Abstract

For given laser output power, object under investigation, and photodiode noise level, we have theoretically compared the signal-to-noise ratios of a heterodyne scanning imager based on a Michelson interferometer and of an autodyne setup based on the laser optical feedback imaging (LOFI) technique. In both cases, the image is obtained point by point. In the heterodyne configuration, the beating between the reference beam and the signal beam is realized outside the laser cavity (i.e., directly on the detector), while in the autodyne configuration, the wave beating takes place inside the laser cavity and therefore is indirectly detected. In the autodyne configuration, where the laser relaxation oscillations play a leading role, we have compared one-dimensional scans obtained by numerical simulations with different lasers' dynamical parameters. Finally, we have determined the best laser for LOFI applications and the experimental conditions for which the LOFI detection setup (autodyne interferometer) is competitive compared to a heterodyne interferometer.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  22. V. Muzet, E. Lacot, O. Hugon, and Y. Gaillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793–799 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
  26. In , we have demonstrated that, far from resonance, the nonlinear phase noise of a LOFI setup is less important. The working condition (F+≈7FR) could be of interest for obtaining a shot noise limited setup for phase measurements with a very high precision level.

2010 (4)

2009 (3)

2008 (3)

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser,” Ultramicroscopy 108, 523–528 (2008).
[CrossRef]

H. Gilles, S. Girard, M. Laroche, and A. Belarouci, “Near-field amplitude and phase measurements using heterodyne optical feedback on solid-state lasers,” Opt. Lett. 33, 1–3 (2008).
[CrossRef]

S. Blaize, B. Bérenguier, I. Stéfanon, A. Bruyant, G. Lerondel, P. Royer, O. Hugon, O. Jacquin, and E. Lacot, Opt. Express 16, 11718–11726 (2008).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

V. Muzet, E. Lacot, O. Hugon, and Y. Gaillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793–799 (2005).
[CrossRef]

2004 (1)

E. Lacot and O. Hugon, “Phase-sensitive laser detection by frequency-shifted optical feedback,” Phys. Rev. A 70, 053824 (2004).
[CrossRef]

2002 (1)

2001 (1)

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

1999 (1)

1996 (1)

1995 (1)

S. Okamoto, H. Takeda, and F. Kannari, “Ultrahighly sensitive laser-Doppler velocity meter with a diode-pumped Nd:YVO4 microchip laser,” Rev. Sci. Instrum. 66, 3116–3120 (1995).
[CrossRef]

1994 (1)

1993 (1)

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

1992 (1)

K. Otsuka, “Highly sensitive measurement of Doppler-shift with a microchip solid-state laser,” Jpn. J. Appl. Phys., Part 2 31, L1546–L1548 (1992).
[CrossRef]

1991 (1)

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, 1991).

1989 (1)

1974 (1)

M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

1973 (1)

Abe, K.

Adler, D.

Aguirre, A. D.

Arai, K.

Belarouci, A.

Bérenguier, B.

Blaize, S.

Boas, D. A.

Bruyant, A.

Cohen, D. W.

Connolly, J. L.

Davidovich, L.

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

Day, R.

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744–746 (1999).
[CrossRef]

Denk, W.

Fabre, C.

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

Fujimoto, J. G.

Gaillard, Y.

V. Muzet, E. Lacot, O. Hugon, and Y. Gaillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793–799 (2005).
[CrossRef]

Giacobino, E.

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

Gilles, H.

Girard, S.

Gorczynska, I.

Gouda, N.

Goullioud, R.

I. Hahn, M. Xeilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81, 045103 (2010).
[CrossRef] [PubMed]

Hahn, I.

I. Hahn, M. Xeilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81, 045103 (2010).
[CrossRef] [PubMed]

Hee, M. R.

Heidmann, S.

Huang, S. W.

Hugon, O.

O. Jacquin, S. Heidmann, E. Lacot, and O. Hugon, “Self aligned setup for laser optical feedback imaging insensitive to parasitic optical feedback,” Appl. Opt. 48, 64–68 (2009).
[CrossRef]

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser,” Ultramicroscopy 108, 523–528 (2008).
[CrossRef]

S. Blaize, B. Bérenguier, I. Stéfanon, A. Bruyant, G. Lerondel, P. Royer, O. Hugon, O. Jacquin, and E. Lacot, Opt. Express 16, 11718–11726 (2008).
[CrossRef] [PubMed]

A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Synthetic aperture laser optical feedback imaging using galvanometric scanning,” Opt. Lett. 31, 3031–3033 (2006).
[CrossRef] [PubMed]

V. Muzet, E. Lacot, O. Hugon, and Y. Gaillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793–799 (2005).
[CrossRef]

E. Lacot and O. Hugon, “Phase-sensitive laser detection by frequency-shifted optical feedback,” Phys. Rev. A 70, 053824 (2004).
[CrossRef]

Izatt, J. A.

Jacquin, O.

Kannari, F.

S. Okamoto, H. Takeda, and F. Kannari, “Ultrahighly sensitive laser-Doppler velocity meter with a diode-pumped Nd:YVO4 microchip laser,” Rev. Sci. Instrum. 66, 3116–3120 (1995).
[CrossRef]

Kempe, M.

Ko, J. Y.

Kobayashi, Y.

Kolobov, M. I.

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

Lacot, E.

O. Jacquin, S. Heidmann, E. Lacot, and O. Hugon, “Self aligned setup for laser optical feedback imaging insensitive to parasitic optical feedback,” Appl. Opt. 48, 64–68 (2009).
[CrossRef]

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser,” Ultramicroscopy 108, 523–528 (2008).
[CrossRef]

S. Blaize, B. Bérenguier, I. Stéfanon, A. Bruyant, G. Lerondel, P. Royer, O. Hugon, O. Jacquin, and E. Lacot, Opt. Express 16, 11718–11726 (2008).
[CrossRef] [PubMed]

A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Synthetic aperture laser optical feedback imaging using galvanometric scanning,” Opt. Lett. 31, 3031–3033 (2006).
[CrossRef] [PubMed]

V. Muzet, E. Lacot, O. Hugon, and Y. Gaillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793–799 (2005).
[CrossRef]

E. Lacot and O. Hugon, “Phase-sensitive laser detection by frequency-shifted optical feedback,” Phys. Rev. A 70, 053824 (2004).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744–746 (1999).
[CrossRef]

Lamb, W. E.

M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

Laroche, M.

Lee, H. C.

Lerondel, G.

Lim, T. S.

Mondelblatt, A.

Mooradian, A.

Muzet, V.

V. Muzet, E. Lacot, O. Hugon, and Y. Gaillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793–799 (2005).
[CrossRef]

Niwa, Y.

Okamoto, S.

S. Okamoto, H. Takeda, and F. Kannari, “Ultrahighly sensitive laser-Doppler velocity meter with a diode-pumped Nd:YVO4 microchip laser,” Rev. Sci. Instrum. 66, 3116–3120 (1995).
[CrossRef]

Otsuka, K.

K. Otsuka, K. Abe, J. Y. Ko, and T. S. Lim, “Real-time nanometer vibration measurement with self-mixing microchip solid-state laser,” Opt. Lett. 27, 1339–1341 (2002).
[CrossRef]

K. Otsuka, “Highly sensitive measurement of Doppler-shift with a microchip solid-state laser,” Jpn. J. Appl. Phys., Part 2 31, L1546–L1548 (1992).
[CrossRef]

Owen, G. M.

Paun, I. A.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser,” Ultramicroscopy 108, 523–528 (2008).
[CrossRef]

Petermann, K.

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, 1991).

Ricard, C.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser,” Ultramicroscopy 108, 523–528 (2008).
[CrossRef]

Royer, P.

Rudolph, W.

Ruvinskaya, S.

Sakadzic, S.

Sakagami, M.

Sargent, M.

M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

Sawatari, T.

Sawinski, J.

Scully, M. O.

M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

Srinivasan, V. J.

Stéfanon, I.

Stoeckel, F.

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744–746 (1999).
[CrossRef]

Swanson, E. A.

Takeda, H.

S. Okamoto, H. Takeda, and F. Kannari, “Ultrahighly sensitive laser-Doppler velocity meter with a diode-pumped Nd:YVO4 microchip laser,” Rev. Sci. Instrum. 66, 3116–3120 (1995).
[CrossRef]

Tsai, T. H.

Ueda, A.

van der Sanden, B.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser,” Ultramicroscopy 108, 523–528 (2008).
[CrossRef]

Wang, X.

I. Hahn, M. Xeilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81, 045103 (2010).
[CrossRef] [PubMed]

Wang, Y.

Welsh, E.

Witomski, A.

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser,” Ultramicroscopy 108, 523–528 (2008).
[CrossRef]

A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Synthetic aperture laser optical feedback imaging using galvanometric scanning,” Opt. Lett. 31, 3031–3033 (2006).
[CrossRef] [PubMed]

Wu, W.

Xeilert, M.

I. Hahn, M. Xeilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81, 045103 (2010).
[CrossRef] [PubMed]

Yamada, Y.

Yano, T.

Zayhowski, J. J.

Zhou, C.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys., Part 2 (1)

K. Otsuka, “Highly sensitive measurement of Doppler-shift with a microchip solid-state laser,” Jpn. J. Appl. Phys., Part 2 31, L1546–L1548 (1992).
[CrossRef]

Opt. Express (3)

Opt. Lett. (7)

Phys. Rev. A (3)

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

E. Lacot and O. Hugon, “Phase-sensitive laser detection by frequency-shifted optical feedback,” Phys. Rev. A 70, 053824 (2004).
[CrossRef]

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

Proc. SPIE (1)

V. Muzet, E. Lacot, O. Hugon, and Y. Gaillard, “Experimental comparison of shearography and laser optical feedback imaging for crack detection in concrete structures,” Proc. SPIE 5856, 793–799 (2005).
[CrossRef]

Rev. Sci. Instrum. (2)

S. Okamoto, H. Takeda, and F. Kannari, “Ultrahighly sensitive laser-Doppler velocity meter with a diode-pumped Nd:YVO4 microchip laser,” Rev. Sci. Instrum. 66, 3116–3120 (1995).
[CrossRef]

I. Hahn, M. Xeilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81, 045103 (2010).
[CrossRef] [PubMed]

Ultramicroscopy (1)

O. Hugon, I. A. Paun, C. Ricard, B. van der Sanden, E. Lacot, O. Jacquin, and A. Witomski, “Cell imaging by coherent backscattering microscopy using frequency shifted optical feedback in a microchip laser,” Ultramicroscopy 108, 523–528 (2008).
[CrossRef]

Other (4)

T.Yoshizawa, ed., Handbook of Optical Metrology: Principles and Applications (CRC Press, 2009).
[CrossRef]

M. Sargent III, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, 1991).

In , we have demonstrated that, far from resonance, the nonlinear phase noise of a LOFI setup is less important. The working condition (F+≈7FR) could be of interest for obtaining a shot noise limited setup for phase measurements with a very high precision level.

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Figures (4)

Fig. 1
Fig. 1

Schematic diagrams (a) of the autodyne interferometer setup and (b) of the heterodyne interferometer setup for scanning microscopy: L 1 , L 2 , and L 3 —lenses; OI, optical Isolator; BS, beam splitter with a power reflectivity R bs ; GS, galvanometric scanner; RM, reference mirror with a unitary power reflectivity R rm = 1 ; FS, frequency shifter with a round trip frequency shift F e ; PD: photodiode with a white noise spectrum. The lock-in amplifier is characterized by a bandwidth Δ F around the reference frequency F e . The laser output power is characterized by p out ( photons / s ) ; the target is characterized by its effective reflectivity R e 1 .

Fig. 2
Fig. 2

Normalized power spectra of the laser output power versus the normalized shift frequency. The power spectra are normalized to the quantum shot noise with Δ F = 600   Hz and p out = 4.7 × 10 16   photons / s ( P out = 8.8   mW at λ = 1064   nm ): (a) saturation level, (b) autodyne signal ( R e = 2 × 10 11 , R bs = 1 / 2 ), (c) autodyne noise (laser quantum noise), (d) heterodyne noise (detection noise), (e) heterodyne signal ( R e = 2 × 10 11 , R bs = 1 / 2 ). Laser dynamical parameters corresponding to a conventional Nd 3 + : YAG microchip laser: γ c / γ 1 η = 5 × 10 3 , η = 2 , γ c = 5 × 10 8 s 1 , γ 1 = 5 × 10 4 s 1 , F R = 796   kHz .

Fig. 3
Fig. 3

Normalized power spectra of the laser output power versus the normalized shift frequency. The power spectra are normalized to the quantum shot noise with Δ F = 600   Hz and p out = 4.7 × 10 16   photons / s ( P out = 8.8   mW at λ = 1064   nm ): (a) saturation level, (b) autodyne signal ( R e = 2 × 10 11 , R bs = 1 / 2 ), (c) autodyne noise (laser quantum noise), (d) heterodyne noise (detection noise), (e) heterodyne signal ( R e = 2 × 10 11 , R bs = 1 / 2 ). Laser dynamical parameters corresponding to the optimum values: γ c / γ 1 η = 146 , η = 2 , γ c = 8.6 × 10 7 s 1 , γ 1 = 2.9 × 10 5 s 1 , F R = 796   kHz .

Fig. 4
Fig. 4

Numerical 1D scans obtained from the measured laser output power MC of an autodyne interferometer, when the laser beam is scanned on reflectivity stairs composed of four steps. Experimental conditions: Laser output power, P out = 117   mW ( p out = 6.25 × 10 17   photons / s at λ = 1064   nm ); laser relaxation frequency, F R = 1.6   MHz ; autodyne modulation frequency, F e = F R . Step 1: (pixels 1–10), R e , 1 = 0 ; step 2: (pixels 11–20), R e , 2 = 10 10 / 100 ; step 3: (pixels 21–30), R e , 3 = 10 10 / 4 ; step 4: (pixels 31–40), R e , 4 = 10 10 . Top row: G ( F R ) = 5 × 10 5 ; bottom row: G ( F R ) = 1 × 10 4 ; left column: Δ F = 1 / ( 2 T ) = 666   Hz ; right column: Δ F = 1 / ( 2 T ) = 66.6   kHz .

Tables (2)

Tables Icon

Table 1 MC and SNR of the LOFI Images (Fig. 4) Obtained with the Laser Having the Lower Value of LOFI Gain [ G ( F R ) = 1 × 10 4 ] a

Tables Icon

Table 2 MC and SNR of the LOFI Images (Fig. 4) Obtained with the Laser Having the Higher Value of LOFI Gain [ G ( F R ) = 5 × 10 5 ] a

Equations (37)

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

d N ( t ) d t = γ 1 N 0 γ 1 N ( t ) B N ( t ) | E c ( t ) | 2 + F N ( t ) ,
d E c ( t ) d t = 1 2 ( B N ( t ) γ c ) E c ( t ) + γ e E c ( t τ e ) cos [ 2 π F e t 2 π ( ν + F e 2 ) τ e ϕ c ( t ) + ϕ c ( t τ e ) ] + F E c ( t ) ,
d ϕ c ( t ) d t = 2 π ( ν c ν ) + γ e E c ( t τ e ) E c ( t ) sin [ 2 π F e t 2 π ( ν + F e 2 ) τ e ϕ c ( t ) + ϕ c ( t τ e ) ] + F Φ c ( t ) .
γ e = γ c R e ( 1 R bs ) ,
p out ( t ) = p out + 2 G ( F e ) p out ( 1 R bs ) R e   cos [ 2 π F e t + φ ] ,
Δ p out ( F e , R e ) p out = 2 G ( F e ) ( 1 R bs ) R e .
G ( F e ) = γ c 2 π Δ F R 2 + F e 2 ( F R 2 F e 2 ) 2 + Δ F R 2 F e 2 ,
S LOFI ( R e , F e ) = R bs Δ p out ( F e , R e ) 2 = 2 G ( F e ) R bs ( 1 R bs ) R e p out 2 .
R Sat ( F e ) = 1 4 1 G 2 ( F e ) 1 ( 1 R bs ) 2 ,
S Sat = S LOFI ( R sat , F e ) = R bs p out 2 .
G ( F R ) = γ c γ 1 η .
PD Laser ( F ) = 2 p out γ c 2 4 π 2 ( Δ F R 2 + F 2 ) ( F R 2 F 2 ) 2 + Δ F R 2 F 2 = 2 p out ( t ) G 2 ( F ) .
N Laser 2 ( F e , Δ F ) = 2 R bs F e Δ F / 2 F e + Δ F / 2 PD Laser ( F ) d F .
N Laser ( F e , Δ F ) = 2 R bs p out ( t ) G ( F e ) Δ F .
S LOFI ( R e , F e ) N Laser ( F e , Δ F ) = R bs p out 2 Δ F R e ( 1 R bs ) ,
R Laser ( Δ F ) = 2 Δ F R bs p out 1 ( 1 R bs ) 2 = 3 × 10 13 .
R Laser ( 1 R bs ) 2 p out 2 Δ F = 1 R bs .
N Detector 2 ( Δ F ) N Laser 2 ( F e , Δ F ) S LOFI 2 ( R e , F e ) S Sat 2 ,
N Detector ( Δ F ) = [ ( 6 × 10 9   W / Hz ) h c / λ ] 2 Δ F .
R Detector ( F e , Δ F ) R Laser ( Δ F ) R e R Sat ( F e ) ,
R Detector ( F e , Δ F ) = 1 2 1 G 2 ( F e ) 1 ( 1 R bs ) 2 R bs 2 N detector 2 ( Δ F ) p out 2 .
R Laser ( 600   Hz ) = 2 × 10 13 R e R Sat ( Ω + ) = 4.5 × 10 5 .
R Laser ( 600   Hz ) = 2 × 10 13 R e R Sat ( F R ) = 4 × 10 8 .
R detector ( 1.5 F R , 600   Hz ) = 3 × 10 13 R e R sat ( 1.5 F R ) = 7 × 10 5 .
N Detector 2 ( Δ F ) = N Laser 2 ( F R , Δ F ) ,
R Detector ( F R , Δ F ) = R Laser ( Δ F ) .
G opt ( F R ) = ( γ c γ 1 η ) opt = N Detector ( Δ F ) 2 R bs p out Δ F = 1 2 R bs [ ( 6 × 10 9   W / Hz ) h c / λ ] p out .
R Laser ( 600   Hz ) = 2 × 10 13 R e R sat ( F R ) = 4.5 × 10 5 ,
R Laser ( 600   Hz ) = 2 × 10 13 R e R sat ( F R ) = 4 × 10 12 .
p ref = R bs ( 1 R bs ) p out ,
p sig = R bs ( 1 R bs ) R e p out ,
S Hetero ( R e ) = 2 p ref p sig 2 = 2 R bs ( 1 R bs ) R e p out 2 ,
S Hetero ( R e ) N Detector ( Δ F ) = R bs ( 1 R bs ) R e p out ( 6 × 10 9   W / Hz h c / λ ) Δ F .
S LOFI ( R e , F e ) N Laser ( F e , Δ F ) S Hetero ( R e ) N Detector ( Δ F ) = 1 2 R bs [ ( 6 × 10 9   W / Hz ) h c / λ ] p out = G opt ( Ω R ) .
S LOFI 2 ( R e = 2 × 10 11 , F e ) N Laser 2 ( F e , Δ F = 600   Hz ) = + 20   dB ,
S Hetero 2 ( R e = 2 × 10 11 ) N Detector 2 ( Δ F = 600   Hz ) = 23.3   dB ,
S LOFI ( R e , F e ) N Laser ( F e , Δ F ) S Hetero ( R e ) N Detector ( Δ F ) = 21.65   dB = 146 = G opt ( Ω R ) ,

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