Abstract

In cavity ring-down spectroscopy (CRDS), residual or stress-induced birefringence (107106rad) of supermirrors will lift the polarization degeneracy of TEM00 modes and generate two new polarization eigenstates in the cavity with small resonant frequency splitting (0.1kHz); the new eigenstates are nearly linearly polarized. When both modes are excited simultaneously, the intracavity polarization state will evolve as the energy decays in the cavity. Without polarization analysis, such mode beating would not be observable. However, real supermirrors have a linear polarization-dependent loss (dichroism) that leads to a change in the loss rate as the polarization state evolves and thus to deviation from the expected single-exponential decay. We develop a model for the evolution of the intracavity polarization state and intensity for a cavity with both birefringence and polarization-dependent loss in the mirrors. We demonstrate, experimentally, that these parameters (both magnitudes and directions) can be extracted from a series of measurements of the cavity decay and depolarization of the transmitted light.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. O'Keefe and D. A. G. Deacon , “Cavity ring-down optical spectrometer for absoption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
    [Crossref]
  2. D. Romanini and K. K. Lehmann , “Ring-down cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287-6301 (1993).
    [Crossref]
  3. K. K. Lehmann , “Ring-down cavity spectroscopy cell using continuous wave excitation for trace species detection,” U.S. patent 5,528,040 (18 June 1996 ).
  4. J. Dudek , P. Rabinowitz , K. K. Lehmann , and A. Velasquez , “Trace gas detection with cw cavity ring-down laser absorption spectroscopy,” presented at 52nd Ohio State University International Symposium on Molecular Spectroscopy, Columbus, Ohio, 16-20 June 1997, paper 36 WG05.
  5. D. Romanini , A. A. Kachanov , N. Sadeghi , and F. Stoeckel , “Cw cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316-322 (1997).
    [Crossref]
  6. H. Huang and K. K. Lehmann , “Noise in cavity ring-down spectroscopy caused by transverse mode coupling,” Opt. Express 15, 8745-8759 (2007).
    [Crossref] [PubMed]
  7. A. E. Siegman , Lasers, 4th ed. (University Science Books, 1986), Sec. 21.7.
  8. D. Jacob , M. Vallet , F. Bretenaker , A. Le Floch , and M. Oger , “Supermirror phase anisotropy measurement,” Opt. Lett. 20, 671-673 (1995).
    [Crossref] [PubMed]
  9. J. L. Hall , J. Ye , and L. Ma , “Measurement of mirror birefringence at the sub-ppm level: proposed application to a test of QED,” Phys. Rev. A 62, 013815 (2000).
    [Crossref]
  10. J. Morville and D. Romanini , “Sensitive birefringence measurement in a high-finesse resonator using diode laser optical self-locking,” Appl. Phys. B 74, 495-501 (2002).
    [Crossref]
  11. J. Y. Lee , H. W. Lee , J. W. Kim , Y. S. Yoo , and J. W. Hahn , “Measurement of ultralow supermirror birefringence by use of the polarimetric differential cavity ringdown technique,” Appl. Opt. 39, 1941-1945 (2000).
    [Crossref]
  12. A. Le Floch and R. Le Naour , “Theoretical existence and experimental evidence of basculating eigenvectors in a Fabry-Perot,” C. R. Acad. Sci. Ser. B 288, 225-228 (1979).
  13. V. Evtuhov and A. E. Siegman , “A 'twisted-mode' technique for obtaining axially uniform energy density in a laser cavity,” Appl. Opt. 4, 142-143 (1965).
    [Crossref]
  14. M. Vallet , F. Bretenaker , A. Le Floch , R. Le Naour , and M. Oger , “The Malus Fabry-Perot interferometer,” Opt. Commun. 168, 423-443 (1999).
    [Crossref]
  15. Y. L. Grand and A. Le Floch , “Sensitive dichroism measurements using eigenstate decay times,” Appl. Opt. 29, 1244-1246 (1990).
    [Crossref] [PubMed]
  16. R. Engeln , G. Berden , E. van den Berg , and G. Meijer , “Polarization dependent cavity ring down spectroscopy,” J. Chem. Phys. 107, 4458-4467 (1997).
    [Crossref]
  17. R. C. Jones , “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488-493 (1941).
    [Crossref]
  18. J. B. Dudek , P. B. Tarsa , A. Velasquez , M. Wladyslawski , P. Rabinowitz , and K. K. Lehmann , “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
    [Crossref] [PubMed]
  19. The output of the cavity for monochromatic excitation will be in a state of definite polarization (i.e., describable by a Jones vector). However, when the finite line width of the excitation source is averaged over, the cavity output will be a partially incoherent superposition of different polarization vectors even if the input is in a definite polarization state independent of frequency.
  20. E. Hecht , Optics4th ed. (Addison-Wesley, 2002), p. 376.
  21. T. Muller , K. Wiberg , and P. Vaccaro , “Cavity ring-down polarimetry (CRDP): a new scheme for probing circular birefringence and circular dichroism in the gas phase,” J. Phys. Chem. A 104, 5959-5968 (2000).
    [Crossref]
  22. C. Z. Tan and J. Arndt , “Wavelength dependence of the Faraday effect in glassy SiO2,” J. Phys. Chem. Solids 60, 1689-1692 (1999).
    [Crossref]

2007 (1)

2003 (1)

J. B. Dudek , P. B. Tarsa , A. Velasquez , M. Wladyslawski , P. Rabinowitz , and K. K. Lehmann , “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[Crossref] [PubMed]

2002 (1)

J. Morville and D. Romanini , “Sensitive birefringence measurement in a high-finesse resonator using diode laser optical self-locking,” Appl. Phys. B 74, 495-501 (2002).
[Crossref]

2000 (3)

J. Y. Lee , H. W. Lee , J. W. Kim , Y. S. Yoo , and J. W. Hahn , “Measurement of ultralow supermirror birefringence by use of the polarimetric differential cavity ringdown technique,” Appl. Opt. 39, 1941-1945 (2000).
[Crossref]

T. Muller , K. Wiberg , and P. Vaccaro , “Cavity ring-down polarimetry (CRDP): a new scheme for probing circular birefringence and circular dichroism in the gas phase,” J. Phys. Chem. A 104, 5959-5968 (2000).
[Crossref]

J. L. Hall , J. Ye , and L. Ma , “Measurement of mirror birefringence at the sub-ppm level: proposed application to a test of QED,” Phys. Rev. A 62, 013815 (2000).
[Crossref]

1999 (2)

M. Vallet , F. Bretenaker , A. Le Floch , R. Le Naour , and M. Oger , “The Malus Fabry-Perot interferometer,” Opt. Commun. 168, 423-443 (1999).
[Crossref]

C. Z. Tan and J. Arndt , “Wavelength dependence of the Faraday effect in glassy SiO2,” J. Phys. Chem. Solids 60, 1689-1692 (1999).
[Crossref]

1997 (2)

R. Engeln , G. Berden , E. van den Berg , and G. Meijer , “Polarization dependent cavity ring down spectroscopy,” J. Chem. Phys. 107, 4458-4467 (1997).
[Crossref]

D. Romanini , A. A. Kachanov , N. Sadeghi , and F. Stoeckel , “Cw cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316-322 (1997).
[Crossref]

1995 (1)

1993 (1)

D. Romanini and K. K. Lehmann , “Ring-down cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287-6301 (1993).
[Crossref]

1990 (1)

1988 (1)

A. O'Keefe and D. A. G. Deacon , “Cavity ring-down optical spectrometer for absoption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
[Crossref]

1979 (1)

A. Le Floch and R. Le Naour , “Theoretical existence and experimental evidence of basculating eigenvectors in a Fabry-Perot,” C. R. Acad. Sci. Ser. B 288, 225-228 (1979).

1965 (1)

1941 (1)

Arndt, J.

C. Z. Tan and J. Arndt , “Wavelength dependence of the Faraday effect in glassy SiO2,” J. Phys. Chem. Solids 60, 1689-1692 (1999).
[Crossref]

Berden, G.

R. Engeln , G. Berden , E. van den Berg , and G. Meijer , “Polarization dependent cavity ring down spectroscopy,” J. Chem. Phys. 107, 4458-4467 (1997).
[Crossref]

Bretenaker, F.

M. Vallet , F. Bretenaker , A. Le Floch , R. Le Naour , and M. Oger , “The Malus Fabry-Perot interferometer,” Opt. Commun. 168, 423-443 (1999).
[Crossref]

D. Jacob , M. Vallet , F. Bretenaker , A. Le Floch , and M. Oger , “Supermirror phase anisotropy measurement,” Opt. Lett. 20, 671-673 (1995).
[Crossref] [PubMed]

Deacon, D. A. G.

A. O'Keefe and D. A. G. Deacon , “Cavity ring-down optical spectrometer for absoption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
[Crossref]

Dudek, J.

J. Dudek , P. Rabinowitz , K. K. Lehmann , and A. Velasquez , “Trace gas detection with cw cavity ring-down laser absorption spectroscopy,” presented at 52nd Ohio State University International Symposium on Molecular Spectroscopy, Columbus, Ohio, 16-20 June 1997, paper 36 WG05.

Dudek, J. B.

J. B. Dudek , P. B. Tarsa , A. Velasquez , M. Wladyslawski , P. Rabinowitz , and K. K. Lehmann , “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[Crossref] [PubMed]

Engeln, R.

R. Engeln , G. Berden , E. van den Berg , and G. Meijer , “Polarization dependent cavity ring down spectroscopy,” J. Chem. Phys. 107, 4458-4467 (1997).
[Crossref]

Evtuhov, V.

Grand, Y. L.

Hahn, J. W.

Hall, J. L.

J. L. Hall , J. Ye , and L. Ma , “Measurement of mirror birefringence at the sub-ppm level: proposed application to a test of QED,” Phys. Rev. A 62, 013815 (2000).
[Crossref]

Hecht, E.

E. Hecht , Optics4th ed. (Addison-Wesley, 2002), p. 376.

Huang, H.

Jacob, D.

Jones, R. C.

Kachanov, A. A.

D. Romanini , A. A. Kachanov , N. Sadeghi , and F. Stoeckel , “Cw cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316-322 (1997).
[Crossref]

Kim, J. W.

Le Floch, A.

M. Vallet , F. Bretenaker , A. Le Floch , R. Le Naour , and M. Oger , “The Malus Fabry-Perot interferometer,” Opt. Commun. 168, 423-443 (1999).
[Crossref]

D. Jacob , M. Vallet , F. Bretenaker , A. Le Floch , and M. Oger , “Supermirror phase anisotropy measurement,” Opt. Lett. 20, 671-673 (1995).
[Crossref] [PubMed]

Y. L. Grand and A. Le Floch , “Sensitive dichroism measurements using eigenstate decay times,” Appl. Opt. 29, 1244-1246 (1990).
[Crossref] [PubMed]

A. Le Floch and R. Le Naour , “Theoretical existence and experimental evidence of basculating eigenvectors in a Fabry-Perot,” C. R. Acad. Sci. Ser. B 288, 225-228 (1979).

Le Naour, R.

M. Vallet , F. Bretenaker , A. Le Floch , R. Le Naour , and M. Oger , “The Malus Fabry-Perot interferometer,” Opt. Commun. 168, 423-443 (1999).
[Crossref]

A. Le Floch and R. Le Naour , “Theoretical existence and experimental evidence of basculating eigenvectors in a Fabry-Perot,” C. R. Acad. Sci. Ser. B 288, 225-228 (1979).

Lee, H. W.

Lee, J. Y.

Lehmann, K. K.

H. Huang and K. K. Lehmann , “Noise in cavity ring-down spectroscopy caused by transverse mode coupling,” Opt. Express 15, 8745-8759 (2007).
[Crossref] [PubMed]

J. B. Dudek , P. B. Tarsa , A. Velasquez , M. Wladyslawski , P. Rabinowitz , and K. K. Lehmann , “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[Crossref] [PubMed]

D. Romanini and K. K. Lehmann , “Ring-down cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287-6301 (1993).
[Crossref]

K. K. Lehmann , “Ring-down cavity spectroscopy cell using continuous wave excitation for trace species detection,” U.S. patent 5,528,040 (18 June 1996 ).

J. Dudek , P. Rabinowitz , K. K. Lehmann , and A. Velasquez , “Trace gas detection with cw cavity ring-down laser absorption spectroscopy,” presented at 52nd Ohio State University International Symposium on Molecular Spectroscopy, Columbus, Ohio, 16-20 June 1997, paper 36 WG05.

Ma, L.

J. L. Hall , J. Ye , and L. Ma , “Measurement of mirror birefringence at the sub-ppm level: proposed application to a test of QED,” Phys. Rev. A 62, 013815 (2000).
[Crossref]

Meijer, G.

R. Engeln , G. Berden , E. van den Berg , and G. Meijer , “Polarization dependent cavity ring down spectroscopy,” J. Chem. Phys. 107, 4458-4467 (1997).
[Crossref]

Morville, J.

J. Morville and D. Romanini , “Sensitive birefringence measurement in a high-finesse resonator using diode laser optical self-locking,” Appl. Phys. B 74, 495-501 (2002).
[Crossref]

Muller, T.

T. Muller , K. Wiberg , and P. Vaccaro , “Cavity ring-down polarimetry (CRDP): a new scheme for probing circular birefringence and circular dichroism in the gas phase,” J. Phys. Chem. A 104, 5959-5968 (2000).
[Crossref]

Oger, M.

M. Vallet , F. Bretenaker , A. Le Floch , R. Le Naour , and M. Oger , “The Malus Fabry-Perot interferometer,” Opt. Commun. 168, 423-443 (1999).
[Crossref]

D. Jacob , M. Vallet , F. Bretenaker , A. Le Floch , and M. Oger , “Supermirror phase anisotropy measurement,” Opt. Lett. 20, 671-673 (1995).
[Crossref] [PubMed]

O'Keefe, A.

A. O'Keefe and D. A. G. Deacon , “Cavity ring-down optical spectrometer for absoption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
[Crossref]

Rabinowitz, P.

J. B. Dudek , P. B. Tarsa , A. Velasquez , M. Wladyslawski , P. Rabinowitz , and K. K. Lehmann , “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[Crossref] [PubMed]

J. Dudek , P. Rabinowitz , K. K. Lehmann , and A. Velasquez , “Trace gas detection with cw cavity ring-down laser absorption spectroscopy,” presented at 52nd Ohio State University International Symposium on Molecular Spectroscopy, Columbus, Ohio, 16-20 June 1997, paper 36 WG05.

Romanini, D.

J. Morville and D. Romanini , “Sensitive birefringence measurement in a high-finesse resonator using diode laser optical self-locking,” Appl. Phys. B 74, 495-501 (2002).
[Crossref]

D. Romanini , A. A. Kachanov , N. Sadeghi , and F. Stoeckel , “Cw cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316-322 (1997).
[Crossref]

D. Romanini and K. K. Lehmann , “Ring-down cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287-6301 (1993).
[Crossref]

Sadeghi, N.

D. Romanini , A. A. Kachanov , N. Sadeghi , and F. Stoeckel , “Cw cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316-322 (1997).
[Crossref]

Siegman, A. E.

Stoeckel, F.

D. Romanini , A. A. Kachanov , N. Sadeghi , and F. Stoeckel , “Cw cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316-322 (1997).
[Crossref]

Tan, C. Z.

C. Z. Tan and J. Arndt , “Wavelength dependence of the Faraday effect in glassy SiO2,” J. Phys. Chem. Solids 60, 1689-1692 (1999).
[Crossref]

Tarsa, P. B.

J. B. Dudek , P. B. Tarsa , A. Velasquez , M. Wladyslawski , P. Rabinowitz , and K. K. Lehmann , “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[Crossref] [PubMed]

Vaccaro, P.

T. Muller , K. Wiberg , and P. Vaccaro , “Cavity ring-down polarimetry (CRDP): a new scheme for probing circular birefringence and circular dichroism in the gas phase,” J. Phys. Chem. A 104, 5959-5968 (2000).
[Crossref]

Vallet, M.

M. Vallet , F. Bretenaker , A. Le Floch , R. Le Naour , and M. Oger , “The Malus Fabry-Perot interferometer,” Opt. Commun. 168, 423-443 (1999).
[Crossref]

D. Jacob , M. Vallet , F. Bretenaker , A. Le Floch , and M. Oger , “Supermirror phase anisotropy measurement,” Opt. Lett. 20, 671-673 (1995).
[Crossref] [PubMed]

van den Berg, E.

R. Engeln , G. Berden , E. van den Berg , and G. Meijer , “Polarization dependent cavity ring down spectroscopy,” J. Chem. Phys. 107, 4458-4467 (1997).
[Crossref]

Velasquez, A.

J. B. Dudek , P. B. Tarsa , A. Velasquez , M. Wladyslawski , P. Rabinowitz , and K. K. Lehmann , “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[Crossref] [PubMed]

J. Dudek , P. Rabinowitz , K. K. Lehmann , and A. Velasquez , “Trace gas detection with cw cavity ring-down laser absorption spectroscopy,” presented at 52nd Ohio State University International Symposium on Molecular Spectroscopy, Columbus, Ohio, 16-20 June 1997, paper 36 WG05.

Wiberg, K.

T. Muller , K. Wiberg , and P. Vaccaro , “Cavity ring-down polarimetry (CRDP): a new scheme for probing circular birefringence and circular dichroism in the gas phase,” J. Phys. Chem. A 104, 5959-5968 (2000).
[Crossref]

Wladyslawski, M.

J. B. Dudek , P. B. Tarsa , A. Velasquez , M. Wladyslawski , P. Rabinowitz , and K. K. Lehmann , “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[Crossref] [PubMed]

Ye, J.

J. L. Hall , J. Ye , and L. Ma , “Measurement of mirror birefringence at the sub-ppm level: proposed application to a test of QED,” Phys. Rev. A 62, 013815 (2000).
[Crossref]

Yoo, Y. S.

Anal. Chem. (1)

J. B. Dudek , P. B. Tarsa , A. Velasquez , M. Wladyslawski , P. Rabinowitz , and K. K. Lehmann , “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[Crossref] [PubMed]

Appl. Opt. (3)

Appl. Phys. B (1)

J. Morville and D. Romanini , “Sensitive birefringence measurement in a high-finesse resonator using diode laser optical self-locking,” Appl. Phys. B 74, 495-501 (2002).
[Crossref]

C. R. Acad. Sci. Ser. B (1)

A. Le Floch and R. Le Naour , “Theoretical existence and experimental evidence of basculating eigenvectors in a Fabry-Perot,” C. R. Acad. Sci. Ser. B 288, 225-228 (1979).

Chem. Phys. Lett. (1)

D. Romanini , A. A. Kachanov , N. Sadeghi , and F. Stoeckel , “Cw cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316-322 (1997).
[Crossref]

J. Chem. Phys. (2)

D. Romanini and K. K. Lehmann , “Ring-down cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287-6301 (1993).
[Crossref]

R. Engeln , G. Berden , E. van den Berg , and G. Meijer , “Polarization dependent cavity ring down spectroscopy,” J. Chem. Phys. 107, 4458-4467 (1997).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. Chem. A (1)

T. Muller , K. Wiberg , and P. Vaccaro , “Cavity ring-down polarimetry (CRDP): a new scheme for probing circular birefringence and circular dichroism in the gas phase,” J. Phys. Chem. A 104, 5959-5968 (2000).
[Crossref]

J. Phys. Chem. Solids (1)

C. Z. Tan and J. Arndt , “Wavelength dependence of the Faraday effect in glassy SiO2,” J. Phys. Chem. Solids 60, 1689-1692 (1999).
[Crossref]

Opt. Commun. (1)

M. Vallet , F. Bretenaker , A. Le Floch , R. Le Naour , and M. Oger , “The Malus Fabry-Perot interferometer,” Opt. Commun. 168, 423-443 (1999).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

J. L. Hall , J. Ye , and L. Ma , “Measurement of mirror birefringence at the sub-ppm level: proposed application to a test of QED,” Phys. Rev. A 62, 013815 (2000).
[Crossref]

Rev. Sci. Instrum. (1)

A. O'Keefe and D. A. G. Deacon , “Cavity ring-down optical spectrometer for absoption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
[Crossref]

Other (5)

A. E. Siegman , Lasers, 4th ed. (University Science Books, 1986), Sec. 21.7.

K. K. Lehmann , “Ring-down cavity spectroscopy cell using continuous wave excitation for trace species detection,” U.S. patent 5,528,040 (18 June 1996 ).

J. Dudek , P. Rabinowitz , K. K. Lehmann , and A. Velasquez , “Trace gas detection with cw cavity ring-down laser absorption spectroscopy,” presented at 52nd Ohio State University International Symposium on Molecular Spectroscopy, Columbus, Ohio, 16-20 June 1997, paper 36 WG05.

The output of the cavity for monochromatic excitation will be in a state of definite polarization (i.e., describable by a Jones vector). However, when the finite line width of the excitation source is averaged over, the cavity output will be a partially incoherent superposition of different polarization vectors even if the input is in a definite polarization state independent of frequency.

E. Hecht , Optics4th ed. (Addison-Wesley, 2002), p. 376.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Setup diagram. Red (thick) lines indicate light, and unlabeled black (thin) lines are signal propagation. A, acousto-optic modulator; AD, acousto-optic modulator driver; CCB, control circuit board; CV, cavity; D, detector; FI, Faraday isolator; GP, CPAD polarizer; HN, He–Ne laser; HWP, half-wave plate; LD, laser diode; M, mirror; MML, mode matching lenses; OPM, off-axis parabolic mirror; PC, computer; PZT, three piezo transducers; SM1, front supermirror; SM2, back supermirror; TS, trigger signal. The Pockels cell is not shown here.

Fig. 2
Fig. 2

τ (right vertical axis) and the standard deviation of τ (left vertical axis) change in the angle of the polarization analyzer at the output of the cell. This data was recorded when the back mirror was at 53 ° and the cavity was under low-stress conditions. The cavity was excited by circularly polarized light. The two angles of quieter data are indicated by the two arrows. Because of the low-stress conditions, no decay signal with reduced χ 2 larger than 1.5 was recorded, which is the criteria for bad decay in the experiments.

Fig. 3
Fig. 3

The polarization plane of one of the two modes changes with back mirror rotation angle when the cavity is under vacuum. The other mode (not shown here) is perpendicular to the one in this figure. The points are measured results, and the curve is calculated from Eq. (7) with ϵ 1 3.3 ϵ 2 .

Fig. 4
Fig. 4

The polarization plane of one of the two modes changes with back mirror rotation angle when the cavity as under low-stress conditions. The other mode (not shown here) is perpendicular to the one in this figure. The points are measured results, and the curve is calculated from Eq. (7) with ϵ 1 1 / 6 ϵ 2 . The discontinuity in the curve arises because the angle is defined between π / 2 and π / 2 .

Fig. 5
Fig. 5

τ from single-exponential fitting changes as a sine function of the polarization angle of the incident laser. The dots are experimental data, and the curves are calculated from Eq. (21). Back mirror angles are (top) 7 ° and (bottom) 53 ° with the cavity under vacuum.

Fig. 6
Fig. 6

Modulation amplitude of τ (blue jagged curve, left vertical axis) and the polarization angle corresponding to maximum τ (smooth pink curve, right vertical axis) for each orientation angle of the back mirror. The cavity was under vacuum. The back mirror was rotated from 133 ° to 47 ° with a step size of 20 ° .

Fig. 7
Fig. 7

The light intensity at the y axis | E y out | 2 is a sine function of the rotation of the x y coordinate. The curves are theoretical predictions. Diamonds, vacuum cavity, fitted to the expression ( 3.37 + 3.33 sin [ ( π θ ) / 45 2.41 ] ) mV . Triangles, 700 torr cavity pressure and screws not loosened, fitted to the expression ( 1.14 + 1.28 sin [ ( π θ ) / 45 2.18 ] ) mV . Stars, low-stress conditions, fitted to the expression ( 0.38 + 0.52 sin [ ( π θ ) / 45 2.86 ] ) mV . Here θ is the rotation angle of the x y coordinate degrees. | E x out | 2 is 200 mV , measured when the system was not triggered for decay signal and both polarization angles of the polarizer and isolator were aligned.

Fig. 8
Fig. 8

Physical picture of the experiment. Inset (upper left corner), schematic diagram of the mirror model.

Equations (21)

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

R ( θ ) = ( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) .
G i = a i R ( β i ) ( 1 + b i 0 0 1 b i ) R ( β i )
a i = r i max + r i min 2 , b i = r i max r i min r i max + r i min .
F i = R ( α i ) ( exp ( j ϵ i / 2 ) 0 0 exp ( j ϵ i / 2 ) ) R ( α i ) ,
M i = F i G i F i .
M = M 1 M 2 .
M = a 1 a 2 [ ( 1 0 0 1 ) + ( b cos ( 2 β ) + j ϵ cos ( 2 α ) b sin ( 2 β ) + j ϵ sin ( 2 α ) b sin ( 2 β ) + j ϵ sin ( 2 α ) ( b cos ( 2 β ) + j ϵ cos ( 2 α ) ) ) ] ,
b 2 = b 1 2 + b 2 2 + 2 b 1 b 2 cos ( 2 ( β 1 β 2 ) ) ,
β = β 1 + β 2 2 + 1 2 tan 1 ( b 1 b 2 b 1 + b 2 tan ( β 1 β 2 ) ) ,
ϵ 2 = ϵ 1 2 + ϵ 2 2 + 2 ϵ 1 ϵ 2 cos ( 2 ( α 1 α 2 ) ) ,
α = α 1 + α 2 2 + 1 2 tan 1 ( ϵ 1 ϵ 2 ϵ 1 + ϵ 2 tan ( α 1 α 2 ) ) .
λ 1 , 2 = a 1 a 2 ( 1 ± [ b 2 ϵ 2 + 2 j b ϵ cos ( 2 ( β α ) ) ] 1 / 2 ) .
δ ν = arg ( λ 1 ) arg ( λ 2 ) 2 π FSR c 2 L 1 π ( [ b 2 ϵ 2 + 2 j b ϵ cos ( 2 ( β α ) ) ] 1 / 2 ) ,
η = | E y out | 2 | E x out | 2 .
T = T 2 n = 0 [ exp ( j ( 2 n + 1 ) ϕ ) M n ] T 1 .
η = F 2 π 2 [ ( ϵ 1 sin ( 2 α 1 ) + ϵ 2 sin ( 2 α 2 ) ) 2 + ( b 1 sin ( 2 β 1 ) + b 2 sin ( 2 β 2 ) ) 2 ] .
η = ϵ 2 2 F 2 π 2 1 + f 2 + 2 f cos ( 2 γ ) 2 [ 1 + sin ( 4 α 1 + ψ ) ] .
τ i = t r 2 ln ( | λ i | ) .
u = [ cos ( θ ) sin ( θ ) ] .
u = M u = M [ cos ( θ ) sin ( θ ) ] .
τ ( θ ) = t r ln ( | u | 2 ) .

Metrics