Abstract

All-reflective interferometer configurations have been proposed for the next generation of gravitational wave detectors, with diffractive elements replacing transmissive optics. However, an additional phase noise creates more stringent conditions for alignment stability. A framework for alignment stability with the use of diffractive elements was required using a Gaussian model. We successfully create such a framework involving modal decomposition to replicate small displacements of the beam (or grating) and show that the modal model does not contain the phase changes seen in an otherwise geometric planewave approach. The modal decomposition description is justified by verifying experimentally that the phase of a diffracted Gaussian beam is independent of the beam shape, achieved by comparing the phase change between a zero-order and first-order mode beam. To interpret our findings we employ a rigorous time-domain simulation to demonstrate that the phase changes resulting from a modal decomposition are correct, provided that the coordinate system which measures the phase is moved simultaneously with the effective beam displacement. This indeed corresponds to the phase change observed in the geometric planewave model. The change in the coordinate system does not instinctively occur within the analytical framework, and therefore requires either a manual change in the coordinate system or an addition of the geometric planewave phase factor.

© 2013 Optical Society of America

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References

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  • Author
  • Publication
  1. G. Harry, “Advanced Ligo: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
    [Crossref]
  2. T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
    [Crossref]
  3. B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
    [Crossref]
  4. R. Drever, “Concepts for extending the ultimate sensitivity of interferometric gravitational wave detectors using non-transmissive optics with diffractive or holographic coupling,” in Proceedings of the Seventh Marcel Grossman Meeting on General Relativity, M. Keiser and R. T. Jantzen, eds. (World Scientific, 1995).
  5. A. Bunkowski, O. Burmeister, T. Clausnitzer, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractive optics for gravitational wave detectors,” J. Phys.: Conf. Ser. 32, 333–338 (2006).
    [Crossref]
  6. A. Bunkowski, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E. Kley, and A. Tünnermann, “Low-loss grating for coupling to a high-finesse cavity,” Opt. Lett. 29, 2342–2344 (2004).
    [Crossref] [PubMed]
  7. M. Britzger, D. Friedrich, S. Kroker, F. Brückner, O. Burmeister, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Pound-Drever-Hall error signals for the length control of three-port grating coupled cavities,” Appl. Opt. 50, 4340–4346 (2011).
    [Crossref] [PubMed]
  8. D. Friedrich, O. Burmeister, A. Bunkowski, T. Clausnitzer, S. Fahr, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractive beam splitter characterization via a power-recycled interferometer,” Opt. Lett. 33, 101–103 (2008).
    [Crossref] [PubMed]
  9. S. Kroker, T. Käsebier, F. Brückner, F. Fuchs, E. Kley, and A. Tünnermann, “Reflective cavity couplers based on resonant waveguide gratings,” Opt. Express 19, 16466–16479 (2011).
    [Crossref] [PubMed]
  10. S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
    [Crossref]
  11. A. Freise, A. Bunkowski, and R. Schnabel, “Phase and alignment noise in grating interferometers,” New J. Phys. 9, 433 (2007).
    [Crossref]
  12. J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
    [Crossref]
  13. A. Freise, G. Heinzel, H. Lück, R. Schilling, B. Willke, and K. Danzmann, “Frequency-domain interferometer simulation with higher-order spatial modes,” Class. Quantum Grav. 21, S1067 (2004).
    [Crossref]
  14. D. Lodhia, F. Brückner, L. Carbone, P. Fulda, K. Kokeyama, and A. Freise, “Phase effects in Gaussian beams on diffraction gratings,” J. Phys.: Conf. Ser. 363012014 (2012).
    [Crossref]
  15. R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B,  31, 97–105 (1983).
    [Crossref]
  16. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (3rd ed., Artech House, 2005).
  17. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [Crossref]
  18. D. Brown, D. Friedrich, Frank Brückner, L. Carbone, R. Schnabel, and A. Freise, “Invariance of waveguide grating mirrors to lateral displacement phase shifts,” Opt. Letters,  38, 1844–1846 (2013).
    [Crossref]

2013 (1)

D. Brown, D. Friedrich, Frank Brückner, L. Carbone, R. Schnabel, and A. Freise, “Invariance of waveguide grating mirrors to lateral displacement phase shifts,” Opt. Letters,  38, 1844–1846 (2013).
[Crossref]

2012 (1)

D. Lodhia, F. Brückner, L. Carbone, P. Fulda, K. Kokeyama, and A. Freise, “Phase effects in Gaussian beams on diffraction gratings,” J. Phys.: Conf. Ser. 363012014 (2012).
[Crossref]

2011 (3)

2010 (1)

G. Harry, “Advanced Ligo: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
[Crossref]

2009 (1)

J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
[Crossref]

2008 (1)

2007 (1)

A. Freise, A. Bunkowski, and R. Schnabel, “Phase and alignment noise in grating interferometers,” New J. Phys. 9, 433 (2007).
[Crossref]

2006 (2)

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

A. Bunkowski, O. Burmeister, T. Clausnitzer, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractive optics for gravitational wave detectors,” J. Phys.: Conf. Ser. 32, 333–338 (2006).
[Crossref]

2005 (1)

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

2004 (2)

A. Bunkowski, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E. Kley, and A. Tünnermann, “Low-loss grating for coupling to a high-finesse cavity,” Opt. Lett. 29, 2342–2344 (2004).
[Crossref] [PubMed]

A. Freise, G. Heinzel, H. Lück, R. Schilling, B. Willke, and K. Danzmann, “Frequency-domain interferometer simulation with higher-order spatial modes,” Class. Quantum Grav. 21, S1067 (2004).
[Crossref]

1983 (1)

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

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

Accadia, T.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Acernese, F.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Ajith, P.

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

Allen, B.

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

Antonucci, F.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Astone, P.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Aufmuth, P.

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

Aulbert, C.

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

Babak, S.

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

Balasubramanian, R.

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

Ballardin, G.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Barone, F.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Barr, B.

J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
[Crossref]

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

Barsuglia, M.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Basti, A.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Bauer, T.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Bebronne, M.

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

Berukoff, S.

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

Beyersdorf, P.

Britzger, M.

Brown, D.

D. Brown, D. Friedrich, Frank Brückner, L. Carbone, R. Schnabel, and A. Freise, “Invariance of waveguide grating mirrors to lateral displacement phase shifts,” Opt. Letters,  38, 1844–1846 (2013).
[Crossref]

Brückner, F.

Brückner, Frank

D. Brown, D. Friedrich, Frank Brückner, L. Carbone, R. Schnabel, and A. Freise, “Invariance of waveguide grating mirrors to lateral displacement phase shifts,” Opt. Letters,  38, 1844–1846 (2013).
[Crossref]

Bunkowski, A.

D. Friedrich, O. Burmeister, A. Bunkowski, T. Clausnitzer, S. Fahr, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractive beam splitter characterization via a power-recycled interferometer,” Opt. Lett. 33, 101–103 (2008).
[Crossref] [PubMed]

A. Freise, A. Bunkowski, and R. Schnabel, “Phase and alignment noise in grating interferometers,” New J. Phys. 9, 433 (2007).
[Crossref]

A. Bunkowski, O. Burmeister, T. Clausnitzer, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractive optics for gravitational wave detectors,” J. Phys.: Conf. Ser. 32, 333–338 (2006).
[Crossref]

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

A. Bunkowski, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E. Kley, and A. Tünnermann, “Low-loss grating for coupling to a high-finesse cavity,” Opt. Lett. 29, 2342–2344 (2004).
[Crossref] [PubMed]

Burmeister, O.

Carbone, L.

D. Brown, D. Friedrich, Frank Brückner, L. Carbone, R. Schnabel, and A. Freise, “Invariance of waveguide grating mirrors to lateral displacement phase shifts,” Opt. Letters,  38, 1844–1846 (2013).
[Crossref]

D. Lodhia, F. Brückner, L. Carbone, P. Fulda, K. Kokeyama, and A. Freise, “Phase effects in Gaussian beams on diffraction gratings,” J. Phys.: Conf. Ser. 363012014 (2012).
[Crossref]

Chelkowski, S.

J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
[Crossref]

Chen, Y.

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

Clausnitzer, T.

Danzmann, K.

Deshpande, A.

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

Drever, R.

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

R. Drever, “Concepts for extending the ultimate sensitivity of interferometric gravitational wave detectors using non-transmissive optics with diffractive or holographic coupling,” in Proceedings of the Seventh Marcel Grossman Meeting on General Relativity, M. Keiser and R. T. Jantzen, eds. (World Scientific, 1995).

Fahr, S.

Ford, G.

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

Freise, A.

D. Brown, D. Friedrich, Frank Brückner, L. Carbone, R. Schnabel, and A. Freise, “Invariance of waveguide grating mirrors to lateral displacement phase shifts,” Opt. Letters,  38, 1844–1846 (2013).
[Crossref]

D. Lodhia, F. Brückner, L. Carbone, P. Fulda, K. Kokeyama, and A. Freise, “Phase effects in Gaussian beams on diffraction gratings,” J. Phys.: Conf. Ser. 363012014 (2012).
[Crossref]

J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
[Crossref]

A. Freise, A. Bunkowski, and R. Schnabel, “Phase and alignment noise in grating interferometers,” New J. Phys. 9, 433 (2007).
[Crossref]

A. Freise, G. Heinzel, H. Lück, R. Schilling, B. Willke, and K. Danzmann, “Frequency-domain interferometer simulation with higher-order spatial modes,” Class. Quantum Grav. 21, S1067 (2004).
[Crossref]

Friedrich, D.

Fuchs, F.

Fulda, P.

D. Lodhia, F. Brückner, L. Carbone, P. Fulda, K. Kokeyama, and A. Freise, “Phase effects in Gaussian beams on diffraction gratings,” J. Phys.: Conf. Ser. 363012014 (2012).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (3rd ed., Artech House, 2005).

Hall, J.

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

Hallam, J.

J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
[Crossref]

Harry, G.

G. Harry, “Advanced Ligo: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
[Crossref]

Heinzel, G.

A. Freise, G. Heinzel, H. Lück, R. Schilling, B. Willke, and K. Danzmann, “Frequency-domain interferometer simulation with higher-order spatial modes,” Class. Quantum Grav. 21, S1067 (2004).
[Crossref]

Hild, S.

J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
[Crossref]

Hough, J.

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

Käsebier, T.

Kley, E.

Kokeyama, K.

D. Lodhia, F. Brückner, L. Carbone, P. Fulda, K. Kokeyama, and A. Freise, “Phase effects in Gaussian beams on diffraction gratings,” J. Phys.: Conf. Ser. 363012014 (2012).
[Crossref]

Kowalski, F.

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

Kroker, S.

Lodhia, D.

D. Lodhia, F. Brückner, L. Carbone, P. Fulda, K. Kokeyama, and A. Freise, “Phase effects in Gaussian beams on diffraction gratings,” J. Phys.: Conf. Ser. 363012014 (2012).
[Crossref]

Lück, H.

A. Freise, G. Heinzel, H. Lück, R. Schilling, B. Willke, and K. Danzmann, “Frequency-domain interferometer simulation with higher-order spatial modes,” Class. Quantum Grav. 21, S1067 (2004).
[Crossref]

Mueller, G.

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

Munley, A.

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

Quetschke, V.

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

Reitze, D.

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

Schilling, R.

A. Freise, G. Heinzel, H. Lück, R. Schilling, B. Willke, and K. Danzmann, “Frequency-domain interferometer simulation with higher-order spatial modes,” Class. Quantum Grav. 21, S1067 (2004).
[Crossref]

Schnabel, R.

D. Brown, D. Friedrich, Frank Brückner, L. Carbone, R. Schnabel, and A. Freise, “Invariance of waveguide grating mirrors to lateral displacement phase shifts,” Opt. Letters,  38, 1844–1846 (2013).
[Crossref]

M. Britzger, D. Friedrich, S. Kroker, F. Brückner, O. Burmeister, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Pound-Drever-Hall error signals for the length control of three-port grating coupled cavities,” Appl. Opt. 50, 4340–4346 (2011).
[Crossref] [PubMed]

J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
[Crossref]

D. Friedrich, O. Burmeister, A. Bunkowski, T. Clausnitzer, S. Fahr, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractive beam splitter characterization via a power-recycled interferometer,” Opt. Lett. 33, 101–103 (2008).
[Crossref] [PubMed]

A. Freise, A. Bunkowski, and R. Schnabel, “Phase and alignment noise in grating interferometers,” New J. Phys. 9, 433 (2007).
[Crossref]

A. Bunkowski, O. Burmeister, T. Clausnitzer, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractive optics for gravitational wave detectors,” J. Phys.: Conf. Ser. 32, 333–338 (2006).
[Crossref]

A. Bunkowski, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E. Kley, and A. Tünnermann, “Low-loss grating for coupling to a high-finesse cavity,” Opt. Lett. 29, 2342–2344 (2004).
[Crossref] [PubMed]

Strain, K.

J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
[Crossref]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (3rd ed., Artech House, 2005).

Tanner, D.

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

Tünnermann, A.

Ward, H.

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

Whiting, B.

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

Willke, B.

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

A. Freise, G. Heinzel, H. Lück, R. Schilling, B. Willke, and K. Danzmann, “Frequency-domain interferometer simulation with higher-order spatial modes,” Class. Quantum Grav. 21, S1067 (2004).
[Crossref]

Wise, S.

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

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

Class. Quantum Grav. (4)

A. Freise, G. Heinzel, H. Lück, R. Schilling, B. Willke, and K. Danzmann, “Frequency-domain interferometer simulation with higher-order spatial modes,” Class. Quantum Grav. 21, S1067 (2004).
[Crossref]

G. Harry, “Advanced Ligo: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
[Crossref]

T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin, F. Barone, M. Barsuglia, A. Basti, T. Bauer, M. Bebronne, and et al., “Status of the Virgo project,” Class. Quantum Grav. 28, 114002 (2011).
[Crossref]

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. Barr, S. Berukoff, A. Bunkowski, and et al., “The GEO-HF project,” Class. Quantum Grav. 23, S207 (2006).
[Crossref]

IEEE Trans. Antennas Propag. (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

J. Opt. A (1)

J. Hallam, S. Chelkowski, A. Freise, S. Hild, B. Barr, K. Strain, O. Burmeister, and R. Schnabel, “Coupling of lateral grating displacement to the output ports of a diffractive Fabry-Perot cavity,” J. Opt. A,  11, 085502 (2009).
[Crossref]

J. Phys.: Conf. Ser. (2)

D. Lodhia, F. Brückner, L. Carbone, P. Fulda, K. Kokeyama, and A. Freise, “Phase effects in Gaussian beams on diffraction gratings,” J. Phys.: Conf. Ser. 363012014 (2012).
[Crossref]

A. Bunkowski, O. Burmeister, T. Clausnitzer, E. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractive optics for gravitational wave detectors,” J. Phys.: Conf. Ser. 32, 333–338 (2006).
[Crossref]

New J. Phys. (1)

A. Freise, A. Bunkowski, and R. Schnabel, “Phase and alignment noise in grating interferometers,” New J. Phys. 9, 433 (2007).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Opt. Letters (1)

D. Brown, D. Friedrich, Frank Brückner, L. Carbone, R. Schnabel, and A. Freise, “Invariance of waveguide grating mirrors to lateral displacement phase shifts,” Opt. Letters,  38, 1844–1846 (2013).
[Crossref]

Phys. Rev. Lett. (1)

S. Wise, V. Quetschke, A. Deshpande, G. Mueller, D. Reitze, D. Tanner, B. Whiting, Y. Chen, A. Tünnermann, E. Kley, and et al., “Phase effects in the diffraction of light: beyond the grating equation,” Phys. Rev. Lett. 95, 13901 (2005).
[Crossref]

Other (2)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (3rd ed., Artech House, 2005).

R. Drever, “Concepts for extending the ultimate sensitivity of interferometric gravitational wave detectors using non-transmissive optics with diffractive or holographic coupling,” in Proceedings of the Seventh Marcel Grossman Meeting on General Relativity, M. Keiser and R. T. Jantzen, eds. (World Scientific, 1995).

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Figures (6)
Fig. 1
Fig. 1 Left: Diffraction of light into the m-th diffraction order when the grating is displaced by amount Δx′ relative to the beam. A grating displacement Δx′ corresponds to a parallel beam displacement h, and induces an output optical path length change of ΔP according to Eq. (2). Right: A displaced zero-order mode beam can be decomposed into non-displaced zero-order and first-order mode beams for fixed coordinate systems.

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Fig. 2
Fig. 2 Beam phase (wave front) for a fundamental non-translated beam (solid blue), translated beam (solid green) and a modally decomposed beam (solid red). For comparison, the dashed lines represent the corresponding beams before grating diffraction. The following parameters are assumed: h = 0.05 mm, λ = 10−6 m, z = 0.5 m, ω0i = 10 mm and ω 0 r x = 8.99   mm.

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Fig. 3
Fig. 3 Layout of a grating Mach-Zehnder interferometer. A square-wave signal is injected into the mode-cleaner, allowing the instrument to lock to zero-order and first-order mode resonances alternately. One arm of the interferometer accommodates the diffraction grating, whilst the length of the other arm is subjected to tiny fluctuations to create interfering fringe signals at the output.

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Fig. 4
Fig. 4 Interference fringe signal during mode-switching. From top to bottom: PZT modulation signal in one of the Mach-Zehnder arms (purple); fringe signal due to interference at the east port (red); output signal from the MC (blue); square-wave signal applied to the MC to ramp between modes (green), where the maximum and minimum part of the signal correspond to the zero-order and first-order mode resonances respectively. As the system switches between the modes, the waveform of the fringe signal continues undisturbed -this effect is only achieved when the phase of each mode is the same. Note that the slight fluctuations visible in the fringe signal (and simultaneously in the output of the MC) are due to the stabilisation effects of the electronics when locking to each mode.

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Fig. 5
Fig. 5 Diffraction patterns formed by a TEM00 beam (left) and a TEM10 (right). The beams propagate through the diffraction grating (dashed grey lines), and the phase for each diffraction order, m = 0 and m = ±1, is measured at the reference planes (solid pink lines).

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Fig. 6
Fig. 6 Phase changes for a displaced beam after grating diffraction, as measured in the m = 0 and m = ±1 diffractive orders. Three cases are considered: beam/grating displacement (solid), modal decomposition with reference planes adjusted vertically (dotted), and modal decomposition with fixed reference planes (dashed). Note that the green dashed line is coincident with the green solid line.

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Tables (1)

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Table 1 Parameter values used to simulate Gaussian beam diffraction by a grating.

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Equations (15)

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sin   α + sin   β m = m λ d ,
Δ P = δ 1 + δ 2 = − Δ x ′ m λ d .
h = Δ x ′ cos   α .
u 0 t ( x , z 0 ) = ( 2 π ) 1 4 1 ω 0 exp   ( − ( x − h ) 2 ω 0 2 ) ,
u 0 d ( x , z 0 ) = u 0 ( x , z 0 ) + h ω 0 u 1 ( x , z 0 ) ,
θ f , t , d = k z − 1 2 Ψ + ϕ t , f , d ,
ω 0 r x = ω 0 i cos   β cos   α .
E ( x , y , z ) = ∑ n m a n m ( x , y , z ) u n m ( x , y , z ) e − i k z .
u n m ( x , y , z ) = ( 2 n + m − 1 n ! m ! π ) 1 2 1 ω ( z ) exp   ( i ( n + m + 1 ) Ψ ( z ) ) H n ( 2 x ω ( z ) ) × H m ( 2 y ω ( z ) ) exp   ( − i k ( x 2 + y 2 ) 2 R C ( z ) − x 2 + y 2 ω 2 ( z ) ) ,
u n ( x , z ) = ( 2 π ) 1 4 ( exp   ( i ( 2 n + 1 ) Ψ ( z ) ) 2 n n ! ω ( z ) ) 1 2 H n ( 2 x ω ( z ) ) exp   ( − i k x 2 2 R C ( z ) − x 2 ω 2 ( z ) ) .
u 0 ( x , z 0 ) = ( 2 π ) 1 4 1 ω 0 exp   ( − x 2 ω 0 2 ) .
u 1 ( x , z 0 ) = ( 2 π ) 1 4 1 2 ω 0 ( 2 2 x ω 0 ) exp   ( − x 2 ω 0 2 ) .
u 1 ( x , z 0 ) = 2 x ω 0 u 0 ( x , z 0 ) .
ϕ f = k x 2 2 R C ,                         ϕ t = k ( x − h ) 2 2 R C ,                         ϕ d = k x 2 2 R C − φ .
φ = arctan ( sin   Ψ cos   Ψ + ( ω   ω 0 2 x h ) ) .

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