Abstract

We demonstrate efficient upconversion of modulated infrared (IR) signals over a wide bandwidth (up to frequencies in excess of 1 GHz) via cavity-enhanced sum-frequency generation (SFG) in a periodically poled LiNbO3. Intensity modulated IR signal is produced by combining beams from two 1547 nm narrow-linewidth lasers in a fiber coupler while tuning their wavelength difference down to 10 pm or less. The SFG crystal is placed inside an Nd:YVO4 ring cavity that provides 1064 nm circulating pump powers of up to 150 W in unidirectional operation. Measured Fabry-Pérot spectrum at 1064 nm confirms the enhanced spectral stability from multiple to single longitudinal mode pumping condition. We describe analytically and demonstrate experimentally the deleterious effects of using a multimode pump to the high-bandwidth RF spectrum of the 630 nm SFG output. Offering enhanced sensitivity without the need for cooling, the GHz-bandwidth upconverter can readily be extended to the mid-IR (2 – 5 μm) as an alternative to cooled low-bandgap semiconductor detectors for applications such as high-speed free-space optical communications.

© 2017 Optical Society of America

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

2017 (1)

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref] [PubMed]

2012 (3)

J. S. Dam, P. Tidemand-Lichtenberg, and C. Pedersen, “Room-temperature mid-infrared single-photon spectral imaging,” Nat. Photonics 6(11), 788–793 (2012).
[Crossref]

A. Rogalski, “History of infrared detectors,” Opto-Electron. Rev. 3, 279–308 (2012).

J. S. Pelc, G.-L. Shentu, Q. Zhang, M. M. Fejer, and J.-W. Pan, “Up-conversion of optical signals with multi-longitudinal-mode pump lasers,” Phys. Rev. A 86(3), 033827 (2012).
[Crossref]

2010 (1)

J. Q. Zhao, Y. Z. Wang, B. Q. Yao, and Y. L. Ju, “High efficiency, single-frequency continuous wave Nd:YVO4/YVO4 ring laser,” Laser Phys. Lett. 7(2), 135–138 (2010).
[Crossref]

2009 (1)

2008 (1)

H. Pan, E. Wu, H. Dong, and H. Zeng, “Single-photon frequency up-conversion with multimode pumping,” Phys. Rev. A 77(3), 033815 (2008).
[Crossref]

2006 (2)

H. Pan, H. Dong, H. Zeng, and W. Lu, “Efficient single-photon counting at 1.55 mm by intracavity frequency upconversion in a unidirectional ring laser,” Appl. Phys. Lett. 89(19), 191108 (2006).
[Crossref]

H. Pan and H. Zeng, “Efficient and stable single-photon counting at 1.55 μm by intracavity frequency upconversion,” Opt. Lett. 31(6), 793–795 (2006).
[Crossref] [PubMed]

2005 (1)

2004 (2)

2001 (1)

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

2000 (1)

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

1986 (1)

P. M. Henry, J. K. Powers, R. L. Rawe, and H. B. Morris, “HgCdTe photodiodes for heterodyne applications,” Proc. SPIE 0663, 145–154 (1986).
[Crossref]

Albota, M. A.

Baillargeon, J. N.

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

Bethea, C. G.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

Büchter, K.-D. F.

Capasso, F.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

Cho, A. Y.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

Curti, F. G.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

Dam, J. S.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref] [PubMed]

J. S. Dam, P. Tidemand-Lichtenberg, and C. Pedersen, “Room-temperature mid-infrared single-photon spectral imaging,” Nat. Photonics 6(11), 788–793 (2012).
[Crossref]

Diamanti, E.

Dong, H.

H. Pan, E. Wu, H. Dong, and H. Zeng, “Single-photon frequency up-conversion with multimode pumping,” Phys. Rev. A 77(3), 033815 (2008).
[Crossref]

H. Pan, H. Dong, H. Zeng, and W. Lu, “Efficient single-photon counting at 1.55 mm by intracavity frequency upconversion in a unidirectional ring laser,” Appl. Phys. Lett. 89(19), 191108 (2006).
[Crossref]

Falciglia, J.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

Fejer, M. M.

Fontana, G.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref] [PubMed]

Gmachl, C.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

Henry, P. M.

P. M. Henry, J. K. Powers, R. L. Rawe, and H. B. Morris, “HgCdTe photodiodes for heterodyne applications,” Proc. SPIE 0663, 145–154 (1986).
[Crossref]

Herrmann, H.

Hutchinson, A. L.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

Ju, Y. L.

J. Q. Zhao, Y. Z. Wang, B. Q. Yao, and Y. L. Ju, “High efficiency, single-frequency continuous wave Nd:YVO4/YVO4 ring laser,” Laser Phys. Lett. 7(2), 135–138 (2010).
[Crossref]

Kurz, J. R.

Langrock, C.

Liu, H. C.

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

Lu, W.

H. Pan, H. Dong, H. Zeng, and W. Lu, “Efficient single-photon counting at 1.55 mm by intracavity frequency upconversion in a unidirectional ring laser,” Appl. Phys. Lett. 89(19), 191108 (2006).
[Crossref]

Mancinelli, M.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref] [PubMed]

Martini, R.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

Morris, H. B.

P. M. Henry, J. K. Powers, R. L. Rawe, and H. B. Morris, “HgCdTe photodiodes for heterodyne applications,” Proc. SPIE 0663, 145–154 (1986).
[Crossref]

Paiella, R.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

Pan, H.

H. Pan, E. Wu, H. Dong, and H. Zeng, “Single-photon frequency up-conversion with multimode pumping,” Phys. Rev. A 77(3), 033815 (2008).
[Crossref]

H. Pan, H. Dong, H. Zeng, and W. Lu, “Efficient single-photon counting at 1.55 mm by intracavity frequency upconversion in a unidirectional ring laser,” Appl. Phys. Lett. 89(19), 191108 (2006).
[Crossref]

H. Pan and H. Zeng, “Efficient and stable single-photon counting at 1.55 μm by intracavity frequency upconversion,” Opt. Lett. 31(6), 793–795 (2006).
[Crossref] [PubMed]

Pan, J.-W.

J. S. Pelc, G.-L. Shentu, Q. Zhang, M. M. Fejer, and J.-W. Pan, “Up-conversion of optical signals with multi-longitudinal-mode pump lasers,” Phys. Rev. A 86(3), 033827 (2012).
[Crossref]

Pavesi, L.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref] [PubMed]

Pedersen, C.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref] [PubMed]

J. S. Dam, P. Tidemand-Lichtenberg, and C. Pedersen, “Room-temperature mid-infrared single-photon spectral imaging,” Nat. Photonics 6(11), 788–793 (2012).
[Crossref]

Pelc, J. S.

J. S. Pelc, G.-L. Shentu, Q. Zhang, M. M. Fejer, and J.-W. Pan, “Up-conversion of optical signals with multi-longitudinal-mode pump lasers,” Phys. Rev. A 86(3), 033827 (2012).
[Crossref]

Piccione, S.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref] [PubMed]

Powers, J. K.

P. M. Henry, J. K. Powers, R. L. Rawe, and H. B. Morris, “HgCdTe photodiodes for heterodyne applications,” Proc. SPIE 0663, 145–154 (1986).
[Crossref]

Rawe, R. L.

P. M. Henry, J. K. Powers, R. L. Rawe, and H. B. Morris, “HgCdTe photodiodes for heterodyne applications,” Proc. SPIE 0663, 145–154 (1986).
[Crossref]

Rogalski, A.

A. Rogalski, “History of infrared detectors,” Opto-Electron. Rev. 3, 279–308 (2012).

Roussev, R. V.

Shentu, G.-L.

J. S. Pelc, G.-L. Shentu, Q. Zhang, M. M. Fejer, and J.-W. Pan, “Up-conversion of optical signals with multi-longitudinal-mode pump lasers,” Phys. Rev. A 86(3), 033827 (2012).
[Crossref]

Sivco, D. L.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

R. Paiella, F. Capasso, C. Gmachl, C. G. Bethea, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Generation and detection of high-speed pulses of mid-infrared radiation with intersubband semiconductor lasers and detectors,” IEEE Photonics Technol. Lett. 12(7), 780–782 (2000).
[Crossref]

Sohler, W.

Takesue, H.

Tidemand-Lichtenberg, P.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref] [PubMed]

J. S. Dam, P. Tidemand-Lichtenberg, and C. Pedersen, “Room-temperature mid-infrared single-photon spectral imaging,” Nat. Photonics 6(11), 788–793 (2012).
[Crossref]

Tredicucci, A.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

Trenti, A.

M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. S. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, “Mid-infrared coincidence measurements on twin photons at room temperature,” Nat. Commun. 8, 15184 (2017).
[Crossref] [PubMed]

Wang, Y. Z.

J. Q. Zhao, Y. Z. Wang, B. Q. Yao, and Y. L. Ju, “High efficiency, single-frequency continuous wave Nd:YVO4/YVO4 ring laser,” Laser Phys. Lett. 7(2), 135–138 (2010).
[Crossref]

Whittaker, E. A.

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

Wong, F. N. C.

Wu, E.

H. Pan, E. Wu, H. Dong, and H. Zeng, “Single-photon frequency up-conversion with multimode pumping,” Phys. Rev. A 77(3), 033815 (2008).
[Crossref]

Yamamoto, Y.

Yao, B. Q.

J. Q. Zhao, Y. Z. Wang, B. Q. Yao, and Y. L. Ju, “High efficiency, single-frequency continuous wave Nd:YVO4/YVO4 ring laser,” Laser Phys. Lett. 7(2), 135–138 (2010).
[Crossref]

Zeng, H.

H. Pan, E. Wu, H. Dong, and H. Zeng, “Single-photon frequency up-conversion with multimode pumping,” Phys. Rev. A 77(3), 033815 (2008).
[Crossref]

H. Pan, H. Dong, H. Zeng, and W. Lu, “Efficient single-photon counting at 1.55 mm by intracavity frequency upconversion in a unidirectional ring laser,” Appl. Phys. Lett. 89(19), 191108 (2006).
[Crossref]

H. Pan and H. Zeng, “Efficient and stable single-photon counting at 1.55 μm by intracavity frequency upconversion,” Opt. Lett. 31(6), 793–795 (2006).
[Crossref] [PubMed]

Zhang, Q.

J. S. Pelc, G.-L. Shentu, Q. Zhang, M. M. Fejer, and J.-W. Pan, “Up-conversion of optical signals with multi-longitudinal-mode pump lasers,” Phys. Rev. A 86(3), 033827 (2012).
[Crossref]

Zhao, J. Q.

J. Q. Zhao, Y. Z. Wang, B. Q. Yao, and Y. L. Ju, “High efficiency, single-frequency continuous wave Nd:YVO4/YVO4 ring laser,” Laser Phys. Lett. 7(2), 135–138 (2010).
[Crossref]

Appl. Phys. Lett. (1)

H. Pan, H. Dong, H. Zeng, and W. Lu, “Efficient single-photon counting at 1.55 mm by intracavity frequency upconversion in a unidirectional ring laser,” Appl. Phys. Lett. 89(19), 191108 (2006).
[Crossref]

Electron. Lett. (1)

R. Martini, C. Gmachl, J. Falciglia, F. G. Curti, C. G. Bethea, F. Capasso, E. A. Whittaker, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “High-speed modulation and free-space optical audio/video transmission using quantum cascade lasers,” Electron. Lett. 37(3), 191–193 (2001).
[Crossref]

IEEE Photonics Technol. Lett. (1)

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Figures (5)

Fig. 1
Fig. 1

(a) Instantaneous pump power Pp (normalized by Pmax) versus time (normalized by the inverse of the mode frequency spacing Δf) for four different combinations of average pump power and contrast that describe the beating of two pump axial modes, (b) corresponding instantaneous conversion efficiency given by Eq. (13), and (c) magnitude of the FFT of conversion efficiency curves shown in (b).

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Fig. 2
Fig. 2

(a) High-bandwidth IR upconversion detector based on a unidirectional ring cavity and (b) intensity modulated signal generator with a central wavelength of 1547 nm and modulation frequency tuning up to GHz.

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Fig. 3
Fig. 3

Scanning FPI spectrum of the 1064 nm pump in (a) multiple and (b) single longitudinal mode operation. The spectra are stable for durations longer than our typical measurement time.

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Fig. 4
Fig. 4

Modulus of the FFT spectra of simultaneously detected powers of (a), (b) the 1547 nm signal, (c), (d) the 1064 nm pump, and (e), (f) the 630 nm SFG output for (a), (c), (e) multimode and (b), (d), (f) single longitudinal mode pump operation. Each inset shows a time series of the detected instantaneous power (i.e. InGaAs or Si detector voltage) for the respective wavelength. Each FFT spectrum is normalized by their zero-frequency component.

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Fig. 5
Fig. 5

Magnitude of the normalized FFT spectra of the 1547 nm input signal and the 630 nm SFG output power for single frequency pump operation and for signal modulation frequency of (a) 611.8 MHz, (b) 954.4 MHz, and (c) 1.19 GHz, and (d) the measured normalized transfer function of our GHz-bandwidth upconversion detector.

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Equations (14)

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∂ a IR (z,t) ∂z =−iκ a p ∗ (t) a up (z,t)exp(−iΔkz),
∂ a up (z,t) ∂z =−iκ a p (t) a IR (z,t)exp(iΔkz).
κ=2π ε 0 d eff Θ [ 2h ν IR ν p ν up Z 0 3 / ( n IR n p n up ) ] 1/2 ,
a IR (L,t)= a IR (0,t)cos[ κ| a p (t) |L ]+ a up (0,t)sin[ κ| a p (t) |L ],
a up (L,t)=− a IR (0,t)sin[ κ| a p (t) |L ]+ a up (0,t)cos[ κ| a p (t) |L ].
a up (L,t)=− a IR (0,t)sin[ κ| a p (t) |L ].
η(t)= | a up (L,t) | 2 / | a IR (0,t) | 2 = sin 2 [ κ| a p (t) |L ].
N ˜ IR (f)= b 1 F{ | a IR (0,t) | 2 },
N ˜ up (f)= b 2 F{ | a up (L,t) | 2 },
N ˜ up (f)= H ˜ (f) N ˜ IR (f).
| H ˜ (f) |= b| F{ | a IR (0,t) | 2 }⊗F{ η(t) } | | F{ | a IR (0,t) | 2 } | ,
P p (t)= P ave [ 1+αcos(2πΔft+φ) ],
η(t)= sin 2 [ π 2 ( P ave P max )| 1+αcos(2πΔft+φ) | ],
FSR=c/ ∑ i n i l i ,

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