Abstract

We report on a simple method of locking a laser to a birefringent cavity using polarization spectroscopy. The birefringence of the resonator permits the simple extraction of an error signal by using one polarization state as a phase reference for another state. No modulation of the light or the resonator is required, reducing the complexity of the laser locking setup. This method of producing an error signal can be used on most birefringent optical resonators, even if the details of birefringence and eigenpolarizations are not known. This technique is particularly well suited for fiber ring resonators due to the inherent birefringence of the fiber and the unknown nature of that birefringence. We present an experimental demonstration of this technique using a fiber ring.

© 2015 Optical Society of America

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References

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2014 (1)

2013 (1)

G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, “Tenfold reduction of Brownian noise in high-reflectivity optical coatings,” Nature Photon. 7, 644–650 (2013).
[Crossref]

2012 (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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

2011 (3)

2009 (2)

Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. 282, 3108–3112 (2009).
[Crossref]

S. Moriwaki, T. Mori, K. Takeno, and N. Mio, “Frequency discrimination method making use of polarization selectivity of triangular optical cavity,” Appl. Phys. Express 2, 016501 (2009).
[Crossref]

2008 (3)

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

P.-H. Merrer, O. Llopis, and G. Cibiel, “Laser stabilization on a fiber ring resonator and application to RF filtering,” IEEE Photon. Technol. Lett. 20, 1399–1401 (2008).
[Crossref]

Z. Xiao-Guang and Z. Yuan, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17, 2509–2513 (2008).
[Crossref]

2007 (1)

2005 (1)

2003 (1)

M. D. Harvey and A. G. White, “Frequency locking by analysis of orthogonal modes,” Opt. Commun. 221, 163–171 (2003).
[Crossref]

2002 (1)

W. F. Brinkman, T. L. Koch, D. V. Lang, and D. P. Wilt, “The lasers behind the communications revolution,” Bell Labs Techn. J. 5, 150–167 (2002).
[Crossref]

1999 (1)

1997 (1)

J. Boon-Engering, W. van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140, 285–288 (1997).
[Crossref]

1993 (1)

1988 (1)

1986 (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 31, 97–105 (1983).
[Crossref]

1982 (3)

1980 (1)

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

1972 (1)

E. Ippen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

1946 (1)

R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Scientific Instrum. 17, 490–505 (1946).
[Crossref]

Anetsberger, G.

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

Arcizet, O.

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

Arndt, M.

Asenbaum, P.

Aspelmeyer, M.

G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, “Tenfold reduction of Brownian noise in high-reflectivity optical coatings,” Nature Photon. 7, 644–650 (2013).
[Crossref]

Baumgartel, L. M.

Bente, E.

J. Boon-Engering, W. van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140, 285–288 (1997).
[Crossref]

Bergh, R. a.

Boon-Engering, J.

J. Boon-Engering, W. van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140, 285–288 (1997).
[Crossref]

Brinkman, W. F.

W. F. Brinkman, T. L. Koch, D. V. Lang, and D. P. Wilt, “The lasers behind the communications revolution,” Bell Labs Techn. J. 5, 150–167 (2002).
[Crossref]

Burns, W.

W. Burns, “Polarization characteristics of single-mode fiber couplers,” IEEE Trans. Microwave Theory Tech. 30, 1577–1588 (1982).
[Crossref]

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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

Cibiel, G.

P.-H. Merrer, O. Llopis, and G. Cibiel, “Laser stabilization on a fiber ring resonator and application to RF filtering,” IEEE Photon. Technol. Lett. 20, 1399–1401 (2008).
[Crossref]

Cole, G. D.

G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, “Tenfold reduction of Brownian noise in high-reflectivity optical coatings,” Nature Photon. 7, 644–650 (2013).
[Crossref]

Couillaud, B.

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

De Natale, P.

De Nicola, S.

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 31, 97–105 (1983).
[Crossref]

Evans, M.

Ferraro, P.

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 31, 97–105 (1983).
[Crossref]

Fukuda, M.

Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. 282, 3108–3112 (2009).
[Crossref]

Gagliardi, G.

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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

Grudinin, I. S.

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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

Hall, J. L.

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 31, 97–105 (1983).
[Crossref]

Hänsch, T.

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

Harvey, M. D.

M. D. Harvey and A. G. White, “Frequency locking by analysis of orthogonal modes,” Opt. Commun. 221, 163–171 (2003).
[Crossref]

Hecht, E.

E. Hecht, Optics (Pearson Education, 2008).

Higashiguchi, M.

Hogervorst, W.

J. Boon-Engering, W. van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140, 285–288 (1997).
[Crossref]

Honda, Y.

Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. 282, 3108–3112 (2009).
[Crossref]

Hotate, K.

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 31, 97–105 (1983).
[Crossref]

Ippen, E.

E. Ippen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

Iwatsuki, K.

Kappe, P.

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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

Kippenberg, T. J.

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

Koch, T. L.

W. F. Brinkman, T. L. Koch, D. V. Lang, and D. P. Wilt, “The lasers behind the communications revolution,” Bell Labs Techn. J. 5, 150–167 (2002).
[Crossref]

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 31, 97–105 (1983).
[Crossref]

Lamouroux, B.

Lang, D. V.

W. F. Brinkman, T. L. Koch, D. V. Lang, and D. P. Wilt, “The lasers behind the communications revolution,” Bell Labs Techn. J. 5, 150–167 (2002).
[Crossref]

Lefevre, H. C.

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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

Lit, J. W. Y.

Llopis, O.

P.-H. Merrer, O. Llopis, and G. Cibiel, “Laser stabilization on a fiber ring resonator and application to RF filtering,” IEEE Photon. Technol. Lett. 20, 1399–1401 (2008).
[Crossref]

Martin, M. J.

G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, “Tenfold reduction of Brownian noise in high-reflectivity optical coatings,” Nature Photon. 7, 644–650 (2013).
[Crossref]

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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

McClelland, D. E.

Menzel, R.

Merrer, P.-H.

P.-H. Merrer, O. Llopis, and G. Cibiel, “Laser stabilization on a fiber ring resonator and application to RF filtering,” IEEE Photon. Technol. Lett. 20, 1399–1401 (2008).
[Crossref]

Miller, J.

Mio, N.

S. Moriwaki, T. Mori, K. Takeno, and N. Mio, “Frequency discrimination method making use of polarization selectivity of triangular optical cavity,” Appl. Phys. Express 2, 016501 (2009).
[Crossref]

Mori, T.

S. Moriwaki, T. Mori, K. Takeno, and N. Mio, “Frequency discrimination method making use of polarization selectivity of triangular optical cavity,” Appl. Phys. Express 2, 016501 (2009).
[Crossref]

Moriwaki, S.

S. Moriwaki, T. Mori, K. Takeno, and N. Mio, “Frequency discrimination method making use of polarization selectivity of triangular optical cavity,” Appl. Phys. Express 2, 016501 (2009).
[Crossref]

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 31, 97–105 (1983).
[Crossref]

Omori, T.

Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. 282, 3108–3112 (2009).
[Crossref]

Orszag, A.

Ostermeyer, M.

Paul, T. J.

Pound, R. V.

R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Scientific Instrum. 17, 490–505 (1946).
[Crossref]

Prade, B.

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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

Rivière, R.

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

Sakai, H.

Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. 282, 3108–3112 (2009).
[Crossref]

Sakaue, K.

Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. 282, 3108–3112 (2009).
[Crossref]

Sasao, N.

Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. 282, 3108–3112 (2009).
[Crossref]

Schliesser, A.

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
[Crossref]

Shaddock, D. A.

Shaw, H. J.

Shimizu, H.

Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. 282, 3108–3112 (2009).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

Strekalov, D. V.

Swanson, E. A.

Takeno, K.

S. Moriwaki, T. Mori, K. Takeno, and N. Mio, “Frequency discrimination method making use of polarization selectivity of triangular optical cavity,” Appl. Phys. Express 2, 016501 (2009).
[Crossref]

Thompson, R. J.

Urakawa, J.

Y. Honda, H. Shimizu, M. Fukuda, T. Omori, J. Urakawa, K. Sakaue, H. Sakai, and N. Sasao, “Stabilization of a non-planar optical cavity using its polarization property,” Opt. Commun. 282, 3108–3112 (2009).
[Crossref]

van der Veer, W.

J. Boon-Engering, W. van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140, 285–288 (1997).
[Crossref]

Wang, W.

W. Wang and J. Xia, “The characterizations of polarization in resonator integrated optic gyroscope,” Opt. Quantum Electron. 42, 313–325 (2011).
[Crossref]

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 31, 97–105 (1983).
[Crossref]

White, A. G.

M. D. Harvey and A. G. White, “Frequency locking by analysis of orthogonal modes,” Opt. Commun. 221, 163–171 (2003).
[Crossref]

Wilt, D. P.

W. F. Brinkman, T. L. Koch, D. V. Lang, and D. P. Wilt, “The lasers behind the communications revolution,” Bell Labs Techn. J. 5, 150–167 (2002).
[Crossref]

Wulfmeyer, V.

Xia, J.

W. Wang and J. Xia, “The characterizations of polarization in resonator integrated optic gyroscope,” Opt. Quantum Electron. 42, 313–325 (2011).
[Crossref]

Xiao-Guang, Z.

Z. Xiao-Guang and Z. Yuan, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17, 2509–2513 (2008).
[Crossref]

Ye, J.

G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, “Tenfold reduction of Brownian noise in high-reflectivity optical coatings,” Nature Photon. 7, 644–650 (2013).
[Crossref]

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,” Nature Photon. 6, 687–692 (2012).
[Crossref]

Yu, N.

Yuan, Z.

Z. Xiao-Guang and Z. Yuan, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17, 2509–2513 (2008).
[Crossref]

Zhang, F.

Zhang, W.

G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, “Tenfold reduction of Brownian noise in high-reflectivity optical coatings,” Nature Photon. 7, 644–650 (2013).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (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 31, 97–105 (1983).
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Appl. Phys. Express (1)

S. Moriwaki, T. Mori, K. Takeno, and N. Mio, “Frequency discrimination method making use of polarization selectivity of triangular optical cavity,” Appl. Phys. Express 2, 016501 (2009).
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Bell Labs Techn. J. (1)

W. F. Brinkman, T. L. Koch, D. V. Lang, and D. P. Wilt, “The lasers behind the communications revolution,” Bell Labs Techn. J. 5, 150–167 (2002).
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Chin. Phys. B (1)

Z. Xiao-Guang and Z. Yuan, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17, 2509–2513 (2008).
[Crossref]

IEEE Photon. Technol. Lett. (1)

P.-H. Merrer, O. Llopis, and G. Cibiel, “Laser stabilization on a fiber ring resonator and application to RF filtering,” IEEE Photon. Technol. Lett. 20, 1399–1401 (2008).
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IEEE Trans. Microwave Theory Tech. (1)

W. Burns, “Polarization characteristics of single-mode fiber couplers,” IEEE Trans. Microwave Theory Tech. 30, 1577–1588 (1982).
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J. Opt. Soc. Am. A (1)

Nature Photon. (2)

G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, “Tenfold reduction of Brownian noise in high-reflectivity optical coatings,” Nature Photon. 7, 644–650 (2013).
[Crossref]

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,” Nature Photon. 6, 687–692 (2012).
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Nature Phys. (1)

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nature Phys. 4, 415–419 (2008).
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Opt. Commun. (4)

M. D. Harvey and A. G. White, “Frequency locking by analysis of orthogonal modes,” Opt. Commun. 221, 163–171 (2003).
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Opt. Express (2)

Opt. Lett. (6)

Opt. Quantum Electron. (1)

W. Wang and J. Xia, “The characterizations of polarization in resonator integrated optic gyroscope,” Opt. Quantum Electron. 42, 313–325 (2011).
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Figures (4)

Fig. 1
Fig. 1 Block diagram of an optical experiment compatible with our locking method. Polarization controllers (PCs) on the input and output of the resonator are given by the Jones matrices ��in and ��out. Matrices ℝ and �� represent reflection and transmission of the coupler, respectively, and �� represents propagation through the resonator. A polarizing beam splitter (PBS) follows at the end.
Fig. 2
Fig. 2 Cavity reflectance rcav as a function of frequency for a single input mode. Blue/Solid: Real, Red/Dashed: Imaginary. Input mirror reflectivity r2 = 1 − t2 = 0.95, cavity losses (1 − α2) = 0.05. Inset: closeup of resonance peak. Note the sharp change in the imaginary component as the frequency moves through a cavity resonance.
Fig. 3
Fig. 3 A schematic of the experimental setup used to produce the error signal. The fiber polarization controllers (FPCs) approximate adjustable waveplates to tune the input and output polarizations to optimize the error signal. The 20-m fiber ring has a finesse of ≈ 60. The sum signal is used to ensure equal input power in each eigenpolarization.
Fig. 4
Fig. 4 Observed and modeled (a) Sum and (b) difference of the two output ports of the PBS for the fiber ring resonator setup. The modeled resonator has a 95:5 coupler with 2% loss and the fiber has 4% loss. (a) Input power is split between cavity eigenpolarizations. (b) The output polarization controller was set to optimize the error signal. Discrepancies between theory and data are likely due to low-passing in the photodiode.

Equations (15)

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E = ( E H E V ) = ( A H A V e i ϕ ) e i ( ω t k z ) ,
( cos 2 θ sin 2 θ sin 2 θ cos 2 θ ) .
cav = 𝕋 𝔽 ( 𝕀 𝔽 ) 1 𝕋 ,
r cav j ( ω ) = r j t j 2 f j ( ω ) 1 r j f j ( ω ) ,
Δ θ mod 2 π > 2 π Δ ν FSR
Δ θ mod 2 π > 2 π Δ ν FSR ,
Δ | E out , 2 | 2 | E out , 1 | 2 ,
E out , 1 = ( 1 0 ) 𝕐 out cav 𝕐 in E laser E out , 2 = ( 0 1 ) 𝕐 out cav 𝕐 in E laser
E in = 𝕐 in E laser = E 0 2 ( E a + e i γ E b ) ,
E refl = cav E in = E 0 2 ( cav E a + e i γ cav E b ) = E 0 2 ( r cav a ( ω ) E a + e i γ r cav b ( ω ) E b ) .
E a 𝕐 out E a = 1 2 ( 1 e i δ )
E b 𝕐 out E b = e i ϕ 2 ( 1 e i δ ) .
E out = 𝕐 out E refl = E 0 2 ( r cav a ( ω ) ( 1 e i δ ) + e i ( γ + ϕ ) r cav b ( ω ) ( 1 e i δ ) )
| E out , 1 | 2 + | E out , 2 | 2 = E 0 2 2 ( | r cav a ( ω ) | 2 + | r cav b ( ω ) | 2 )
| E out , 2 | 2 | E out , 1 | 2 = E 0 2 Re { ( r cav a ( ω ) ) * r cav b ( ω ) e i ( γ + ϕ ) } ,

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