Abstract

Chirped pulse heterodyne is proposed to maximize the signal-to-noise ratio (SNR) when measuring the beat note between an optical frequency comb and a continuous wave (CW) laser. The noise model reveals that all the comb power within the largest possible detection bandwidth can be used to increase the SNR. The chirped comb/CW interference experiment is shown to be equivalent to CW/CW interference, using the comb’s spectrally available power. The approach can also greatly alleviate dynamic range issues when detected pulsed heterodyne signals. A beat note SNR of 68.3 dB in a 100 kHz bandwidth is achieved.

© 2015 Optical Society of America

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    [Crossref]

2013 (3)

F. Quinlan, T. M. Fortier, H. Jiang, A. Hati, C. Nelson, Y. Fu, J. C. Campbell, and S. A. Diddams, “Exploiting shot noise correlations in the photodetection of ultrashort optical pulse trains,” Nat. Photonics 7(4), 290–293 (2013).
[Crossref]

F. Quinlan, T. M. Fortier, H. Jiang, and S. A. Diddams, “Analysis of shot noise in the detection of ultrashort optical pulse trains,” J. Opt. Soc. Am. B 30(6), 1775–1785 (2013).
[Crossref]

J.-D. Deschênes and J. Genest, “Heterodyne beats between a continuous-wave laser and a frequency comb beyond the shot noise limit of a single comb line,” Phys. Rev. A 87(2), 023802 (2013).
[Crossref]

2012 (1)

2011 (2)

F. Quinlan, T. M. Fortier, M. S. Kirchner, J. A. Taylor, M. J. Thorpe, N. Lemke, A. D. Ludlow, Y. Jiang, and S. A. Diddams, “Ultralow phase noise microwave generation with an Er:fiber-based optical frequency divider,” Opt. Lett. 36(16), 3260–3262 (2011).
[Crossref] [PubMed]

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

2010 (3)

Y. Nakajima, H. Inaba, K. Hosaka, K. Minoshima, A. Onae, M. Yasuda, T. Kohno, S. Kawato, T. Kobayashi, T. Katsuyama, and F. L. Hong, “A multi-branch, fiber-based frequency comb with millihertz-level relative linewidths using an intra-cavity electro-optic modulator,” Opt. Express 18(2), 1667–1676 (2010).
[Crossref] [PubMed]

S. Gupta, A. Parsa, E. Perret, R. V. Snyder, R. J. Wenzel, and C. Caloz, “Group-delay engineered noncommensurate transmission line all-pass network for analog signal processing,” IEEE Trans. Microw. Theory Tech. 58(9), 2392–2407 (2010).
[Crossref]

I. Ozdur, M. Akbulut, N. Hoghooghi, D. Mandridis, S. Ozharar, F. Quinlan, and P. J. Delfyett, “A Semiconductor-Based 10-GHz Optical Comb Source With Sub 3-fs Shot-Noise-Limited Timing Jitter and 500-Hz Comb Linwidth,” Photon. Technol. Lett. 22(6), 431–433 (2010).
[Crossref]

2009 (2)

2008 (2)

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
[Crossref]

P. Malara, P. Maddaloni, G. Gagliardi, and P. De Natale, “Absolute frequency measurement of molecular transitions by a direct link to a comb generated around 3-microm,” Opt. Express 16(11), 8242–8249 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (1)

2004 (1)

2002 (1)

2001 (2)

2000 (1)

1999 (3)

J. Reichert, R. Holzwarth, T. Udem, and T. W. Hänsch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. 172(1-6), 59–68 (1999).
[Crossref]

H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, “Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation,” Appl. Phys. B 69(4), 327–332 (1999).
[Crossref]

K. Imai, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, “12-THz frequency difference measurements and noise analysis of an optical frequency comb in optical fibers,” IEEE J. Quantum Electron. 35(4), 559–564 (1999).
[Crossref]

Adler, F.

Akbulut, M.

I. Ozdur, M. Akbulut, N. Hoghooghi, D. Mandridis, S. Ozharar, F. Quinlan, and P. J. Delfyett, “A Semiconductor-Based 10-GHz Optical Comb Source With Sub 3-fs Shot-Noise-Limited Timing Jitter and 500-Hz Comb Linwidth,” Photon. Technol. Lett. 22(6), 431–433 (2010).
[Crossref]

Baumann, E.

Bender, C. F.

Benko, C.

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

Bergquist, J. C.

Botzer, B.

Caloz, C.

S. Gupta, A. Parsa, E. Perret, R. V. Snyder, R. J. Wenzel, and C. Caloz, “Group-delay engineered noncommensurate transmission line all-pass network for analog signal processing,” IEEE Trans. Microw. Theory Tech. 58(9), 2392–2407 (2010).
[Crossref]

Campbell, J. C.

F. Quinlan, T. M. Fortier, H. Jiang, A. Hati, C. Nelson, Y. Fu, J. C. Campbell, and S. A. Diddams, “Exploiting shot noise correlations in the photodetection of ultrashort optical pulse trains,” Nat. Photonics 7(4), 290–293 (2013).
[Crossref]

Chen, L.

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

Coddington, I.

Cossel, K. C.

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

F. Adler, K. C. Cossel, M. J. Thorpe, I. Hartl, M. E. Fermann, and J. Ye, “Phase-stabilized, 1.5 W frequency comb at 2.8-4.8 microm,” Opt. Lett. 34(9), 1330–1332 (2009).
[Crossref] [PubMed]

Cundiff, S. T.

Curtis, E. A.

Daimon, Y.

De Natale, P.

Delfyett, P. J.

I. Ozdur, M. Akbulut, N. Hoghooghi, D. Mandridis, S. Ozharar, F. Quinlan, and P. J. Delfyett, “A Semiconductor-Based 10-GHz Optical Comb Source With Sub 3-fs Shot-Noise-Limited Timing Jitter and 500-Hz Comb Linwidth,” Photon. Technol. Lett. 22(6), 431–433 (2010).
[Crossref]

Deschênes, J.-D.

J.-D. Deschênes and J. Genest, “Heterodyne beats between a continuous-wave laser and a frequency comb beyond the shot noise limit of a single comb line,” Phys. Rev. A 87(2), 023802 (2013).
[Crossref]

Diddams, S. A.

F. Quinlan, T. M. Fortier, H. Jiang, A. Hati, C. Nelson, Y. Fu, J. C. Campbell, and S. A. Diddams, “Exploiting shot noise correlations in the photodetection of ultrashort optical pulse trains,” Nat. Photonics 7(4), 290–293 (2013).
[Crossref]

F. Quinlan, T. M. Fortier, H. Jiang, and S. A. Diddams, “Analysis of shot noise in the detection of ultrashort optical pulse trains,” J. Opt. Soc. Am. B 30(6), 1775–1785 (2013).
[Crossref]

G. G. Ycas, F. Quinlan, S. A. Diddams, S. Osterman, S. Mahadevan, S. Redman, R. Terrien, L. Ramsey, C. F. Bender, B. Botzer, and S. Sigurdsson, “Demonstration of on-sky calibration of astronomical spectra using a 25 GHz near-IR laser frequency comb,” Opt. Express 20(6), 6631–6643 (2012).
[Crossref] [PubMed]

F. Quinlan, T. M. Fortier, M. S. Kirchner, J. A. Taylor, M. J. Thorpe, N. Lemke, A. D. Ludlow, Y. Jiang, and S. A. Diddams, “Ultralow phase noise microwave generation with an Er:fiber-based optical frequency divider,” Opt. Lett. 36(16), 3260–3262 (2011).
[Crossref] [PubMed]

K. R. Vogel, S. A. Diddams, C. W. Oates, E. A. Curtis, R. J. Rafac, W. M. Itano, J. C. Bergquist, R. W. Fox, W. D. Lee, J. S. Wells, and L. Hollberg, “Direct comparison of two cold-atom-based optical frequency standards by using a femtosecond-laser comb,” Opt. Lett. 26(2), 102–104 (2001).
[Crossref] [PubMed]

S. A. Diddams, D. J. Jones, L. S. Ma, S. T. Cundiff, and J. L. Hall, “Optical frequency measurement across a 104-THz gap with a femtosecond laser frequency comb,” Opt. Lett. 25(3), 186–188 (2000).
[Crossref] [PubMed]

Diels, J. C.

R. J. Jones and J. C. Diels, “Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency synthesis,” Phys. Rev. Lett. 86(15), 3288–3291 (2001).
[Crossref] [PubMed]

Dong, L.

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

Dudley, J. M.

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

Dunlop, A. E.

H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, “Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation,” Appl. Phys. B 69(4), 327–332 (1999).
[Crossref]

Fermann, M. E.

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

F. Adler, K. C. Cossel, M. J. Thorpe, I. Hartl, M. E. Fermann, and J. Ye, “Phase-stabilized, 1.5 W frequency comb at 2.8-4.8 microm,” Opt. Lett. 34(9), 1330–1332 (2009).
[Crossref] [PubMed]

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
[Crossref]

T. R. Schibli, K. Minoshima, F. L. Hong, H. Inaba, A. Onae, H. Matsumoto, I. Hartl, and M. E. Fermann, “Frequency metrology with a turnkey all-fiber system,” Opt. Lett. 29(21), 2467–2469 (2004).
[Crossref] [PubMed]

Fortier, T. M.

Fox, R. W.

Fu, Y.

F. Quinlan, T. M. Fortier, H. Jiang, A. Hati, C. Nelson, Y. Fu, J. C. Campbell, and S. A. Diddams, “Exploiting shot noise correlations in the photodetection of ultrashort optical pulse trains,” Nat. Photonics 7(4), 290–293 (2013).
[Crossref]

Gagliardi, G.

Genest, J.

J.-D. Deschênes and J. Genest, “Heterodyne beats between a continuous-wave laser and a frequency comb beyond the shot noise limit of a single comb line,” Phys. Rev. A 87(2), 023802 (2013).
[Crossref]

Giorgetta, F. R.

Gupta, S.

S. Gupta, A. Parsa, E. Perret, R. V. Snyder, R. J. Wenzel, and C. Caloz, “Group-delay engineered noncommensurate transmission line all-pass network for analog signal processing,” IEEE Trans. Microw. Theory Tech. 58(9), 2392–2407 (2010).
[Crossref]

Hall, J.

Hall, J. L.

Hänsch, T. W.

J. Reichert, R. Holzwarth, T. Udem, and T. W. Hänsch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. 172(1-6), 59–68 (1999).
[Crossref]

Hartl, I.

A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
[Crossref]

F. Adler, K. C. Cossel, M. J. Thorpe, I. Hartl, M. E. Fermann, and J. Ye, “Phase-stabilized, 1.5 W frequency comb at 2.8-4.8 microm,” Opt. Lett. 34(9), 1330–1332 (2009).
[Crossref] [PubMed]

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
[Crossref]

T. R. Schibli, K. Minoshima, F. L. Hong, H. Inaba, A. Onae, H. Matsumoto, I. Hartl, and M. E. Fermann, “Frequency metrology with a turnkey all-fiber system,” Opt. Lett. 29(21), 2467–2469 (2004).
[Crossref] [PubMed]

Hati, A.

F. Quinlan, T. M. Fortier, H. Jiang, A. Hati, C. Nelson, Y. Fu, J. C. Campbell, and S. A. Diddams, “Exploiting shot noise correlations in the photodetection of ultrashort optical pulse trains,” Nat. Photonics 7(4), 290–293 (2013).
[Crossref]

Hirano, M.

Hoghooghi, N.

I. Ozdur, M. Akbulut, N. Hoghooghi, D. Mandridis, S. Ozharar, F. Quinlan, and P. J. Delfyett, “A Semiconductor-Based 10-GHz Optical Comb Source With Sub 3-fs Shot-Noise-Limited Timing Jitter and 500-Hz Comb Linwidth,” Photon. Technol. Lett. 22(6), 431–433 (2010).
[Crossref]

Hollberg, L.

Holzwarth, R.

J. Reichert, R. Holzwarth, T. Udem, and T. W. Hänsch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. 172(1-6), 59–68 (1999).
[Crossref]

Hong, F. L.

Hosaka, K.

Imai, K.

K. Imai, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, “12-THz frequency difference measurements and noise analysis of an optical frequency comb in optical fibers,” IEEE J. Quantum Electron. 35(4), 559–564 (1999).
[Crossref]

Inaba, H.

Itano, W. M.

Jiang, H.

F. Quinlan, T. M. Fortier, H. Jiang, A. Hati, C. Nelson, Y. Fu, J. C. Campbell, and S. A. Diddams, “Exploiting shot noise correlations in the photodetection of ultrashort optical pulse trains,” Nat. Photonics 7(4), 290–293 (2013).
[Crossref]

F. Quinlan, T. M. Fortier, H. Jiang, and S. A. Diddams, “Analysis of shot noise in the detection of ultrashort optical pulse trains,” J. Opt. Soc. Am. B 30(6), 1775–1785 (2013).
[Crossref]

Jiang, Y.

Jones, D. J.

Jones, R. J.

R. J. Jones and J. C. Diels, “Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency synthesis,” Phys. Rev. Lett. 86(15), 3288–3291 (2001).
[Crossref] [PubMed]

Jost, J.

Katsuyama, T.

Kawato, S.

Keller, U.

H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, “Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation,” Appl. Phys. B 69(4), 327–332 (1999).
[Crossref]

Kirchner, M. S.

Kobayashi, T.

Kohno, T.

Kourogi, M.

K. Imai, B. Widiyatmoko, M. Kourogi, and M. Ohtsu, “12-THz frequency difference measurements and noise analysis of an optical frequency comb in optical fibers,” IEEE J. Quantum Electron. 35(4), 559–564 (1999).
[Crossref]

Lee, W. D.

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T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
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A. Ruehl, M. J. Martin, K. C. Cossel, L. Chen, H. McKay, B. Thomas, C. Benko, L. Dong, J. M. Dudley, M. E. Fermann, I. Hartl, and J. Ye, “Ultrabroadband coherent supercontinuum frequency comb,” Phys. Rev. A 84(1), 011806 (2011).
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I. Ozdur, M. Akbulut, N. Hoghooghi, D. Mandridis, S. Ozharar, F. Quinlan, and P. J. Delfyett, “A Semiconductor-Based 10-GHz Optical Comb Source With Sub 3-fs Shot-Noise-Limited Timing Jitter and 500-Hz Comb Linwidth,” Photon. Technol. Lett. 22(6), 431–433 (2010).
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Appl. Phys. B (1)

H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, “Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation,” Appl. Phys. B 69(4), 327–332 (1999).
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IEEE J. Quantum Electron. (1)

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[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

S. Gupta, A. Parsa, E. Perret, R. V. Snyder, R. J. Wenzel, and C. Caloz, “Group-delay engineered noncommensurate transmission line all-pass network for analog signal processing,” IEEE Trans. Microw. Theory Tech. 58(9), 2392–2407 (2010).
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J. Opt. Soc. Am. B (2)

Nat. Photonics (2)

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Illustrating the scaling between the chirped and un-chirped pulses.
Fig. 2
Fig. 2 Illustrating the time-frequency representation of the chirped optical pulses from the comb as the blue diagonal lines indicating linear chirp, and the orange horizontal line depicting the CW laser signal.
Fig. 3
Fig. 3 Filtering the chirped beat signal (blue lines) with a chirp-compensating filter yields a transform-limited pulse every Tr period (orange lines). This signal can then be time-gated to remove noise while keeping signal unaffected.
Fig. 4
Fig. 4 Multiplying the chirped beat signal (blue diagonal lines) with a repeating template of the complex conjugate of the chirp collapses the chirp to a single frequency (orange horizontal lines). Most of the energy in the signal is now concentrated into a single frequency. All that is left is the amplitude fluctuation of the beat signal, which can be averaged out by filtering.
Fig. 5
Fig. 5 Three different implementations of chirped pulse heterodyne. In A), the comb pulses are recorded on the oscilloscope to use as a gate. Optical fibers are solid lines and dashed lines are electrical coaxial cable. BPF: Optical bandpass filter, PC: polarization controller, DL: Adjustable optical delay line, D: wideband photodetector, LPF: low-pass filter, LNA: Low-noise amplifier. cFBG: Chirped fiber Bragg grating. h*(-t): Matched filter. B) and C) show two possible hardware implementations of the technique and the signal flow.
Fig. 6
Fig. 6 Fiber Bragg grating impulse response characterisation, shown in both the time-domain (bottom) and the time-frequency domain (top) using a short-time Fourier transform.
Fig. 7
Fig. 7 Raw chirped pulse heterodyne signal. On top: time-frequency representation and bottom: time-domain signal.
Fig. 8
Fig. 8 Chirped pulse heterodyne results after processing by the correlation approach. The bottom blue trace shows the signal after multiplication with the chirp template, while the green trace is the output result, low-pass filtered in the Nyquist range of the comb, 50 MHz.
Fig. 9
Fig. 9 Chirped pulse heterodyne results after processing by the matched filter approach. The bottom blue trace shows the signal after de-chirping, the red dots show the gated/sampled points, while the green signal is the gated and low-pass filtered signal.
Fig. 10
Fig. 10 Spectrum of the chirped pulse heterodyne signal, before and after processing using the correlation approach. The processed spectrum shows 9.5 dB of improvement over the single comb mode SNR limit, reaching 55.9 dB in a 1.7 MHz resolution bandwidth. The raw beat spectrum shows indication of excess noise, most likely being ASE from the comb itself.
Fig. 11
Fig. 11 Power spectral density of the demodulated beat signals. Dashed lines indicate the signal demodulated at a single beating mode, while solid lines indicate demodulated phase and amplitude noise on the CPH-processed signal. Blue lines show the amplitude noise of the beat signal, dominated by additive noise, as an indication of measurement noise floor. Red lines show phase noise and are dominated by laser frequency noise or measurement noise. Finally, the black solid line is the PSD of the difference of the phase between the single-mode demodulation and CPH-processed signal, sitting on top of the noise floor of the single-mode demodulation.

Tables (2)

Tables Icon

Table 1 Parameters for the derivation of the SNR in the chirped pulse heterodyne.

Tables Icon

Table 2 Comparison of the comb-CW beat SNRs reported in the literature.

Equations (15)

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A unchirped ( t )={ P unchirped exp( j2π f c t ) 0<t< T unchirped 0 elsewhere,
A chirped ( t )={ P unchirped k exp( j2π f c t+j ϕ chirp ( t ) ) 0<t<k T unchirped 0 elsewhere,
A CW ( t )= P CW exp( j2π f c t+j ϕ 0 ),
P( t )=η | A CW ( t ) | 2 +η | A chirped ( t ) | 2 +2ηRe{ A chirped ( t ) A CW * ( t ) },
E = 0 k T unchirped P( t )exp( j ϕ chirp ( t ) )dt .
E beat = 0 k T unchirped 2ηRe{ A chirped ( t ) A CW * ( t ) }exp( j ϕ chirp ( t ) )dt .
E beat =η 0 k T unchirped P CW P unchirped k [ exp( j ϕ 0 )+exp( 2j ϕ chirp ( t )j ϕ 0 ) ]dt .
E beat =η T unchirped P CW P unchirped k exp( j ϕ 0 ).
| E beat | 2 ¯ = η 2 P CW P unchirped k T unchirped 2 .
E sources = 0 k T unchirped η | A CW ( t ) | 2 +η | A chirped ( t ) | 2 dt ==η( k P CW + P unchirped ) T unchirped ,
σ shot 2 = E photon E sources =η E photon ( k P CW + P unchirped ) T unchirped ,
SNR= | E beat | 2 ¯ σ shot 2 = η 2 P CW P unchirped k T unchirped 2 η E photon ( k P CW + P unchirped ) T unchirped = η P CW P unchirped T unchirped E photon ( P CW + 1 k P unchirped ) .
lim k { SNR }= η P CW P unchirped T unchirped E photon P CW = η E unchirped E photon =η N unchirped ,
SNR= η P CW P unchirped T unchirped E photon ( P CW + T unchirped T r P unchirped ) .
SNR= η P CW P SAP T r E photon ( P CW + P SAP ) ,

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