A. C. Lindsay, G. A. Knight, and S. T. Winnall, “Photonic mixers for wide bandwidth RF receiver application,” IEEE Trans. Microw. Theory Tech. 43(9), 2311–2317 (1995).
[Crossref]

Liu, L.

Liu, T.

Lu, H.

Moeller, R. P.

G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995).
[Crossref]

Monroy, I. T.

Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Murphy, T. E.

Myslivets, E.

Newbury, N. R.

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
[Crossref] [PubMed]

Niu, J.

X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
[Crossref]

Novak, D.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Orazi, R. J.

C. K. Sun, R. J. Orazi, S. A. Pappert, and W. K. Bums, “A photonic-link millimeter-wave mixer using cascaded optical modulators and harmonic carrier generation,” IEEE Photon. Technol. Lett. 8(9), 1166–1168 (1996).
[Crossref]

Pagán, V. R.

Pang, X. D.

Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Pappert, S. A.

C. K. Sun, R. J. Orazi, S. A. Pappert, and W. K. Bums, “A photonic-link millimeter-wave mixer using cascaded optical modulators and harmonic carrier generation,” IEEE Photon. Technol. Lett. 8(9), 1166–1168 (1996).
[Crossref]

Park, S. G.

Radic, S.

Rodriguez, J. D.

Schiller, S.

Sun, C. K.

C. K. Sun, R. J. Orazi, S. A. Pappert, and W. K. Bums, “A photonic-link millimeter-wave mixer using cascaded optical modulators and harmonic carrier generation,” IEEE Photon. Technol. Lett. 8(9), 1166–1168 (1996).
[Crossref]

Swann, W. C.

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
[Crossref] [PubMed]

Tavik, G. C.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Tong, Z.

Wang, R. X.

X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
[Crossref]

Weiner, A. M.

Wiberg, A. O.

Williams, C.

Williams, K. J.

G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995).
[Crossref]

Winnall, S. T.

A. C. Lindsay, G. A. Knight, and S. T. Winnall, “Photonic mixers for wide bandwidth RF receiver application,” IEEE Trans. Microw. Theory Tech. 43(9), 2311–2317 (1995).
[Crossref]

Wu, D. C.

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

Xie, X. J.

X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
[Crossref]

Xu, K.

X. Yang, K. Xu, J. Yin, Y. Dai, F. Yin, J. Li, H. Lu, T. Liu, and Y. Ji, “Optical frequency comb based multi-band microwave frequency conversion for satellite applications,” Opt. Express 22(1), 869–877 (2014).
[Crossref] [PubMed]

X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
[Crossref]

Yan, L.

X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
[Crossref]

Yang, X.

Yao, J. P.

Yin, F.

Yin, J.

Yu, X. B.

Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Zhao, Y. G.

Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

Zheng, X. P.

Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

IEEE Photon. Technol. Lett. (2)

C. K. Sun, R. J. Orazi, S. A. Pappert, and W. K. Bums, “A photonic-link millimeter-wave mixer using cascaded optical modulators and harmonic carrier generation,” IEEE Photon. Technol. Lett. 8(9), 1166–1168 (1996).
[Crossref]

Y. G. Zhao, X. D. Pang, L. Deng, X. B. Yu, X. P. Zheng, and I. T. Monroy, “Ultra-broadband photonic harmonic mixer based on optical comb generation,” IEEE Photon. Technol. Lett. 24(7), 620 (2012).
[Crossref]

IEEE Photonics Journal (1)

X. J. Xie, Y. T. Dai, K. Xu, J. Niu, R. X. Wang, L. Yan, and J. T. Lin, “Broadband photonic RF channelization based on coherent optical frequency combs and I/Q demodulators,” IEEE Photonics Journal 4(4), 1196–1202 (2012).
[Crossref]

IEEE Trans. Microw. Theory Tech. (5)

G. K. Gopalakrishnan, R. P. Moeller, M. M. Howerton, W. K. Burns, K. J. Williams, and R. D. Esman, “A low-loss downconverting analog fiber-optic link,” IEEE Trans. Microw. Theory Tech. 43(9), 2318–2323 (1995).
[Crossref]

A. C. Lindsay, G. A. Knight, and S. T. Winnall, “Photonic mixers for wide bandwidth RF receiver application,” IEEE Trans. Microw. Theory Tech. 43(9), 2311–2317 (1995).
[Crossref]

G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech. 53(3), 1009–1020 (2005).
[Crossref]

A. K. M. Lam, M. Fairburn, and N. A. F. Jaeger, “Wide-band electrooptic intensity modulator frequency response measurement using an optical heterodyne down-conversion technique,” IEEE Trans. Microw. Theory Tech. 54(1), 240–246 (2006).
[Crossref]

E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic filters based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010).
[Crossref]

J. Lightwave Technol. (3)

Nat. Photonics (1)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008).
[Crossref] [PubMed]

Other (3)

P. H. Gruchala and M. Czyzewski, “The instantaneous frequency measurement receiver in the complex electromagnetic environment,” in Proc. Int. Conf. Microwave, Radar, and Wireless Communications (MIKON2004), May 2004, 1, 155–158.
[Crossref]

L. Po, Z. Yue, and C. Z. Ping, “Design and implementation of a wideband digital reconnaissance receiver, ” in Proc. Int. Conf. Radar (IET2009), April 2009, pp. 1–5.

J. Ye and S. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005).

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic diagram of photonic system of down-converting microwave signals. (b) Basic scheme of the method that is used to derive out the frequency of received microwave signals:
Fig. 2
Fig. 2 Schematic diagram of four down-converting cases.
Fig. 3
Fig. 3 Experimental setup of the photonics down-converting system. CW: continuous laswer, IM: intensity modulator, PM: phase modulator, PS: microwave phase shifter, PA: microwave power amplifier, PD: photo detector, EF: low-pass electrical filter
Fig. 4
Fig. 4 Measured IF signals that different received RF signals down-convert to. (a) f r e = 2.5 G H z , (b) f r e = 7.6 G H z , (c) f r e = 13.6 G H z , (d) f r e = 19.5 G H z
Fig. 5
Fig. 5 Measured SNRs of IF signals that received microwave signals downconvert to, and the frequency range of received RF signals is 2~20GHz.
Fig. 6
Fig. 6 Measured IF signals that verify the rightness of frequency-calculating method. (a) f r e = 14.7 G H z , FS of OFC1 = 2 G H z , FS of OFC2 = 2.01 G H z . (b) f r e = 15.6 G H z , FS of OFC1 = 2 G H z , FS of OFC2 = 2.01 G H z , (c) f r e = 14.05 G H z , FS of OFC1 = 2 G H z , FS of OFC2 = 2.01 G H z , (d) f r e = 15.05 G H z , FS of OFC1 = 2 G H z , FS of OFC2 = 2.01 G H z .

Equations (11)

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

E a ( t ) P c e j ω c t .
E b P c 2 e j π v s V π cos ( ω s t ) + c . c .
E c n = 0 N 1 P n e j ( ω n t + ϕ n ) + c . c .
E d n = 0 N 1 P n e j ( ω n t + ϕ n ) + P c 2 e j π v s V π cos ( ω s t ) + c . c .
i ( t ) R | E d ( t ) | 2 .
i I F 2 R P n 1 P c J m ( π v s V π ) e j ( ω I F + ϕ n 1 )
ω I F [ 0 , Δ ω 2 ]
ω s = n 1 Δ ω + ω I F .
ω s = s i g n ( ω I F 1 ω I F 2 ) ( ω I F 1 + ω I F 1 ω I F 2 Δ ω ' Δ ω )
ω s = ω I F 1 + ω I F 1 + ω I F 2 Δ ω ' Δ ω
ω s = Δ ω ( ω I F 1 + ω I F 2 ) + Δ ω ' Δ ω ' Δ ω ω I F 1

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