Abstract

Real-time electrical spectrum analysis is of great significance for applications involving radio astronomy and electronic warfare, e.g. the dynamic spectrum monitoring of outer space signal, and the instantaneous capture of frequency from other electronic systems. However, conventional electrical spectrum analyzer (ESA) has limited operation speed and observation bandwidth due to the electronic bottleneck. Therefore, a variety of photonics-assisted methods have been extensively explored due to the bandwidth advantage of the optical domain. Alternatively, we proposed and experimentally demonstrated an ultrafast ESA based on all-optical Fourier transform and temporal magnification in this paper. The radio-frequency (RF) signal under test is temporally multiplexed to the spectrum of an ultrashort pulse, thus the frequency information is converted to the time axis. Moreover, since the bandwidth of this ultrashort pulse is far beyond that of the state-of-the-art photo-detector, a temporal magnification system is applied to stretch the time axis, and capture the RF spectrum with 1-GHz resolution. The observation bandwidth of this ultrafast ESA is over 20 GHz, limited by that of the electro-optic modulator. Since all the signal processing is in the optical domain, the acquisition frame rate can be as high as 50 MHz. This ultrafast ESA scheme can be further improved with better dispersive engineering, and is promising for some ultrafast spectral information acquisition applications.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]

2015 (1)

2013 (9)

H. Chi, Y. Chen, Y. Mei, X. Jin, S. Zheng, and X. Zhang, “Microwave spectrum sensing based on photonic time stretch and compressive sampling,” Opt. Lett. 38(2), 136–138 (2013).
[Crossref] [PubMed]

B. T. Bosworth and M. A. Foster, “High-speed ultrawideband photonically enabled compressed sensing of sparse radio frequency signals,” Opt. Lett. 38(22), 4892–4895 (2013).
[Crossref] [PubMed]

C. Wang and J. P. Yao, “Ultrahigh-resolution photonic-assisted microwave frequency identification based on temporal channelization,” IEEE Trans. Microw. Theory Tech. 61(12), 4275–4282 (2013).
[Crossref]

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

A. A. Adnani, J. Duplicy, and L. Philips, “Spectrum analyzers today and tomorrow: part 1 towards filter banks-enabled real-time spectrum analysis,” IEEE Instrum. Meas. Mag. 16(5), 6–11 (2013).
[Crossref]

D. Marpaung, “On-chip photonic-assisted instantaneous microwave frequency measurement system,” IEEE Photonics Technol. Lett. 25(9), 837–840 (2013).
[Crossref]

C. Zhang, J. Xu, P. C. Chui, and K. K. Y. Wong, “Parametric spectro-temporal analyzer (PASTA) for real-time optical spectrum observation,” Sci. Rep. 3, 2064 (2013).
[PubMed]

R. Salem, M. A. Foster, and A. L. Gaeta, “Application of space-time duality to ultrahigh-speed optical signal processing,” Adv. Opt. Photonics 5(3), 274–317 (2013).
[Crossref]

C. Zhang, P. C. Chui, and K. K. Y. Wong, “Comparison of state-of-art phase modulators and parametric mixers in time-lens applications under different repetition rates,” Appl. Opt. 52(36), 8817–8826 (2013).
[Crossref] [PubMed]

2012 (1)

E. Palushani, H. C. H. Mulvad, M. Galili, H. Hu, L. K. Oxenlowe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18(2), 681–688 (2012).
[Crossref]

2011 (1)

M. Li and J. P. Yao, “All-optical short-time Fourier transform based on a temporal pulse shaping system incorporating an array of cascaded linearly chirped fiber Bragg gratings,” IEEE Photonics Technol. Lett. 23(20), 1439–1441 (2011).
[Crossref]

2009 (3)

2008 (1)

H. Chi, X. Zou, and J. Yao, “An approach to the measurement of microwave frequency based on optical power monitoring,” IEEE Photonics Technol. Lett. 20(14), 1249–1251 (2008).
[Crossref]

2007 (1)

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

2006 (1)

S. T. Winnall, A. C. Lindsay, M. W. Austin, J. Canning, and A. Mitchell, “A microwave channelizer and spectroscope based on an integrated optical Bragg-grating Fabry-Perot and integrated hybrid Fresnel lens system,” IEEE Trans. Microw. Theory Tech. 54(2), 868–872 (2006).
[Crossref]

2004 (1)

2001 (1)

W. Wang, R. Davis, T. Jung, R. Lodenkamper, L. Lembo, J. Brook, and M. Wu, “Characterization of a coherent optical RF channelizer based on a diffraction grating,” IEEE Trans. Microw. Theory Tech. 49(10), 1996–2001 (2001).
[Crossref]

2000 (1)

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part I: System configurations,” IEEE J. Quantum Electron. 36(4), 430–437 (2000).
[Crossref]

1999 (1)

F. Coppinger, A. S. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory 47(7), 1309–1314 (1999).
[Crossref]

1998 (1)

J. M. Heaton, C. D. Watson, S. B. Jones, M. M. Bourke, C. M. Boyne, G. W. Smith, and D. R. Wight, “Sixteen channel (1 to 16 GHz) microwave spectrum analyzer device based on phased-array of GaAs-AlGaAs electrooptic waveguide delay lines,” Proc. SPIE 3278, 245–251 (1998).
[Crossref]

1994 (2)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[Crossref]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
[Crossref]

1989 (1)

Aditya, S.

Adnani, A. A.

A. A. Adnani, J. Duplicy, and L. Philips, “Spectrum analyzers today and tomorrow: part 1 towards filter banks-enabled real-time spectrum analysis,” IEEE Instrum. Meas. Mag. 16(5), 6–11 (2013).
[Crossref]

Austin, M. W.

S. T. Winnall, A. C. Lindsay, M. W. Austin, J. Canning, and A. Mitchell, “A microwave channelizer and spectroscope based on an integrated optical Bragg-grating Fabry-Perot and integrated hybrid Fresnel lens system,” IEEE Trans. Microw. Theory Tech. 54(2), 868–872 (2006).
[Crossref]

Bennett, C. V.

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part I: System configurations,” IEEE J. Quantum Electron. 36(4), 430–437 (2000).
[Crossref]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
[Crossref]

Bhushan, A. S.

F. Coppinger, A. S. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory 47(7), 1309–1314 (1999).
[Crossref]

Bosworth, B. T.

Bourke, M. M.

J. M. Heaton, C. D. Watson, S. B. Jones, M. M. Bourke, C. M. Boyne, G. W. Smith, and D. R. Wight, “Sixteen channel (1 to 16 GHz) microwave spectrum analyzer device based on phased-array of GaAs-AlGaAs electrooptic waveguide delay lines,” Proc. SPIE 3278, 245–251 (1998).
[Crossref]

Boyne, C. M.

J. M. Heaton, C. D. Watson, S. B. Jones, M. M. Bourke, C. M. Boyne, G. W. Smith, and D. R. Wight, “Sixteen channel (1 to 16 GHz) microwave spectrum analyzer device based on phased-array of GaAs-AlGaAs electrooptic waveguide delay lines,” Proc. SPIE 3278, 245–251 (1998).
[Crossref]

Brook, J.

W. Wang, R. Davis, T. Jung, R. Lodenkamper, L. Lembo, J. Brook, and M. Wu, “Characterization of a coherent optical RF channelizer based on a diffraction grating,” IEEE Trans. Microw. Theory Tech. 49(10), 1996–2001 (2001).
[Crossref]

Bui, L. A.

Canning, J.

S. T. Winnall, A. C. Lindsay, M. W. Austin, J. Canning, and A. Mitchell, “A microwave channelizer and spectroscope based on an integrated optical Bragg-grating Fabry-Perot and integrated hybrid Fresnel lens system,” IEEE Trans. Microw. Theory Tech. 54(2), 868–872 (2006).
[Crossref]

Capmany, J.

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

Chen, Y.

Chi, H.

H. Chi, Y. Chen, Y. Mei, X. Jin, S. Zheng, and X. Zhang, “Microwave spectrum sensing based on photonic time stretch and compressive sampling,” Opt. Lett. 38(2), 136–138 (2013).
[Crossref] [PubMed]

H. Chi, X. Zou, and J. Yao, “An approach to the measurement of microwave frequency based on optical power monitoring,” IEEE Photonics Technol. Lett. 20(14), 1249–1251 (2008).
[Crossref]

Chin, S.

Chui, P. C.

C. Zhang, J. Xu, P. C. Chui, and K. K. Y. Wong, “Parametric spectro-temporal analyzer (PASTA) for real-time optical spectrum observation,” Sci. Rep. 3, 2064 (2013).
[PubMed]

C. Zhang, P. C. Chui, and K. K. Y. Wong, “Comparison of state-of-art phase modulators and parametric mixers in time-lens applications under different repetition rates,” Appl. Opt. 52(36), 8817–8826 (2013).
[Crossref] [PubMed]

Clausen, A. T.

E. Palushani, H. C. H. Mulvad, M. Galili, H. Hu, L. K. Oxenlowe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18(2), 681–688 (2012).
[Crossref]

Coppinger, F.

F. Coppinger, A. S. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory 47(7), 1309–1314 (1999).
[Crossref]

Davis, R.

W. Wang, R. Davis, T. Jung, R. Lodenkamper, L. Lembo, J. Brook, and M. Wu, “Characterization of a coherent optical RF channelizer based on a diffraction grating,” IEEE Trans. Microw. Theory Tech. 49(10), 1996–2001 (2001).
[Crossref]

Duplicy, J.

A. A. Adnani, J. Duplicy, and L. Philips, “Spectrum analyzers today and tomorrow: part 1 towards filter banks-enabled real-time spectrum analysis,” IEEE Instrum. Meas. Mag. 16(5), 6–11 (2013).
[Crossref]

Eggleton, B. J.

Emami, H.

Fainman, Y.

Foster, M. A.

Fu, S.

Gaeta, A. L.

R. Salem, M. A. Foster, and A. L. Gaeta, “Application of space-time duality to ultrahigh-speed optical signal processing,” Adv. Opt. Photonics 5(3), 274–317 (2013).
[Crossref]

Galili, M.

E. Palushani, H. C. H. Mulvad, M. Galili, H. Hu, L. K. Oxenlowe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18(2), 681–688 (2012).
[Crossref]

Goda, K.

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

Heaton, J. M.

J. M. Heaton, C. D. Watson, S. B. Jones, M. M. Bourke, C. M. Boyne, G. W. Smith, and D. R. Wight, “Sixteen channel (1 to 16 GHz) microwave spectrum analyzer device based on phased-array of GaAs-AlGaAs electrooptic waveguide delay lines,” Proc. SPIE 3278, 245–251 (1998).
[Crossref]

Hu, H.

E. Palushani, H. C. H. Mulvad, M. Galili, H. Hu, L. K. Oxenlowe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18(2), 681–688 (2012).
[Crossref]

Jalali, B.

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

F. Coppinger, A. S. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory 47(7), 1309–1314 (1999).
[Crossref]

Jeppesen, P.

E. Palushani, H. C. H. Mulvad, M. Galili, H. Hu, L. K. Oxenlowe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18(2), 681–688 (2012).
[Crossref]

Jin, X.

Jones, S. B.

J. M. Heaton, C. D. Watson, S. B. Jones, M. M. Bourke, C. M. Boyne, G. W. Smith, and D. R. Wight, “Sixteen channel (1 to 16 GHz) microwave spectrum analyzer device based on phased-array of GaAs-AlGaAs electrooptic waveguide delay lines,” Proc. SPIE 3278, 245–251 (1998).
[Crossref]

Jung, T.

W. Wang, R. Davis, T. Jung, R. Lodenkamper, L. Lembo, J. Brook, and M. Wu, “Characterization of a coherent optical RF channelizer based on a diffraction grating,” IEEE Trans. Microw. Theory Tech. 49(10), 1996–2001 (2001).
[Crossref]

Kolner, B. H.

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part I: System configurations,” IEEE J. Quantum Electron. 36(4), 430–437 (2000).
[Crossref]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[Crossref]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
[Crossref]

B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14(12), 630–632 (1989).
[Crossref] [PubMed]

Lembo, L.

W. Wang, R. Davis, T. Jung, R. Lodenkamper, L. Lembo, J. Brook, and M. Wu, “Characterization of a coherent optical RF channelizer based on a diffraction grating,” IEEE Trans. Microw. Theory Tech. 49(10), 1996–2001 (2001).
[Crossref]

Li, J.

Li, M.

M. Li and J. P. Yao, “All-optical short-time Fourier transform based on a temporal pulse shaping system incorporating an array of cascaded linearly chirped fiber Bragg gratings,” IEEE Photonics Technol. Lett. 23(20), 1439–1441 (2011).
[Crossref]

Lindsay, A. C.

S. T. Winnall, A. C. Lindsay, M. W. Austin, J. Canning, and A. Mitchell, “A microwave channelizer and spectroscope based on an integrated optical Bragg-grating Fabry-Perot and integrated hybrid Fresnel lens system,” IEEE Trans. Microw. Theory Tech. 54(2), 868–872 (2006).
[Crossref]

Lodenkamper, R.

W. Wang, R. Davis, T. Jung, R. Lodenkamper, L. Lembo, J. Brook, and M. Wu, “Characterization of a coherent optical RF channelizer based on a diffraction grating,” IEEE Trans. Microw. Theory Tech. 49(10), 1996–2001 (2001).
[Crossref]

Marpaung, D.

D. Marpaung, “On-chip photonic-assisted instantaneous microwave frequency measurement system,” IEEE Photonics Technol. Lett. 25(9), 837–840 (2013).
[Crossref]

Mei, Y.

Mitchell, A.

L. A. Bui, M. D. Pelusi, T. D. Vo, N. Sarkhosh, H. Emami, B. J. Eggleton, and A. Mitchell, “Instantaneous frequency measurement system using optical mixing in highly nonlinear fiber,” Opt. Express 17(25), 22983–22991 (2009).
[Crossref] [PubMed]

S. T. Winnall, A. C. Lindsay, M. W. Austin, J. Canning, and A. Mitchell, “A microwave channelizer and spectroscope based on an integrated optical Bragg-grating Fabry-Perot and integrated hybrid Fresnel lens system,” IEEE Trans. Microw. Theory Tech. 54(2), 868–872 (2006).
[Crossref]

Mulvad, H. C. H.

E. Palushani, H. C. H. Mulvad, M. Galili, H. Hu, L. K. Oxenlowe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18(2), 681–688 (2012).
[Crossref]

Nazarathy, M.

Novak, D.

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

Oxenlowe, L. K.

E. Palushani, H. C. H. Mulvad, M. Galili, H. Hu, L. K. Oxenlowe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18(2), 681–688 (2012).
[Crossref]

Palushani, E.

E. Palushani, H. C. H. Mulvad, M. Galili, H. Hu, L. K. Oxenlowe, A. T. Clausen, and P. Jeppesen, “OTDM-to-WDM conversion based on time-to-frequency mapping by time-domain optical fourier transformation,” IEEE J. Sel. Top. Quantum Electron. 18(2), 681–688 (2012).
[Crossref]

Panasenko, D.

Pelusi, M. D.

Philips, L.

A. A. Adnani, J. Duplicy, and L. Philips, “Spectrum analyzers today and tomorrow: part 1 towards filter banks-enabled real-time spectrum analysis,” IEEE Instrum. Meas. Mag. 16(5), 6–11 (2013).
[Crossref]

Salem, R.

R. Salem, M. A. Foster, and A. L. Gaeta, “Application of space-time duality to ultrahigh-speed optical signal processing,” Adv. Opt. Photonics 5(3), 274–317 (2013).
[Crossref]

Saperstein, R. E.

Sarkhosh, N.

Scott, R. P.

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
[Crossref]

Shum, P. P.

Smith, G. W.

J. M. Heaton, C. D. Watson, S. B. Jones, M. M. Bourke, C. M. Boyne, G. W. Smith, and D. R. Wight, “Sixteen channel (1 to 16 GHz) microwave spectrum analyzer device based on phased-array of GaAs-AlGaAs electrooptic waveguide delay lines,” Proc. SPIE 3278, 245–251 (1998).
[Crossref]

Stroud, J. R.

Sun, X.

Tran, D. N.

Tran, T. D.

Vo, T. D.

Wang, C.

C. Wang and J. P. Yao, “Ultrahigh-resolution photonic-assisted microwave frequency identification based on temporal channelization,” IEEE Trans. Microw. Theory Tech. 61(12), 4275–4282 (2013).
[Crossref]

Wang, W.

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[Crossref]

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[Crossref]

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[Crossref]

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S. T. Winnall, A. C. Lindsay, M. W. Austin, J. Canning, and A. Mitchell, “A microwave channelizer and spectroscope based on an integrated optical Bragg-grating Fabry-Perot and integrated hybrid Fresnel lens system,” IEEE Trans. Microw. Theory Tech. 54(2), 868–872 (2006).
[Crossref]

Wong, K. K. Y.

C. Zhang, J. Xu, P. C. Chui, and K. K. Y. Wong, “Parametric spectro-temporal analyzer (PASTA) for real-time optical spectrum observation,” Sci. Rep. 3, 2064 (2013).
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[Crossref]

Xia, L.

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C. Zhang, J. Xu, P. C. Chui, and K. K. Y. Wong, “Parametric spectro-temporal analyzer (PASTA) for real-time optical spectrum observation,” Sci. Rep. 3, 2064 (2013).
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[Crossref]

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Figures (7)
Fig. 1
Fig. 1 Schematic of the proposed ultrafast ESA. (a) The temporal ray diagram of the system. A point light source (pulse source) is diverged by propagating a certain distance, before diffracted by a sinusoidal grating (different density corresponds to different RF frequency), and followed by an opposite propagation. Finally, the diffracted spots are amplified by a magnification system. (b) The temporal counterpart of the system. A time-lens is introduced to implement the temporal magnification system, which helps a microwave photonics based spectrum analyzer achieve high frame rate without degrading the resolution.

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Fig. 2
Fig. 2 Simulation results of the system. (a) Resolution performance under ideal conditions (red), limited time-lens window (blue), high order dispersions (black). (b) Measurement results of chirped frequency with chirp rate changing from 0 to 80 MHz/ns, with 20 MHz/ns spacing.

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Fig. 3
Fig. 3 Experimental setup of the proposed ultrafast ESA. The time-lens of the temporal magnification system is based on the parametric mixing process, and a linear chirped swept pump provides a quadratic phase modulation. To synchronize the pump and the signal of the parametric process, they are filtered from an identical wideband pulse source.

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Fig. 4
Fig. 4 The spectrum and waveform of the pulse source. (a) The intensity spectrum of the pulse source, with 3-dB bandwidth of 35 nm and center wavelength located at 1565nm. (b) The waveform of a single pulse. The pulsewidth is largely broadened due to the bandwidth limitation of the PD (40 GHz) and the oscilloscope (16 GHz).

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Fig. 5
Fig. 5 The normalized temporal waveforms before (a) and after (b) the amplitude modulator. The temporal shape with 13-ns duration resembles its spectrum. The inset shows the zoom-in fringes after the modulator with 10-GHz sinusoidal signal.

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Fig. 6
Fig. 6 (a) The spectra before (blue dash-dotted line) and after (red solid line) FWM process. (b) The real-time acquisition of the RF spectrum with a 10-GHz sinusoidal signal under test, it achieves 50-MHz acquisition frame rate. (c) Single period of (b), with inset exhibited the tested spectrum. (d) Under 1 V drive voltage, the experiment results with different bias voltages (blue: 3.2 V; red: 3.5 V).

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Fig. 7
Fig. 7 The performance of the proposed ultrafast ESA. (a) The response of the frequencies from 2 GHz to 20 GHz in a single observation period. (b) The dynamic range measurement, with the red markers represent the experimental data and the blue dashed line represents a fitted curve. Insets: the output profiles changes with the increasing drive voltage.

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Equations (6)

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

A 2 ( τ )= A 1 ( τ )×cos[ π 2 V π ( V bias +f( τ ) ) ]≈− π 2 V π A 1 ( τ )×f( τ )
A 3 ( τ )= ℑ −1 { − 1 4 V π [ U 0 ( ω ) G 1 ( ω )⊗F( ω ) ] G 2 ( ω ) } = 1 8π V π Φ 0 exp( i τ 2 2 Φ 0 )×[ A 0 ( τ )exp( − i τ 2 2 Φ 0 )⊗F( − τ Φ 0 ) ]
I 3 ( τ )= a 2 64 V π 2 [ I 0 ( τ+ Φ 0 ω 0 )+ I 0 ( τ− Φ 0 ω 0 ) ]
A 6 ( τ )= ℑ −1 { [ U 3 ∗ ( −ω ) G in * ( ω )⊗H( ω ) ]⋅ G out ( ω ) }
A 6 ( τ )= Φ f Φ f + Φ out exp[ − i τ 2 2( Φ out + Φ f ) ]× 1 2π ∫ U 3 ∗ ( −S )exp( −i τ Φ f S Φ out + Φ f )dS = 1 M exp[ − i τ 2 2M Φ f ] A 3 ∗ ( τ M )
I 6 ( τ )∝ I 0 ( τ M + Φ 0 ω 0 )+ I 0 ( τ M − Φ 0 ω 0 )

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