Abstract

We present a method to analyze the coupling of lateral displacements in nanoscale structures, in particular waveguide grating mirrors (WGMs), into the phase of a reflected Gaussian beam using a finite-difference time-domain simulation. Such phase noise is of interest for using WGMs in high-precision interferometry. We show that, to the precision of our simulations (107rad), waveguide mirrors do not couple lateral displacement into phase noise of a reflected beam and that WGMs are therefore not subject to the same stringent alignment requirements as previously proposed layouts using diffraction gratings.

© 2013 Optical Society of America

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  1. H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Phys. Rev. Lett. 100, 013601 (2008).
    [CrossRef]
  2. K. Numata, A. Kemery, and J. Camp, Phys. Rev. Lett. 93, 250602 (2004).
    [CrossRef]
  3. G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
    [CrossRef]
  4. S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
    [CrossRef]
  5. K.-X. Sun and R. L. Byer, Opt. Lett. 23, 567 (1998).
    [CrossRef]
  6. A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, Classical Quantum Gravity 23, 7297 (2006).
    [CrossRef]
  7. F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, Opt. Lett. 33, 264 (2008).
    [CrossRef]
  8. S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
    [CrossRef]
  9. D. Lodhia, D. Brown, F. Brückner, and A. Freise, “Phase effects from diffraction gratings for alignment stability,” (in preparation).
  10. A. Freise, A. Bunkowski, and R. Schnabel, New J. Phys. 9, 433 (2007).
    [CrossRef]
  11. A. Sharon, D. Rosenblatt, and A. A. Friesem, J. Opt. Soc. Am. A 14, 2985 (1997).
    [CrossRef]
  12. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  13. P. Sysel and P. Rajmic, EURASIP J. Adv. Signal Process. 2012, 1 (2012).
    [CrossRef]

2012

P. Sysel and P. Rajmic, EURASIP J. Adv. Signal Process. 2012, 1 (2012).
[CrossRef]

2009

S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
[CrossRef]

2008

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef]

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, Opt. Lett. 33, 264 (2008).
[CrossRef]

2007

A. Freise, A. Bunkowski, and R. Schnabel, New J. Phys. 9, 433 (2007).
[CrossRef]

2006

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, Classical Quantum Gravity 23, 7297 (2006).
[CrossRef]

2005

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

2004

K. Numata, A. Kemery, and J. Camp, Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

2002

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

1998

1997

Brown, D.

D. Lodhia, D. Brown, F. Brückner, and A. Freise, “Phase effects from diffraction gratings for alignment stability,” (in preparation).

Brückner, F.

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, Opt. Lett. 33, 264 (2008).
[CrossRef]

D. Lodhia, D. Brown, F. Brückner, and A. Freise, “Phase effects from diffraction gratings for alignment stability,” (in preparation).

Bunkowski, A.

A. Freise, A. Bunkowski, and R. Schnabel, New J. Phys. 9, 433 (2007).
[CrossRef]

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, Classical Quantum Gravity 23, 7297 (2006).
[CrossRef]

Burmeister, O.

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, Opt. Lett. 33, 264 (2008).
[CrossRef]

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, Classical Quantum Gravity 23, 7297 (2006).
[CrossRef]

Byer, R. L.

Cagnoli, G.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Camp, J.

K. Numata, A. Kemery, and J. Camp, Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

Chelkowski, S.

S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
[CrossRef]

Chen, Y.

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef]

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Clausnitzer, T.

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, Opt. Lett. 33, 264 (2008).
[CrossRef]

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Crooks, D. R. M.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Danzmann, K.

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, Opt. Lett. 33, 264 (2008).
[CrossRef]

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef]

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, Classical Quantum Gravity 23, 7297 (2006).
[CrossRef]

Deshpande, A. J.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Fejer, M. M.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Freise, A.

S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
[CrossRef]

A. Freise, A. Bunkowski, and R. Schnabel, New J. Phys. 9, 433 (2007).
[CrossRef]

D. Lodhia, D. Brown, F. Brückner, and A. Freise, “Phase effects from diffraction gratings for alignment stability,” (in preparation).

Friedrich, D.

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, Opt. Lett. 33, 264 (2008).
[CrossRef]

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, Classical Quantum Gravity 23, 7297 (2006).
[CrossRef]

Friesem, A. A.

Gretarsson, A. M.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Hagness, S. C.

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

Harry, G. M.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Hild, S.

S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
[CrossRef]

Hough, J.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Kemery, A.

K. Numata, A. Kemery, and J. Camp, Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

Kittelberger, S. E.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Kley, E.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Kley, E.-B.

Lodhia, D.

D. Lodhia, D. Brown, F. Brückner, and A. Freise, “Phase effects from diffraction gratings for alignment stability,” (in preparation).

Mueller, G.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Müller-Ebhardt, H.

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef]

Nakagawa, N.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Numata, K.

K. Numata, A. Kemery, and J. Camp, Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

Penn, S. D.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Quetschke, V.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Rajmic, P.

P. Sysel and P. Rajmic, EURASIP J. Adv. Signal Process. 2012, 1 (2012).
[CrossRef]

Rehbein, H.

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef]

Reitze, D. H.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Rosenblatt, D.

Rowan, S.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Saulson, P. R.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Schnabel, R.

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, Opt. Lett. 33, 264 (2008).
[CrossRef]

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef]

A. Freise, A. Bunkowski, and R. Schnabel, New J. Phys. 9, 433 (2007).
[CrossRef]

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, Classical Quantum Gravity 23, 7297 (2006).
[CrossRef]

Sharon, A.

Startin, W. J.

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

Sun, K.-X.

Sysel, P.

P. Sysel and P. Rajmic, EURASIP J. Adv. Signal Process. 2012, 1 (2012).
[CrossRef]

Taflove, A.

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

Tanner, D. B.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Tünnermann, A.

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, Opt. Lett. 33, 264 (2008).
[CrossRef]

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Whiting, B. F.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Wise, S.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

Classical Quantum Gravity

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, Classical Quantum Gravity 19, 897 (2002).
[CrossRef]

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, Classical Quantum Gravity 23, 7297 (2006).
[CrossRef]

EURASIP J. Adv. Signal Process.

P. Sysel and P. Rajmic, EURASIP J. Adv. Signal Process. 2012, 1 (2012).
[CrossRef]

J. Opt. Soc. Am. A

New J. Phys.

A. Freise, A. Bunkowski, and R. Schnabel, New J. Phys. 9, 433 (2007).
[CrossRef]

Opt. Lett.

Phys. Rev. D

S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
[CrossRef]

Phys. Rev. Lett.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, B. F. Whiting, Y. Chen, A. Tünnermann, E. Kley, and T. Clausnitzer, Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef]

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef]

K. Numata, A. Kemery, and J. Camp, Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

Other

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

D. Lodhia, D. Brown, F. Brückner, and A. Freise, “Phase effects from diffraction gratings for alignment stability,” (in preparation).

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Figures (4)

Fig. 1.
Fig. 1.

Incident beam (black) is coupled into the waveguide layer by the grating into m=±1. Orders ±1T propagate along the waveguide coupling back into the vacuum (blue) to interfere with the initially reflected light, 0R, picking up ΔΦm phase terms with each interaction with the grating. Further coupling into the vacuum is also possible (magenta), which involved further ΔΦm terms. The “*” superscript refers to the number of diffractions the beam has undergone. R=Reflection fromgrating, T=Transmission from grating, first number is order of diffraction m.

Fig. 2.
Fig. 2.

Schematic layout of 2D FDTD simulation for testing WGM shift invariance. Gaussian beam injected along TFSF boundary onto WGM (red). The reflected beam (green) then propagates to the measurement line where the phase is measured. The CPML absorbs outgoing waves to reduce reflections from boundaries.

Fig. 3.
Fig. 3.

Central plot shows phase change in the reflected beams wavefront Δϕp(x) against WGM displacement, δx. Top plot shows cross section of phase at δx=350nm, where the phase change is maximum. The right plot shows the variation in the phase of at x=0 against δx. Bottom plot shows the intensity of the reflected beam along x.

Fig. 4.
Fig. 4.

Maximum change in phase at beam peak intensity as the WGM is displaced from δx=0d with increasing distance from the WGM surface and different FFT windowing functions. Large near-field phase shifts are seen close to the WGM and then a flat noise. Increasing the spatial and temporal resolution of FDTD ΔxΔx/2 did not appear to offer reductions in numerical noise caused by FDTD.

Equations (1)

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ΔΦm=2πmδx/d,

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