Abstract

We present a robust sideband laser locking technique ideally suited for applications requiring low probe power and heterodyne readout. By feeding back to a high-bandwidth voltage-controlled oscillator, we lock a first-order phase-modulation sideband to a high-finesse Fabry-Perot cavity in ambient conditions, achieving a closed-loop bandwidth of 3.5 MHz (with a single integrator) limited fundamentally by the signal delay. The measured transfer function of the closed loop agrees with a simple model based on ideal system components, and from this we suggest a modified design that should achieve a bandwidth exceeding 6 MHz with a near-causally limited feedback gain as high as 4 × 107 at 1 kHz. The off-resonance optical carrier enables alignment-free heterodyne readout, alleviating the need for additional lasers or optical modulators.

© 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]

2016 (4)

B. P. Abbott and Others, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

C. Reinhardt, T. Muller, A. Bourassa, and J. C. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

R. A. Norte, J. P. Moura, and S. Groblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref] [PubMed]

J. F. S. Brachmann, H. Kaupp, T. W. Hansch, and D. Hunger, “Photothermal effects in ultra-precisely stabilized tunable microcavities,” Opt. Express 24, 21205 (2016).
[Crossref] [PubMed]

2015 (3)

D. Gatti, R. Gotti, T. Sala, N. Coluccelli, M. Belmonte, M. Prevedelli, P. Laporta, and M. Marangoni, “Wide-bandwidth Pound-Drever-Hall locking through a single-sideband modulator,” Opt. Lett. 40, 5176 (2015).
[Crossref] [PubMed]

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

2014 (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

2012 (4)

S. L. Danilishin and F. Y. Khalili, “Quantum measurement theory in gravitational-wave detectors,” Living Rev. Relativ. 15, 1–147 (2012).
[Crossref]

R. Kohlhaas, T. Vanderbruggen, S. Bernon, A. Bertoldi, A. Landragin, and P. Bouyer, “Robust laser frequency stabilization by serrodyne modulation,” Opt. Lett. 37, 1005 (2012).
[Crossref] [PubMed]

A. H. Safavi Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

2010 (2)

C. Chou, D. Hume, J. Koelemeij, D. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al + optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

T. C. Briles, D. C. Yost, A. Cingoz, J. Ye, and T. R. Schibli, “Simple piezoelectric-actuated mirror with 180 kHz servo bandwidth,” Opt. Express 18, 9739 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

2005 (1)

J. Bechhoefer, “Feedback for physicists: A tutorial essay on control,” Rev. Mod. Phys. 77, 783–836 (2005).
[Crossref]

2003 (1)

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281–324 (2003).
[Crossref]

2002 (1)

M. Rakhmanov, R. Savage, D. Reitze, and D. Tanner, “Dynamic resonance of light in Fabry-Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[Crossref]

2001 (2)

1999 (2)

1998 (1)

1984 (1)

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophysics Laser Chem. 31, 97–105 (1983).
[Crossref]

1982 (1)

1980 (3)

T. W. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reference cavity,” Opt. Commun. 35, 441–444 (1980).
[Crossref]

K. Dschao, M. Glaser, and J. Helmcke, “I2 Stabilized He-Ne Lasers at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 354–357 (1980).
[Crossref]

P. Cerez, A. Brillet, C. N. Man-Pichot, and R. Felder, “He-Ne Lasers Stabilized by Saturated Absorption in Iodine at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 352–354 (1980).
[Crossref]

1973 (1)

R. L. Barger, “Frequency stabilization of a cw dye laser,” Appl. Phys. Lett. 22, 573 (1973).
[Crossref]

1946 (1)

R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. 17, 490 (1946).
[Crossref] [PubMed]

Abbott, B. P.

B. P. Abbott and Others, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Barger, R. L.

R. L. Barger, “Frequency stabilization of a cw dye laser,” Appl. Phys. Lett. 22, 573 (1973).
[Crossref]

Bechhoefer, J.

J. Bechhoefer, “Feedback for physicists: A tutorial essay on control,” Rev. Mod. Phys. 77, 783–836 (2005).
[Crossref]

Belmonte, M.

Bergquist, J. C.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Bernon, S.

Bertoldi, A.

Black, E. D.

E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys 69, 79–87 (2001).
[Crossref]

Blatt, R.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281–324 (2003).
[Crossref]

Borkje, K.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Bourassa, A.

C. Reinhardt, T. Muller, A. Bourassa, and J. C. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

Bouyer, P.

Brachmann, J. F. S.

Briles, T. C.

Brillet, A.

P. Cerez, A. Brillet, C. N. Man-Pichot, and R. Felder, “He-Ne Lasers Stabilized by Saturated Absorption in Iodine at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 352–354 (1980).
[Crossref]

Brusch, A.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Byer, R. L.

Cerez, P.

P. Cerez, A. Brillet, C. N. Man-Pichot, and R. Felder, “He-Ne Lasers Stabilized by Saturated Absorption in Iodine at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 352–354 (1980).
[Crossref]

Chan, C.

Chan, J.

A. H. Safavi Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

Chen, L.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

Chou, C.

C. Chou, D. Hume, J. Koelemeij, D. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al + optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

Chou, C. W.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Cicak, K.

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

Cingoz, A.

Coluccelli, N.

Couillaud, B.

T. W. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reference cavity,” Opt. Commun. 35, 441–444 (1980).
[Crossref]

Danilishin, S. L.

S. L. Danilishin and F. Y. Khalili, “Quantum measurement theory in gravitational-wave detectors,” Living Rev. Relativ. 15, 1–147 (2012).
[Crossref]

Diddams, S. A.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophysics Laser Chem. 31, 97–105 (1983).
[Crossref]

Drullinger, R. E.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Dschao, K.

K. Dschao, M. Glaser, and J. Helmcke, “I2 Stabilized He-Ne Lasers at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 354–357 (1980).
[Crossref]

Felder, R.

P. Cerez, A. Brillet, C. N. Man-Pichot, and R. Felder, “He-Ne Lasers Stabilized by Saturated Absorption in Iodine at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 352–354 (1980).
[Crossref]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophysics Laser Chem. 31, 97–105 (1983).
[Crossref]

Fortier, T. M.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Gatti, D.

Gilbert, S. L.

Girvin, S. M.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Glaser, M.

K. Dschao, M. Glaser, and J. Helmcke, “I2 Stabilized He-Ne Lasers at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 354–357 (1980).
[Crossref]

Gotti, R.

Gray, M. B.

Grebing, C.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

Groblacher, S.

R. A. Norte, J. P. Moura, and S. Groblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref] [PubMed]

Grunert, J.

Gustafson, E. K.

Hagemann, C.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

Hall, J. L.

J. Ye, L. S. Ma, and J. L. Hall, “Ultrasensitive detections in atomic and Mol. Phys.: demonstration in molecular overtone spectroscopy,” J. Opt. Soc. Am. B 15, 6 (1998).
[Crossref]

J. L. Hall and T. W. Hansch, “External dye-laser frequency stabilizer,” Opt. Lett. 9, 502 (1984).
[Crossref] [PubMed]

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophysics Laser Chem. 31, 97–105 (1983).
[Crossref]

Hansch, T. W.

Harris, J. G. E.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Helmcke, J.

K. Dschao, M. Glaser, and J. Helmcke, “I2 Stabilized He-Ne Lasers at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 354–357 (1980).
[Crossref]

Hemmerich, A.

Heurs, M.

M. Heurs, I. R. Petersen, M. R. James, and E. H. Huntington, “Homodyne locking of a squeezer,” in 2010 Conference on Lasers and Electro-Optics (CLEO) and Quant. Electron. and Laser Science Conference (QELS, 2010), pp. 1–2.

Hill, J. T.

A. H. Safavi Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophysics Laser Chem. 31, 97–105 (1983).
[Crossref]

Houtz, R.

Hume, D.

C. Chou, D. Hume, J. Koelemeij, D. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al + optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

Hume, D. B.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Hunger, D.

Huntington, E. H.

M. Heurs, I. R. Petersen, M. R. James, and E. H. Huntington, “Homodyne locking of a squeezer,” in 2010 Conference on Lasers and Electro-Optics (CLEO) and Quant. Electron. and Laser Science Conference (QELS, 2010), pp. 1–2.

Husman, M. E.

Itano, W. M.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

James, M. R.

M. Heurs, I. R. Petersen, M. R. James, and E. H. Huntington, “Homodyne locking of a squeezer,” in 2010 Conference on Lasers and Electro-Optics (CLEO) and Quant. Electron. and Laser Science Conference (QELS, 2010), pp. 1–2.

Jiang, L.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Kampel, N. S.

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

Kaupp, H.

Kessler, T.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

Khalili, F. Y.

S. L. Danilishin and F. Y. Khalili, “Quantum measurement theory in gravitational-wave detectors,” Living Rev. Relativ. 15, 1–147 (2012).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Koelemeij, J.

C. Chou, D. Hume, J. Koelemeij, D. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al + optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

Kohlhaas, R.

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophysics Laser Chem. 31, 97–105 (1983).
[Crossref]

Krause, A.

A. H. Safavi Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

Landragin, A.

Laporta, P.

Lawrence, M. J.

Lee, D.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Legero, T.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

Leibfried, D.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281–324 (2003).
[Crossref]

Lorini, L.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Ma, L. S.

Man-Pichot, C. N.

P. Cerez, A. Brillet, C. N. Man-Pichot, and R. Felder, “He-Ne Lasers Stabilized by Saturated Absorption in Iodine at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 352–354 (1980).
[Crossref]

Marangoni, M.

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Martin, M. J.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

Mason, D.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Mayer Alegre, T. P.

A. H. Safavi Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

McClelland, D. E.

Monroe, C.

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281–324 (2003).
[Crossref]

Moura, J. P.

R. A. Norte, J. P. Moura, and S. Groblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref] [PubMed]

Muller, H.

Muller, T.

C. Reinhardt, T. Muller, A. Bourassa, and J. C. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophysics Laser Chem. 31, 97–105 (1983).
[Crossref]

Newbury, N. R.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Norte, R. A.

R. A. Norte, J. P. Moura, and S. Groblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref] [PubMed]

Oskay, W. H.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Painter, O.

A. H. Safavi Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

Petersen, I. R.

M. Heurs, I. R. Petersen, M. R. James, and E. H. Huntington, “Homodyne locking of a squeezer,” in 2010 Conference on Lasers and Electro-Optics (CLEO) and Quant. Electron. and Laser Science Conference (QELS, 2010), pp. 1–2.

Peterson, R. W.

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

Pound, R. V.

R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. 17, 490 (1946).
[Crossref] [PubMed]

Prevedelli, M.

Protopopov, V. V.

V. V. Protopopov, Laser Heterodyning, vol. 149 of Springer Ser. Opt. Sci. (SpringerBerlin Heidelberg, Berlin, Heidelberg, 2009).
[Crossref]

Purdy, T. P.

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

Rakhmanov, M.

M. Rakhmanov, R. Savage, D. Reitze, and D. Tanner, “Dynamic resonance of light in Fabry-Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[Crossref]

Regal, C. A.

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

Reinhardt, C.

C. Reinhardt, T. Muller, A. Bourassa, and J. C. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

Reitze, D.

M. Rakhmanov, R. Savage, D. Reitze, and D. Tanner, “Dynamic resonance of light in Fabry-Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[Crossref]

Riehle, F.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

Ritter, S.

Rosenband, T.

C. Chou, D. Hume, J. Koelemeij, D. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al + optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Safavi Naeini, A. H.

A. H. Safavi Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

Sala, T.

Sankey, J. C.

C. Reinhardt, T. Muller, A. Bourassa, and J. C. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

Savage, R.

M. Rakhmanov, R. Savage, D. Reitze, and D. Tanner, “Dynamic resonance of light in Fabry-Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[Crossref]

Schibli, T. R.

Schmidt, P. O.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Schoof, A.

Shaddock, D. A.

Shkarin, A. B.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Simmonds, R. W.

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

Stalnaker, J. E.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Sterr, U.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

Swann, W. C.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Tanner, D.

M. Rakhmanov, R. Savage, D. Reitze, and D. Tanner, “Dynamic resonance of light in Fabry-Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[Crossref]

Underwood, M.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Vanderbruggen, T.

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophysics Laser Chem. 31, 97–105 (1983).
[Crossref]

Wieman, C. E.

Willke, B.

Wineland, D.

C. Chou, D. Hume, J. Koelemeij, D. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al + optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281–324 (2003).
[Crossref]

Wineland, D. J.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Xu, H.

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Ye, J.

Yost, D. C.

Yu, P.-L.

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

Am. J. Phys (1)

E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys 69, 79–87 (2001).
[Crossref]

Appl. Phys. B Photophysics Laser Chem. (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophysics Laser Chem. 31, 97–105 (1983).
[Crossref]

Appl. Phys. Lett. (1)

R. L. Barger, “Frequency stabilization of a cw dye laser,” Appl. Phys. Lett. 22, 573 (1973).
[Crossref]

IEEE Trans. Instrumentation Measurement (2)

K. Dschao, M. Glaser, and J. Helmcke, “I2 Stabilized He-Ne Lasers at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 354–357 (1980).
[Crossref]

P. Cerez, A. Brillet, C. N. Man-Pichot, and R. Felder, “He-Ne Lasers Stabilized by Saturated Absorption in Iodine at 612 nm,” IEEE Trans. Instrumentation Measurement 29, 352–354 (1980).
[Crossref]

J. Opt. Soc. Am. B (2)

Living Rev. Relativ. (1)

S. L. Danilishin and F. Y. Khalili, “Quantum measurement theory in gravitational-wave detectors,” Living Rev. Relativ. 15, 1–147 (2012).
[Crossref]

Nat. Photonics (1)

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[Crossref]

Opt. Commun. (1)

T. W. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reference cavity,” Opt. Commun. 35, 441–444 (1980).
[Crossref]

Opt. Express (3)

Opt. Lett. (6)

Phys. Lett. A (1)

M. Rakhmanov, R. Savage, D. Reitze, and D. Tanner, “Dynamic resonance of light in Fabry-Perot cavities,” Phys. Lett. A 305, 239–244 (2002).
[Crossref]

Phys. Rev. A (2)

T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).
[Crossref]

M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris, “Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime,” Phys. Rev. A 92, 061801 (2015).
[Crossref]

Phys. Rev. Lett. (4)

A. H. Safavi Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

R. A. Norte, J. P. Moura, and S. Groblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref] [PubMed]

C. Chou, D. Hume, J. Koelemeij, D. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al + optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

B. P. Abbott and Others, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

Phys. Rev. X (1)

C. Reinhardt, T. Muller, A. Bourassa, and J. C. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).

Rev. Mod. Phys. (3)

J. Bechhoefer, “Feedback for physicists: A tutorial essay on control,” Rev. Mod. Phys. 77, 783–836 (2005).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281–324 (2003).
[Crossref]

Rev. Sci. Instrum. (1)

R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. 17, 490 (1946).
[Crossref] [PubMed]

Science (1)

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808 (2008).
[Crossref] [PubMed]

Other (2)

V. V. Protopopov, Laser Heterodyning, vol. 149 of Springer Ser. Opt. Sci. (SpringerBerlin Heidelberg, Berlin, Heidelberg, 2009).
[Crossref]

M. Heurs, I. R. Petersen, M. R. James, and E. H. Huntington, “Homodyne locking of a squeezer,” in 2010 Conference on Lasers and Electro-Optics (CLEO) and Quant. Electron. and Laser Science Conference (QELS, 2010), pp. 1–2.

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Figures (4)

Fig. 1
Fig. 1 Feedback stabilization. (a) Generic control loop for stabilizing a laser’s detuning δ from the resonance of an optical cavity. Noise δn enters, is converted to an optical signal by the cavity (transfer function C), collected by a diode (D), manipulated by electronics (−A) and sent to a “feedback” port (F). (b) Practical implementation using Pound-Drever-Hall readout. Straight red lines represent optical paths, straight black lines represent electrical paths, and dashed gray lines show potential feedback paths. The laser is phase-modulated, lands on a beam splitter (BS) and interacts with the cavity, which converts phase to amplitude modulation. This is recorded with a photodiode and mixed (demodulated) with a local oscillator. Inset shows the resulting steady-state voltage VY (δ), with a red dot indicating a stable lock point. The manipulated signal can be fed back to (i) the cavity length or (ii) the laser frequency. Feeding back to (iii) the oscillator frequency only adjusts the sidebands.
Fig. 2
Fig. 2 Sideband locking with heterodyne readout. Many parts are from Thorlabs (TL), Newport (NP) and Minicircuits (MC). (a) A VCO (MC ZX95-1600W-S+) signal is split (MC ZX-10-2-20-S+) and amplified (MC ZX60-4016E-S+) feeding an EOM (TL LN65-10-P-A-A-BNL-Kr with shortened output fiber) and the LO (“L”) ports of two mixers (MC ZFM-5X-S) for quadrature readout. Laser (Koheras Adjustik E15) feeds a 14.2-dBm (26.3 mW) carrier through the EOM, producing a 10.8 dBm carrier and -2.2 dBm (5%) sidebands, set by a variable attenuator (MC ZX73-2500-S+, ~15 dB) leading to the EOM. Once collimated (TL F260APC-1550), the beam passes through a beam splitter (BS, TL BS018 50:50), mode-matching lenses (−5 cm and 10 cm focal length, see (b)), and steering mirrors (M) before landing on a cavity comprising a flat (NP 10CM00SR.70F) and curved (NP 10CV00SR.70F) supermirror, the second of whose position is swept by a piezo mirror mount (TL K1PZ). The transmitted beam is focused on a photodiode (PD, TL PDA10CF), while the reflected beam is rerouted by the BS, passes through an isolator (TL IO-2.5-1550-VLP), and is focused upon a 2-GHz photodiode (PD, Femto HSA-X-S-2G-IN). Low-frequencies signals (< 20 MHz) are eliminated with a high-pass (MC SHP-20+), before amplification (MC ZX60-P105LN+) and splitting by a π/2 splitter (MC ZX10Q-2-13-S+). The phase-shifted signals are fed to the mixers’ RF (R) ports and demodulated to the IF (I) ports. The “phase” quadrature (VY) is fed to a PI amplifier (−A, NP LB1005) for feedback to the VCO. Inset shows the steady-state voltage of the “amplitude” quadrature VX (δ). (b) Photograph of optics. (c) Simultaneously acquired VX and VY, for three different VCO controls: 0 V (lightest), 0.8 V, and 1.8 V (darkest).
Fig. 3
Fig. 3 (a) Detuning noise power spectral density (PSD) before and after lock, recorded while locked. The pre-feedback noise (red) is inferred from the proportional-integral (PI) amplifier output and the VCO conversion factor 52 MHz/V, while the post-feedback noise (blue) is inferred from the PI output referred back to its input and the independently measured slope of the error function (388 ± 40 mV/MHz) at the lock point. (b) Measured (blue) and modeled (red) closed-loop transfer function. The model includes the cavity (green, ring-down time τ = 1.1 µs), PI amplifier (yellow, ωPI =110 kHz, and g=105), and a delay (brown, 70 ns, 52 ns from the PI amplifier). Transfer functions of other components are assumed to be “flat” on this scale. The gray dashed line shows a closed-loop gain that could be achieved with optimizations: replacement of the PI amplifier and further delay reductions to 10 ns and two PI filters, one with ωPI/2π = 70 kHz, 1/g = 0), and the other with ωPI/2π = 15 MHz and g = 105.
Fig. 4
Fig. 4 Quadrature amplitudes of the photodiode signal for 11 different VCO voltages.

Equations (44)

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δ = δ n 1 + C D A F .
r ( δ ) t 1 2 τ c / L 1 + i 2 τ δ 1 .
E = E l cos ( ω l t + ϕ e sin ω e t + ϕ n sin ω n t )
V ˜ Y Ω ˜ 2 β τ 2 E l 2 ϕ e 1 + 2 i τ ω n
A P I = G 0 1 + i ω n / ω P I 1 / g + i ω n / ω P I
V ˜ Y Ω ˜ A P I = β τ 2 E l 2 ϕ e G 0 i τ ω n .
V ˜ Y , ± Ω ˜ β τ 2 E l 2 ϕ e 1 + 2 i τ ω n
V ˜ X , ± ϵ ˜ β τ E l 2 ϕ e 1 + i τ ω n 1 + 2 i τ ω n
E r E i n = r 1 + t 1 2 r 2 e 2 i ω l L / c + t 1 2 r 2 e 2 i ω l L / c ( r 1 r 2 e 2 i ω l L / c ) + t 1 2 r 2 e 2 i ω l L / c ( r 1 r 2 e 2 i ω l L / c ) 2 +
= r 1 + t 1 2 r 2 e 2 i ω l L / c 1 r 1 r 2 e 2 i ω l L / c
r ( δ ) E r E i n t 1 2 τ c / L 1 + i 2 τ δ 1 ,
C a , S a cos ( ω a t ) , sin ( ω a t )
C a ± b , S a ± b cos ( ω a t ± ω b t ) , sin ( ω a t ± ω b t )
C a ± b ψ , S a ± b ψ cos ( ω a t ± ( ω b t + ψ ) ) , sin ( ω a t ± ( ω b t + ψ ) )
C a ± 2 b ψ , S a ± 2 b ψ cos ( ω a t ± ( ω b t + ψ ) ) , sin ( ω a t ± ( 2 ω b t + ψ ) )
ϵ ( t ) = ϵ n cos ( ω n t )
Ω ( t ) = Ω n cos ( ω n t )
E ( t ) = P l ( 1 + ϵ n C n + ϵ e S e ) cos ( ω l t + ϕ n S n + ϕ e S e ) ,
E ( t ) P l ( 1 1 4 ϕ e 2 1 4 ϕ n 2 ) C l + 1 2 ϵ n ( C l + n + C l n ) + 1 2 ϕ n ( C l + n C l n ) + 1 2 ϵ e ( S l + e S l e ) + 1 2 ϕ e ( C l + e C l e ) + 1 8 ϕ n 2 ( C l + 2 n + C l 2 n ) + 1 4 ϕ n ϵ n ( C l + 2 n C l 2 n ) + 1 8 ϕ e 2 ( C l + 2 e + C l 2 e ) + 1 4 ϕ e ϵ e ( 2 S l + S l + 2 e + S l 2 e ) + 1 4 ϕ e ϕ n ( C l + e + n C l + e n C l e + n + C l e n ) + 1 4 ϕ n ϵ e ( S l + e + n S l + e n S l e + n + S l e n ) + 1 4 ϕ e ϵ n ( C l + e + n + C l + e n C l e + n C l e n ) .
E ( t ) r 1 P l = ( 1 1 4 ϕ e 2 1 4 ϕ n 2 ) ρ A 0 C l + 1 2 ϵ n ( ρ A n C l + n ψ A + ρ A n C l n ψ A ) + 1 2 ϕ n ( ρ A n C l + n ψ A ρ A n C l n ψ A ) + 1 2 ϵ e ( ρ B 0 S l + e α S l e ) + 1 2 ϕ e ( ρ B 0 C l + e α C l e ) + 1 8 ϕ n 2 ( ρ A 2 n C l + 2 n ϕ A + ρ A 2 n C l 2 n ϕ A ) + 1 4 ϕ n ϵ n ( ρ A 2 n C l + 2 n ϕ A ρ A 2 n C l 2 n ϕ A ) + 1 8 ϕ e 2 ( C l + 2 e + C l 2 e ) + 1 4 ϕ e ϵ e ( 2 S l + S l + 2 e + S l 2 e ) + 1 4 ϕ e ϕ n ( ρ B n C l + e + n ψ B ρ B n C l + e n ψ B α C l e + n + α C l e n ) + 1 4 ϕ n ϵ e ( ρ B n S l + e + n ψ B ρ B n S l + e n ψ B α S l e + n + α S l e n ) + 1 4 ϕ e ϵ n ( ρ B n C l + e + n ψ B + ρ B n C l + e n ψ B α C l e + n α C l e n ) .
ρ N ( δ ) 1 t 1 2 τ c / L 1 + i 2 τ δ
ρ N 0 ρ N ( 0 )
ρ N n | ρ N ( ω n ) |
ψ N arg [ ρ ( ω n ) ] = arg [ ρ ( ω n ) ]
ρ N 2 n | ρ N ( 2 ω n ) |
ϕ N arg [ ρ ( 2 ω n ) ] = arg [ ρ ( 2 ω n ) ]
4 V P D ( t ) η P D r 1 2 P l + ρ A 0 ϕ e ϕ n ( ρ B n C e + n ψ B ρ B n C e n ψ B + α C e + n α C e n ) + ρ A 0 ϵ e ϕ n ( ρ B n S e + n ψ B ρ B n S e n ψ B + α S e n α S e + n ) + ρ A 0 ϕ e ϵ n ( ρ B n C e + n ψ B + ρ B n C e n ψ B α C e n α C e + n ) + ρ A n ϵ e ϵ n ( ρ B 0 + α ) ( S e n ψ A + S e + n ψ A ) + ρ A n ϕ e ϵ n ( ρ B 0 α ) ( C e n ψ A + C e + n ψ A ) + ρ A n ϵ e ϕ n ( ρ B 0 α ) ( S e n ψ A S e + n ψ A ) + ρ A n ϕ e ϕ n ( ρ B 0 + α ) ( C e n ψ A C e + n ψ A )
4 V Y ( t ) η M η P D r 1 2 P l ϕ e ϕ n ρ A 0 ( ρ B n S n ψ B + α S n ) + ϕ e ϕ n ( ρ B 0 + α ) ρ A n S n ψ A + ϵ e ϵ n ( ρ B 0 + α ) ρ A n C n ψ A
4 V X ( t ) η M η P D r 1 2 P l ϵ e ϕ n ρ A 0 ( ρ B n α ) S n ψ B + ϕ e ϵ n ρ A 0 ( ρ B n C n ψ B α C n ) + ϕ e ϵ n ρ A n ( ρ B 0 α ) C n ψ A ϵ e ϕ n ρ A n ( ρ B 0 α ) S n ψ A .
V Y ( t ) P l P e 8 η M η P D r 1 2 ϕ n ( 1 + α ) ( ρ A n S n ψ A ρ A 0 S n )
V X ( t ) 0 .
V Y = P l P e 2 η M η P D r 1 2 ϕ n ( ρ A n S n ψ A ρ A 0 S n ) .
V ˜ Y i P l P e 2 η M η P D r 1 2 ϕ n ( ρ A ( ω n ) e i ω n t ρ A ( 0 ) e i ω n t ) .
V ˜ Y Ω ˜ = 2 P l P e η M η P D r 1 2 t 1 2 c τ 2 / L 1 + i 2 τ ω n = 2 ϕ e E l 2 β τ 2 1 + i 2 τ ω n .
V Y ( t ) P l P e 8 η M η P D r 1 2 ϕ n ( ρ B n S n ψ B ρ B 0 S n )
V X ( t ) 0 .
V ˜ Y Ω ˜ = P l P e 2 η M η P D c τ 2 r 1 2 t 1 2 / L 1 + i 2 τ ω n = ϕ e E l 2 β τ 2 1 + i 2 τ ω n .  
V Y ( t ) 0
V X ( t ) P l P e 8 η M η P D r 1 2 ϵ n ( 1 α ) ( ρ A 0 C n + ρ A n C n ψ A )
V ˜ X ϵ ˜ = P l P e 2 η M η P D r 1 2 ( 1 α ) ( 1 c τ t 1 2 L ( 1 + i τ ω n 1 + i 2 τ ω n ) ) .
V Y ( t ) 0
V X ( t ) P l P e 8 η M η P D r 1 2 ϵ n ( ρ B n C n ψ B + ρ B 0 C n 2 α C n )
V ˜ X ϵ ˜ = P l P e 2 η M η P D r 1 2 ( 1 α c τ t 1 2 L ( 1 + i τ ω n 1 + i 2 τ ω n ) )
V ˜ X ϵ ˜ = ϕ e E l 2 β τ ( 1 + i τ ω n 1 + i 2 τ ω n ) .

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