Abstract

Optically heterodyned laser interferometry, as applied to measuring linear displacements, requires different optical frequencies to be encoded onto unique polarization states. To eliminate non-linear contributions to the interferometer signal, the frequency difference must be introduced after beam splitting and the interfering beams must be recombined via spatially separated paths. The polarization jitter of the frequency-shifted beams still originates a noise in the beat-signal phase. A formula is given expressing the noise amplitude in terms of the illuminating beam’s extinction ratio.

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References

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  1. K.-N. Joo, J. D. Ellis, J. W. Spronck, P. J. M. van Kan, and R. H. M. Schmidt, “Simple heterodyne laser interferometer with subnanometer periodic errors,” Opt. Lett.34, 386–388 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. J. Krempel, A new spectrometer to measure the molar Planck constant (Ludwig-Maximilians Universität München, Fakultät für Physik, 2011).
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    [CrossRef]
  11. C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: Fringe 2009, 6th International Workshop on Advanced Optical Metrology, W. Osten and M. Kujawinska, eds. (SpringerBerlin Heidelberg, 2009) 263–268.
  12. S. Cosijns, Displacement laser interferometry with sub-nanometer uncertainty (Technische Universiteit Eindhoven, 2004).
  13. S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Indian Acad. Sci. A44, 247–262 (1956); reprinted in: Collected Works of S. Pancharatnam, G. W. Series ED. (Oxford University, 1975)
  14. M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature326, 277–278 (1987).
    [CrossRef]
  15. T. van Dijk, H. F. Schouten, W. Ubachs, and T. D. Visser, “The Pancharatnam-Berry phase for non-cyclic polarization changes,” Opt. Express18, 10796–10804 (2010).
    [CrossRef] [PubMed]

2012

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012).
[CrossRef]

2010

2009

2005

G. Cavagnero, G. Mana, and E. Massa, “Effect of recycled light in two-beam interferometry,” Rev. Sci. Instrum.76, 053106 (2005).
[CrossRef]

2002

T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Ångstrom-level periodic error,” J. Mod. Opt.49, 2105–2114 (2002).
[CrossRef]

2000

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000).
[CrossRef]

1999

1992

A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt.39, 2053–2074 (1992).
[CrossRef]

1987

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature326, 277–278 (1987).
[CrossRef]

1956

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Indian Acad. Sci. A44, 247–262 (1956); reprinted in: Collected Works of S. Pancharatnam, G. W. Series ED. (Oxford University, 1975)

Andreas, B.

C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012).
[CrossRef]

Balling, P.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Bancone, N.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Beckwith, J. F.

T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Ångstrom-level periodic error,” J. Mod. Opt.49, 2105–2114 (2002).
[CrossRef]

Bergamin, A.

A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt.39, 2053–2074 (1992).
[CrossRef]

Berry, M. V.

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature326, 277–278 (1987).
[CrossRef]

Birlikseven, C.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Bosse, H.

C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: Fringe 2009, 6th International Workshop on Advanced Optical Metrology, W. Osten and M. Kujawinska, eds. (SpringerBerlin Heidelberg, 2009) 263–268.

Buice, E. S.

Cavagnero, G.

G. Cavagnero, G. Mana, and E. Massa, “Effect of recycled light in two-beam interferometry,” Rev. Sci. Instrum.76, 053106 (2005).
[CrossRef]

A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt.39, 2053–2074 (1992).
[CrossRef]

Çelik, M.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Cosijns, S.

S. Cosijns, Displacement laser interferometry with sub-nanometer uncertainty (Technische Universiteit Eindhoven, 2004).

Deslattes, R. D.

Ellis, J. D.

Fluegge, J.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012).
[CrossRef]

Flügge, J.

C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: Fringe 2009, 6th International Workshop on Advanced Optical Metrology, W. Osten and M. Kujawinska, eds. (SpringerBerlin Heidelberg, 2009) 263–268.

Hamid, R.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Joo, K.-N.

Kessler, E.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000).
[CrossRef]

Koechert, P.

C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012).
[CrossRef]

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Koening, R.

C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012).
[CrossRef]

Köning, R.

C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: Fringe 2009, 6th International Workshop on Advanced Optical Metrology, W. Osten and M. Kujawinska, eds. (SpringerBerlin Heidelberg, 2009) 263–268.

Krempel, J.

J. Krempel, A new spectrometer to measure the molar Planck constant (Ludwig-Maximilians Universität München, Fakultät für Physik, 2011).

Kren, P.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Kuetgens, U.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012).
[CrossRef]

Lassila, A.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Lawall, J.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000).
[CrossRef]

C. M. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt.38, 4089–4094 (1999).
[CrossRef]

Mana, G.

G. Cavagnero, G. Mana, and E. Massa, “Effect of recycled light in two-beam interferometry,” Rev. Sci. Instrum.76, 053106 (2005).
[CrossRef]

A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt.39, 2053–2074 (1992).
[CrossRef]

Massa, E.

G. Cavagnero, G. Mana, and E. Massa, “Effect of recycled light in two-beam interferometry,” Rev. Sci. Instrum.76, 053106 (2005).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Indian Acad. Sci. A44, 247–262 (1956); reprinted in: Collected Works of S. Pancharatnam, G. W. Series ED. (Oxford University, 1975)

Picotto, G. B.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Pisani, M.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Sahin, E.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Schmidt, R. H. M.

Schmitz, T. L.

T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Ångstrom-level periodic error,” J. Mod. Opt.49, 2105–2114 (2002).
[CrossRef]

Schouten, H. F.

Seppä, J.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Spronck, J. W.

Tedaldi, M.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Tutsch, R.

C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: Fringe 2009, 6th International Workshop on Advanced Optical Metrology, W. Osten and M. Kujawinska, eds. (SpringerBerlin Heidelberg, 2009) 263–268.

Ubachs, W.

van Dijk, T.

van Kan, P. J. M.

Visser, T. D.

Weichert, C.

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012).
[CrossRef]

C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: Fringe 2009, 6th International Workshop on Advanced Optical Metrology, W. Osten and M. Kujawinska, eds. (SpringerBerlin Heidelberg, 2009) 263–268.

Wu, C. M.

Yacoot, A.

C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012).
[CrossRef]

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

A. Bergamin, G. Cavagnero, and G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt.39, 2053–2074 (1992).
[CrossRef]

T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Ångstrom-level periodic error,” J. Mod. Opt.49, 2105–2114 (2002).
[CrossRef]

Meas. Sci. Technol.

C. Weichert, P. Koechert, R. Koening, J. Fluegge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol.23, 094005 (2012).
[CrossRef]

Metrologia

M. Pisani, A. Yacoot, P. Balling, N. Bancone, C. Birlikseven, M. Çelik, J. Fluegge, R. Hamid, P. Koechert, P. Kren, U. Kuetgens, A. Lassila, G. B. Picotto, E. Sahin, J. Seppä, M. Tedaldi, and C. Weichert, “Comparison of the performance of the next generation of optical interferometers,” Metrologia49, 155–167 (2012).
[CrossRef]

Nature

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature326, 277–278 (1987).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. Indian Acad. Sci. A

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Indian Acad. Sci. A44, 247–262 (1956); reprinted in: Collected Works of S. Pancharatnam, G. W. Series ED. (Oxford University, 1975)

Rev. Sci. Instrum.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum.71, 2669–2676 (2000).
[CrossRef]

G. Cavagnero, G. Mana, and E. Massa, “Effect of recycled light in two-beam interferometry,” Rev. Sci. Instrum.76, 053106 (2005).
[CrossRef]

Other

C. Weichert, J. Flügge, R. Köning, H. Bosse, and R. Tutsch, “Aspects of design and the characterization of a high resolution heterodyne displacement interferometer,” in: Fringe 2009, 6th International Workshop on Advanced Optical Metrology, W. Osten and M. Kujawinska, eds. (SpringerBerlin Heidelberg, 2009) 263–268.

S. Cosijns, Displacement laser interferometry with sub-nanometer uncertainty (Technische Universiteit Eindhoven, 2004).

J. Krempel, A new spectrometer to measure the molar Planck constant (Ludwig-Maximilians Universität München, Fakultät für Physik, 2011).

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Figures (6)

Fig. 1
Fig. 1

Topology of a laser interferometer applying optical heterodyne. Low- and high-frequency photons (red and blue) enter the interferometer in A and B, where they are split and delivered via spatially separated paths to the output ports C and D. We can equivalently say that a beam is split in C, delivered through the reference ( DBC ¯) and measurement ( DAC ¯) arms, and recombined in D. The phase difference between the beat signal detected in C and D is proportional to the difference between the optical lengths of the measurement and reference arms.

Fig. 2
Fig. 2

Interferometer layout. Retro reflectors are mounted at the ends of beams embedded in the spindles that rotate the Si crystals A and B; γ rays propagate from right to left. The low- and high-frequency beams enter the “low” and “high” ports and interfere in the detectors D1 and D2. The optical elements (grey) have plane-parallel surfaces with coatings: mirror (solid line), polarization splitting (dotted line), non-polarization splitting (dashed line), anti-reflection (no line). Retarder plates are shown by thick lines (orange, λ/4) and double thin lines (blue, λ/2). LMM and UMM are the low-middle and up-middle mirrors; CP are compensating plates. The reference roof prism RRP raises the back-reflected beams of the auxiliary monolithic arms to a parallel plane; the λ/2 plates in front of it intercept only the incoming beams in the lowest plane. The detectors of the auxiliary monolithic arms, on the top of D1 and D2, are not indicated.

Fig. 3
Fig. 3

Observed Lissajous curve of the measurement beat signal vs. the reference one. For an ideal system it would have been a perfect ellipse.

Fig. 4
Fig. 4

Model of the heterodyne interferometer. PBS is the polarizing beam-splitter, D are the detectors of the beat signals, P are linear polarizers oriented at 45° with respect to the reflection plane, νlow and νhigh are the low- and high-frequency beam entering the interferometer.

Fig. 5
Fig. 5

Lissajous curves of Ir(t) vs. Im(t) with α1 varying from zero to 2π. Parameter values are α2 = 0, ϕ = 0, δ1 = δ2 = 0.1, and x = 0 (left), π/4 (middle), and π/2 right.

Fig. 6
Fig. 6

Circuits of the polarization states on the Poincaré sphere. Left: ideal beam splitting and recombination. Right: aberrated beam splitting and recombination. |u1〉 and |u1〉 are the polarization states entering the interferometer. The split states – the north and south poles |||〉 and |⊥〉 – are horizontal and vertical linear polarizations. The detection state – the intersection of the zero meridian with the equator, |∠〉 – is a linear polarization at 45° with respect to the reflection plane. The beat signals acquires a geometric phase-shift equal to half the solid angle subtended by the polarization circuit at the origin of the sphere.

Equations (22)

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

| u 1 = R ( π / 4 ) [ 1 δ 1 exp ( i α 1 ) ] exp [ i ( ϕ + Ω t ) ] ,
| u 2 = R ( π / 4 ) [ 1 δ 2 exp ( i α 2 ) ] ,
R ( π / 4 ) = 1 2 [ 1 1 1 1 ]
| v r = P ( P | | | u 1 + P | u 2 ) .
P = 1 2 [ 1 1 1 1 ] ,
P | | = [ 1 0 0 0 ] ,
P = [ 0 0 0 1 ]
| v m = P [ P | u 1 + P | | exp ( i x ) | u 2 ] ,
| P P | | | u 1 [ 1 δ 1 cos ( α 1 ) ] exp [ i δ 1 sin ( α 1 ) ] / 2 ,
| P P | u 2 [ 1 + δ 2 cos ( α 2 ) ] exp { + i [ Ω t + ϕ + δ 2 sin ( α 2 ) ] } / 2 ,
| P P | u 1 [ 1 + δ 1 cos ( α 1 ) ] exp [ + i δ 1 sin ( α 1 ) ] / 2 ,
| P P | | exp ( i x ) | u 2 [ 1 δ 2 cos ( α 2 ) ] exp { + i [ Ω t + ϕ + x δ 2 sin ( α 2 ) ] } / 2 ,
2 I r [ 1 δ 1 cos ( α 1 ) + δ 2 cos ( α 2 ) ] { 1 + cos [ Ω t + ϕ + δ 1 sin ( α 1 ) + δ 2 sin ( α 2 ) ] } ,
2 I m [ 1 + δ 1 cos ( α 1 ) δ 2 cos ( α 2 ) ] { 1 + cos [ Ω t + ϕ + x δ 1 sin ( α 1 ) δ 2 sin ( α 2 ) ] } .
2 I r = 1 + cos ( Ω t + ϕ ) ,
2 I m = 1 + cos ( Ω t + ϕ + x )
x m x 2 [ δ 1 sin ( α 1 ) + δ 2 sin ( α 2 ) ] .
| v r = P ( P | u 1 + P | u 2 ) ,
| v m = P [ P | | | u 1 + P | | exp ( i x ) | u 2 ] .
| v r = P ( | u 1 + | u 2 ) 2 ,
| v m = P [ | u 1 + exp ( i x ) | u 2 ] 2 .
σ x = 2 δ .

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