Abstract

We present an analytical model for single mode, multiply reflected, external cavity, optical fiber Fabry–Perot interferometers in the low finesse regime using simple geometry and the Gaussian beam approximation. The multiple reflection model predicts attenuation of the peak-to-peak interference as the fiber to mirror distance approaches zero, as well as fringe asymmetry in the presence of nonabsorbing mirrors. A series of experiments are conducted in which a series of fiber Fabry–Perot cavities are constructed using uncoated, single mode glass fibers, and mirrors of varying reflectivity. The cavity length is swept, and the predictions of the model are found to be in good agreement with the experimental interferograms.

© 2011 Optical Society of America

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References

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  1. D. Rugar, H. J. Mamin, P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
    [CrossRef]
  2. A. Oral, R. A. Grimble, H. O. Ozer, J. B. Pethica, “High-sensitivity noncontact atomic force microscope/scanning tunneling microscope (nc-AFM/STM) operating at subangstrom oscillation amplitudes for atomic resolution imaging and force spectroscopy,” Rev. Sci. Instrum. 74, 3656–3663 (2003).
    [CrossRef]
  3. N. Suehira, Y. Tomiyoshi, Y. Sugawara, S. Morita, “Low-temperature noncontact atomic-force microscope with quick sample and cantilever exchange mechanism,” Rev. Sci. Instrum. 72, 2971–2976 (2001).
    [CrossRef]
  4. H. I. Rasool, P. R. Wilkinson, A. Z. Stieg, J. K. Gimzewski, Rev. Sci. Instrum. 81, 023703 (2010).
    [CrossRef] [PubMed]
  5. H. Holscher, P. Milde, U. Zerweck, L. M. Eng, R. Hoffman, “The effective quality factor at low temperature in dynamic force microscopes with Fabry–Perot interferometer detection,” Appl. Phys. Lett. 94, 223514 (2009).
    [CrossRef]
  6. D. T. Smith, J. R. Pratt, L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low- frequency displacement measurement,” Rev. Sci. Instrum. 80, 035105 (2009).
    [CrossRef] [PubMed]
  7. T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
    [CrossRef]
  8. S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
    [CrossRef]
  9. S. Yuan, N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt. 38, 3214–3222 (1999).
    [CrossRef]
  10. Y. St. Amant, D. Gariepy, D. Rancourt, “Intrinsic properties of the optical coupling between axisymmetric Gaussian beams,” Appl. Opt. 43, 5691–5703 (2004).
    [CrossRef]
  11. V. Arya, M. De Vries, K. Murphy, A. Wang, R. O. Claus, “Exact analysis of the extrinsic Fabry–Perot interferometric optical fiber sensor using Kirchoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
    [CrossRef]
  12. V. Arya, M. De Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the effect of imperfect fiber endfaces on the performance of optical fiber sensors,” Opt. Eng. 35, 2262–2264 (1996).
    [CrossRef]
  13. M. Han, A. Wang, “Exact analysis of low-finesse multimode fiber extrinsic Fabry–Perot interferometers,” Appl. Opt. 43, 4659–4666 (2004).
    [CrossRef] [PubMed]
  14. K. Chin, “Interference of fiber-coupled Gaussian beam multiply reflected between two planar interfaces,” IEEE Photon. Technol. Lett. 19, 1643–1645 (2007).
    [CrossRef]
  15. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–719 (1977).
  16. Note that some Gaussian beam models define a spot radius in reference to the intensity of the beam, whereas others incorporate the radius of the electric field of the mode. The relation between the two is sintensity=selectric/√2.
  17. C. R. Gouy, “Sur une propiete nouvelles des ondes lumineuses,” C. R. Acad. Sci. Paris Ser. IV 110, 1251–1253 (1890).
  18. M. Han, Y. Zhang, F. Shen, G. Pickrell, A. Wang, “Signal processing algorithm for white-light optical fiber extrinsic Fabry-Perot interferometric sensors,” Opt. Lett. 29, 1736–1738 (2004).
    [CrossRef] [PubMed]
  19. J. R. Lawall, “Fabry–Perot metrology for displacements up to 50 mm,” J. Opt. Soc. Am. A 22, 2786–2798 (2005).
    [CrossRef]
  20. Commercial equipment and materials are identified in order to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by the National Institute of Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

2010

H. I. Rasool, P. R. Wilkinson, A. Z. Stieg, J. K. Gimzewski, Rev. Sci. Instrum. 81, 023703 (2010).
[CrossRef] [PubMed]

2009

H. Holscher, P. Milde, U. Zerweck, L. M. Eng, R. Hoffman, “The effective quality factor at low temperature in dynamic force microscopes with Fabry–Perot interferometer detection,” Appl. Phys. Lett. 94, 223514 (2009).
[CrossRef]

D. T. Smith, J. R. Pratt, L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low- frequency displacement measurement,” Rev. Sci. Instrum. 80, 035105 (2009).
[CrossRef] [PubMed]

2007

K. Chin, “Interference of fiber-coupled Gaussian beam multiply reflected between two planar interfaces,” IEEE Photon. Technol. Lett. 19, 1643–1645 (2007).
[CrossRef]

2005

2004

2003

A. Oral, R. A. Grimble, H. O. Ozer, J. B. Pethica, “High-sensitivity noncontact atomic force microscope/scanning tunneling microscope (nc-AFM/STM) operating at subangstrom oscillation amplitudes for atomic resolution imaging and force spectroscopy,” Rev. Sci. Instrum. 74, 3656–3663 (2003).
[CrossRef]

2001

N. Suehira, Y. Tomiyoshi, Y. Sugawara, S. Morita, “Low-temperature noncontact atomic-force microscope with quick sample and cantilever exchange mechanism,” Rev. Sci. Instrum. 72, 2971–2976 (2001).
[CrossRef]

1999

1997

T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

1996

V. Arya, M. De Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the effect of imperfect fiber endfaces on the performance of optical fiber sensors,” Opt. Eng. 35, 2262–2264 (1996).
[CrossRef]

1995

V. Arya, M. De Vries, K. Murphy, A. Wang, R. O. Claus, “Exact analysis of the extrinsic Fabry–Perot interferometric optical fiber sensor using Kirchoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

1989

D. Rugar, H. J. Mamin, P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

1979

S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

1977

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–719 (1977).

1890

C. R. Gouy, “Sur une propiete nouvelles des ondes lumineuses,” C. R. Acad. Sci. Paris Ser. IV 110, 1251–1253 (1890).

Amant, Y. St.

Arya, V.

V. Arya, M. De Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the effect of imperfect fiber endfaces on the performance of optical fiber sensors,” Opt. Eng. 35, 2262–2264 (1996).
[CrossRef]

V. Arya, M. De Vries, K. Murphy, A. Wang, R. O. Claus, “Exact analysis of the extrinsic Fabry–Perot interferometric optical fiber sensor using Kirchoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Athreya, M.

V. Arya, M. De Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the effect of imperfect fiber endfaces on the performance of optical fiber sensors,” Opt. Eng. 35, 2262–2264 (1996).
[CrossRef]

Botkin, D.

T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Chin, K.

K. Chin, “Interference of fiber-coupled Gaussian beam multiply reflected between two planar interfaces,” IEEE Photon. Technol. Lett. 19, 1643–1645 (2007).
[CrossRef]

Claus, R. O.

V. Arya, M. De Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the effect of imperfect fiber endfaces on the performance of optical fiber sensors,” Opt. Eng. 35, 2262–2264 (1996).
[CrossRef]

V. Arya, M. De Vries, K. Murphy, A. Wang, R. O. Claus, “Exact analysis of the extrinsic Fabry–Perot interferometric optical fiber sensor using Kirchoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

De Vries, M.

V. Arya, M. De Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the effect of imperfect fiber endfaces on the performance of optical fiber sensors,” Opt. Eng. 35, 2262–2264 (1996).
[CrossRef]

V. Arya, M. De Vries, K. Murphy, A. Wang, R. O. Claus, “Exact analysis of the extrinsic Fabry–Perot interferometric optical fiber sensor using Kirchoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Eng, L. M.

H. Holscher, P. Milde, U. Zerweck, L. M. Eng, R. Hoffman, “The effective quality factor at low temperature in dynamic force microscopes with Fabry–Perot interferometer detection,” Appl. Phys. Lett. 94, 223514 (2009).
[CrossRef]

Gariepy, D.

Gimzewski, J. K.

H. I. Rasool, P. R. Wilkinson, A. Z. Stieg, J. K. Gimzewski, Rev. Sci. Instrum. 81, 023703 (2010).
[CrossRef] [PubMed]

Gouy, C. R.

C. R. Gouy, “Sur une propiete nouvelles des ondes lumineuses,” C. R. Acad. Sci. Paris Ser. IV 110, 1251–1253 (1890).

Grimble, R. A.

A. Oral, R. A. Grimble, H. O. Ozer, J. B. Pethica, “High-sensitivity noncontact atomic force microscope/scanning tunneling microscope (nc-AFM/STM) operating at subangstrom oscillation amplitudes for atomic resolution imaging and force spectroscopy,” Rev. Sci. Instrum. 74, 3656–3663 (2003).
[CrossRef]

Guethner, P.

D. Rugar, H. J. Mamin, P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

Han, M.

Hoffman, R.

H. Holscher, P. Milde, U. Zerweck, L. M. Eng, R. Hoffman, “The effective quality factor at low temperature in dynamic force microscopes with Fabry–Perot interferometer detection,” Appl. Phys. Lett. 94, 223514 (2009).
[CrossRef]

Holscher, H.

H. Holscher, P. Milde, U. Zerweck, L. M. Eng, R. Hoffman, “The effective quality factor at low temperature in dynamic force microscopes with Fabry–Perot interferometer detection,” Appl. Phys. Lett. 94, 223514 (2009).
[CrossRef]

Howard, L. P.

D. T. Smith, J. R. Pratt, L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low- frequency displacement measurement,” Rev. Sci. Instrum. 80, 035105 (2009).
[CrossRef] [PubMed]

Kenny, T. W.

T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Lawall, J. R.

Makimoto, T.

S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Mamin, H. J.

D. Rugar, H. J. Mamin, P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–719 (1977).

Milde, P.

H. Holscher, P. Milde, U. Zerweck, L. M. Eng, R. Hoffman, “The effective quality factor at low temperature in dynamic force microscopes with Fabry–Perot interferometer detection,” Appl. Phys. Lett. 94, 223514 (2009).
[CrossRef]

Morita, S.

N. Suehira, Y. Tomiyoshi, Y. Sugawara, S. Morita, “Low-temperature noncontact atomic-force microscope with quick sample and cantilever exchange mechanism,” Rev. Sci. Instrum. 72, 2971–2976 (2001).
[CrossRef]

Murphy, K.

V. Arya, M. De Vries, K. Murphy, A. Wang, R. O. Claus, “Exact analysis of the extrinsic Fabry–Perot interferometric optical fiber sensor using Kirchoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Nemoto, S.

S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Oral, A.

A. Oral, R. A. Grimble, H. O. Ozer, J. B. Pethica, “High-sensitivity noncontact atomic force microscope/scanning tunneling microscope (nc-AFM/STM) operating at subangstrom oscillation amplitudes for atomic resolution imaging and force spectroscopy,” Rev. Sci. Instrum. 74, 3656–3663 (2003).
[CrossRef]

Ozer, H. O.

A. Oral, R. A. Grimble, H. O. Ozer, J. B. Pethica, “High-sensitivity noncontact atomic force microscope/scanning tunneling microscope (nc-AFM/STM) operating at subangstrom oscillation amplitudes for atomic resolution imaging and force spectroscopy,” Rev. Sci. Instrum. 74, 3656–3663 (2003).
[CrossRef]

Pethica, J. B.

A. Oral, R. A. Grimble, H. O. Ozer, J. B. Pethica, “High-sensitivity noncontact atomic force microscope/scanning tunneling microscope (nc-AFM/STM) operating at subangstrom oscillation amplitudes for atomic resolution imaging and force spectroscopy,” Rev. Sci. Instrum. 74, 3656–3663 (2003).
[CrossRef]

Pickrell, G.

Pratt, J. R.

D. T. Smith, J. R. Pratt, L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low- frequency displacement measurement,” Rev. Sci. Instrum. 80, 035105 (2009).
[CrossRef] [PubMed]

Rancourt, D.

Rasool, H. I.

H. I. Rasool, P. R. Wilkinson, A. Z. Stieg, J. K. Gimzewski, Rev. Sci. Instrum. 81, 023703 (2010).
[CrossRef] [PubMed]

Riza, N. A.

Rugar, D.

T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

D. Rugar, H. J. Mamin, P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

Shen, F.

Smith, D. T.

D. T. Smith, J. R. Pratt, L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low- frequency displacement measurement,” Rev. Sci. Instrum. 80, 035105 (2009).
[CrossRef] [PubMed]

Stieg, A. Z.

H. I. Rasool, P. R. Wilkinson, A. Z. Stieg, J. K. Gimzewski, Rev. Sci. Instrum. 81, 023703 (2010).
[CrossRef] [PubMed]

Stowe, T. D.

T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Suehira, N.

N. Suehira, Y. Tomiyoshi, Y. Sugawara, S. Morita, “Low-temperature noncontact atomic-force microscope with quick sample and cantilever exchange mechanism,” Rev. Sci. Instrum. 72, 2971–2976 (2001).
[CrossRef]

Sugawara, Y.

N. Suehira, Y. Tomiyoshi, Y. Sugawara, S. Morita, “Low-temperature noncontact atomic-force microscope with quick sample and cantilever exchange mechanism,” Rev. Sci. Instrum. 72, 2971–2976 (2001).
[CrossRef]

Tomiyoshi, Y.

N. Suehira, Y. Tomiyoshi, Y. Sugawara, S. Morita, “Low-temperature noncontact atomic-force microscope with quick sample and cantilever exchange mechanism,” Rev. Sci. Instrum. 72, 2971–2976 (2001).
[CrossRef]

Wago, K.

T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Wang, A.

M. Han, A. Wang, “Exact analysis of low-finesse multimode fiber extrinsic Fabry–Perot interferometers,” Appl. Opt. 43, 4659–4666 (2004).
[CrossRef] [PubMed]

M. Han, Y. Zhang, F. Shen, G. Pickrell, A. Wang, “Signal processing algorithm for white-light optical fiber extrinsic Fabry-Perot interferometric sensors,” Opt. Lett. 29, 1736–1738 (2004).
[CrossRef] [PubMed]

V. Arya, M. De Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the effect of imperfect fiber endfaces on the performance of optical fiber sensors,” Opt. Eng. 35, 2262–2264 (1996).
[CrossRef]

V. Arya, M. De Vries, K. Murphy, A. Wang, R. O. Claus, “Exact analysis of the extrinsic Fabry–Perot interferometric optical fiber sensor using Kirchoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Wilkinson, P. R.

H. I. Rasool, P. R. Wilkinson, A. Z. Stieg, J. K. Gimzewski, Rev. Sci. Instrum. 81, 023703 (2010).
[CrossRef] [PubMed]

Yasumura, K.

T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Yuan, S.

Zerweck, U.

H. Holscher, P. Milde, U. Zerweck, L. M. Eng, R. Hoffman, “The effective quality factor at low temperature in dynamic force microscopes with Fabry–Perot interferometer detection,” Appl. Phys. Lett. 94, 223514 (2009).
[CrossRef]

Zhang, Y.

Appl. Opt.

Appl. Phys. Lett.

D. Rugar, H. J. Mamin, P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55, 2588–2590 (1989).
[CrossRef]

H. Holscher, P. Milde, U. Zerweck, L. M. Eng, R. Hoffman, “The effective quality factor at low temperature in dynamic force microscopes with Fabry–Perot interferometer detection,” Appl. Phys. Lett. 94, 223514 (2009).
[CrossRef]

T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997).
[CrossRef]

Bell Syst. Tech. J.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–719 (1977).

C. R. Acad. Sci. Paris Ser. IV

C. R. Gouy, “Sur une propiete nouvelles des ondes lumineuses,” C. R. Acad. Sci. Paris Ser. IV 110, 1251–1253 (1890).

IEEE Photon. Technol. Lett.

K. Chin, “Interference of fiber-coupled Gaussian beam multiply reflected between two planar interfaces,” IEEE Photon. Technol. Lett. 19, 1643–1645 (2007).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

V. Arya, M. De Vries, M. Athreya, A. Wang, R. O. Claus, “Analysis of the effect of imperfect fiber endfaces on the performance of optical fiber sensors,” Opt. Eng. 35, 2262–2264 (1996).
[CrossRef]

Opt. Fiber Technol.

V. Arya, M. De Vries, K. Murphy, A. Wang, R. O. Claus, “Exact analysis of the extrinsic Fabry–Perot interferometric optical fiber sensor using Kirchoff’s diffraction formalism,” Opt. Fiber Technol. 1, 380–384 (1995).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Rev. Sci. Instrum.

N. Suehira, Y. Tomiyoshi, Y. Sugawara, S. Morita, “Low-temperature noncontact atomic-force microscope with quick sample and cantilever exchange mechanism,” Rev. Sci. Instrum. 72, 2971–2976 (2001).
[CrossRef]

Rev. Sci. Instrum.

H. I. Rasool, P. R. Wilkinson, A. Z. Stieg, J. K. Gimzewski, Rev. Sci. Instrum. 81, 023703 (2010).
[CrossRef] [PubMed]

A. Oral, R. A. Grimble, H. O. Ozer, J. B. Pethica, “High-sensitivity noncontact atomic force microscope/scanning tunneling microscope (nc-AFM/STM) operating at subangstrom oscillation amplitudes for atomic resolution imaging and force spectroscopy,” Rev. Sci. Instrum. 74, 3656–3663 (2003).
[CrossRef]

D. T. Smith, J. R. Pratt, L. P. Howard, “A fiber-optic interferometer with subpicometer resolution for dc and low- frequency displacement measurement,” Rev. Sci. Instrum. 80, 035105 (2009).
[CrossRef] [PubMed]

Other

Commercial equipment and materials are identified in order to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by the National Institute of Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

Note that some Gaussian beam models define a spot radius in reference to the intensity of the beam, whereas others incorporate the radius of the electric field of the mode. The relation between the two is sintensity=selectric/√2.

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Figures (13)

Fig. 1
Fig. 1

Fiber coupling geometry defined by a mirror-to-fiber distance z m and an angular misalignment θ m . The gray region represents the optical fiber and the solid black line represents the mirror plane against which the fiber is aligned.

Fig. 2
Fig. 2

(a) Geometry of the misaligned fiber coupling problem defined by axial, transverse, and angular offsets, z, x, and θ, respectively. (b) Equivalent representation of the problem in Fig. 1 where the receiving fiber is considered to be the reflection of the transmission fiber across the mirror plane.

Fig. 3
Fig. 3

Geometry of the multiple reflection problem, from which Eq. (11) can be calculated. The gray regions represent optical fibers, with f n representing the nth reflection image of the fiber, whereas the solid black lines represent the mirror surface and reflections thereof.

Fig. 4
Fig. 4

(a) An interferogram and (b) the associated interference parameters Δ and α for a perfectly aligned fiber ( θ m = 0 ) against a perfectly reflecting mirror ( r m = 1 ) using the N = 1 approximation. λ = 1550 nm and s = 3.7 μm .

Fig. 5
Fig. 5

(a) An interferogram and (b) the associated interference parameters Δ and α for a misaligned ( θ m = 0.02 radians) fiber against a perfectly reflecting mirror ( r m = 1 ) using the N = 1 approximation. λ = 1550 nm and s = 3.7 μm .

Fig. 6
Fig. 6

The coupled power of the nth reflection from a perfectly reflecting surface ( r m = 1 ) as a function z m , when (a) perfectly aligned ( θ m = 0 ), and (b) misaligned ( θ m = 0.02 radians). λ = 1550 nm and s = 3.7 μm .

Fig. 7
Fig. 7

Interference fringes ( r m = 1 , N = 5 ) (a) in the range below z r as well as (b) near 2 z r . The blue curve corresponds to the perfectly aligned case ( θ m = 0 ). The red curve corresponds to an angular offset of θ m = 0.02 radians. λ = 1550 nm and s = 3.7 μm .

Fig. 8
Fig. 8

An upper bound for the finesse of an EFPI system due to the fiber coupling problem and angular misalignments. λ = 1550 nm and s = 3.7 μm . Scattering and transmission losses are not included and will further degrade the finesse.

Fig. 9
Fig. 9

Experimental setup designed depicting the cavity length sweeps described in the text. For each experiment, the fiber was aligned against a mirror mounted on a piezo stage, and the mirror was translated in the direction of the fiber axis.

Fig. 10
Fig. 10

Experimental (red) and theoretical (blue) interferograms for a glass fiber aligned against a glass mirror coated with a 30 nm thick Ag film (a) the N = 1 model and (b) the N = 5 model. Insets show the data for each curve near 20 μm cavity length. λ = 1550 nm and s = 3.7 μm .

Fig. 11
Fig. 11

Experimental (red) and theoretical (blue) interferograms for a glass fiber aligned against a Si mirror (a) the N = 1 model and (b) the N = 5 model. Insets show the data for each curve near 20 μm cavity length. λ = 1550 nm and s = 3.7 μm .

Fig. 12
Fig. 12

Experimental (red) and theoretical (blue) interferograms for a glass fiber aligned against a glass mirror (a) the N = 1 model and (b) the N = 5 model. Insets show the data for each curve near 20 μm cavity length. λ = 1550 nm and s = 3.7 μm .

Fig. 13
Fig. 13

Experimental (red) and the theoretical curve fit (blue) for a wavelength sweep of a cavity formed by a glass fiber and a highly reflective R = 99.99 % mirror. The cavity length determined from the fit is 46.322 μm ± 0.0005 μm . s = 3.7 μm .

Equations (25)

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

Δ P = P z Δ z .
Δ P = ± P o α 4 π n c λ Δ z ,
E = n = 0 E n ,
E n = 0 = r o C n = 0 E o , E n 0 = t o t o r m n r o n 1 C n E o ,
r = E E o = r o C n = 0 + t o t o n = 1 r m n r o n 1 C n .
C = 1 1 + z ¯ 2 / 4 exp [ k z R ( u + i v ) 2 ( 4 + z ¯ 2 ) i ( k z tan 1 ( z ¯ / 2 ) ) ] ,
u = 2 x ¯ 2 + 2 x ¯ z ¯ sin θ + ( z ¯ 2 + 2 ) sin 2 θ , v = x ¯ 2 z ¯ 4 x ¯ sin θ z ¯ sin 2 θ ,
x ¯ = x / z R , z ¯ = z / z R ,
C = Λ exp [ i β ] exp [ i Θ ] , Λ = 1 1 + z ¯ 2 / 4 exp [ k z R 2 ( 4 + z ¯ 2 ) u ] , Θ = tan 1 ( z ¯ / 2 ) + k z R 2 ( 4 + z ¯ 2 ) v ,
N < θ c 2 θ m
θ n = 2 n θ m , z n = z m θ m sin ( 2 n θ m ) , x n = z m θ m ( 1 cos ( 2 n θ m ) ) .
Λ n = 1 1 + n 2 z ¯ m 2 exp [ k z R ( 1 + 5 n 2 z ¯ m 2 ) 1 + n 2 z ¯ m 2 n 2 θ m 2 ] , Θ n = tan 1 ( n z ¯ m ) + k z R ( 3 n z ¯ m n 3 z ¯ m 3 ) 1 + n 2 z ¯ m 2 n 2 θ m 2 , β n = 2 n k z m .
Λ n , θ m = 0 = 1 1 + n 2 z ¯ m 2 , Θ n , θ m = 0 = tan 1 ( n z ¯ m ) , β n = 2 n k z m .
r = r o C n = 0 + t o 2 r o n = 1 ( r m r o ) n C n e i π .
R = r o 2 + t o 4 r o 2 n = 1 N ( r m r o ) 2 n Λ n 2 + 2 t o 2 n = 1 N ( r m r o ) n Λ n cos ( β n + Θ n + π ) + 2 t o 4 r o 2 l = 1 N n = l + 1 N ( r m r o ) n + l Λ n Λ l cos ( β n β l + Θ n Θ l ) ,
R = Δ + α cos ( β n = 1 + Θ n = 1 + π ) ,
Δ = Δ n = 0 + Δ n = 1 = r o 2 + t 0 4 r m 2 Λ n = 1 2 , α = 2 r o r m t o 2 Λ n = 1
Δ P = ± P o α 4 π n c λ ( 1 ± K ) Δ z m ,
K = λ Λ 2 4 π n c ( 1 z R t o 2 r m z m r o Λ z R 2 ) ,
Δ = n = 0 Δ n ,
Δ n = 0 = r o 2 , Δ n 0 = t o 2 t o 2 r m 2 n r o 2 ( n 1 ) Λ n 2 .
F = 2 π Γ j ,
Γ ang > 2 θ m θ c .
Γ coup > 1 Λ n = 1 2 = z ¯ m 2 1 + z ¯ m 2 .
F < 2 π ( 2 θ m θ c + z ¯ m 2 1 + z ¯ m 2 ) 1 .

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