Abstract

We demonstrate an optical cavity that supports an eigenmode with a flattop spatial profile—a profile that has been proposed for the cavities in the Advanced Laser Interferometer Gravitational Wave Observatory, the second-generation laser interferometric gravitational wave observatory—because it provides better averaging of the spatially dependent displacement noise on the surface of the mirror than a Gaussian beam. We describe the deformable mirror that we fabricated to tailor the shape of the eigenmode of the cavity and show that this cavity is a factor of 2 more sensitive to misalignments than a comparable cavity with spherical mirrors supporting an eigenmode with a Gaussian profile.

© 2006 Optical Society of America

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References

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  1. P. Fritschel, "The second generation LIGO interferometers," AIP Conf. Proc. 575, 15-23 (2001).
    [CrossRef]
  2. Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses," Phys. Rev. D 62, 122002 (2000).
    [CrossRef]
  3. E. D'Ambrosio, R. O'Shaughnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin, "Advanced LIGO: non-Gaussian beams," Class. Quantum Grav. 21, S867-S873 (2004).
    [CrossRef]
  4. E. D'Ambrosio, R. O'Shaughnessy, S. Strigin, K. S. Thorne, and S. Vyatchanin, "Reducing thermoelastic noise in gravitational-wave interferometers by flattening the light beams," arXiv.org e-Print archive, gr-qc/0409075, 20 September 2004, http://arxiv.org/abs/gr-qc/0409075.

2004 (1)

E. D'Ambrosio, R. O'Shaughnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin, "Advanced LIGO: non-Gaussian beams," Class. Quantum Grav. 21, S867-S873 (2004).
[CrossRef]

2001 (1)

P. Fritschel, "The second generation LIGO interferometers," AIP Conf. Proc. 575, 15-23 (2001).
[CrossRef]

2000 (1)

Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses," Phys. Rev. D 62, 122002 (2000).
[CrossRef]

D'Ambrosio, E.

E. D'Ambrosio, R. O'Shaughnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin, "Advanced LIGO: non-Gaussian beams," Class. Quantum Grav. 21, S867-S873 (2004).
[CrossRef]

E. D'Ambrosio, R. O'Shaughnessy, S. Strigin, K. S. Thorne, and S. Vyatchanin, "Reducing thermoelastic noise in gravitational-wave interferometers by flattening the light beams," arXiv.org e-Print archive, gr-qc/0409075, 20 September 2004, http://arxiv.org/abs/gr-qc/0409075.

Fritschel, P.

P. Fritschel, "The second generation LIGO interferometers," AIP Conf. Proc. 575, 15-23 (2001).
[CrossRef]

Liu, Y. T.

Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses," Phys. Rev. D 62, 122002 (2000).
[CrossRef]

O'Shaughnessy, R.

E. D'Ambrosio, R. O'Shaughnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin, "Advanced LIGO: non-Gaussian beams," Class. Quantum Grav. 21, S867-S873 (2004).
[CrossRef]

E. D'Ambrosio, R. O'Shaughnessy, S. Strigin, K. S. Thorne, and S. Vyatchanin, "Reducing thermoelastic noise in gravitational-wave interferometers by flattening the light beams," arXiv.org e-Print archive, gr-qc/0409075, 20 September 2004, http://arxiv.org/abs/gr-qc/0409075.

Strigin, S.

E. D'Ambrosio, R. O'Shaughnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin, "Advanced LIGO: non-Gaussian beams," Class. Quantum Grav. 21, S867-S873 (2004).
[CrossRef]

E. D'Ambrosio, R. O'Shaughnessy, S. Strigin, K. S. Thorne, and S. Vyatchanin, "Reducing thermoelastic noise in gravitational-wave interferometers by flattening the light beams," arXiv.org e-Print archive, gr-qc/0409075, 20 September 2004, http://arxiv.org/abs/gr-qc/0409075.

Thorne, K.

E. D'Ambrosio, R. O'Shaughnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin, "Advanced LIGO: non-Gaussian beams," Class. Quantum Grav. 21, S867-S873 (2004).
[CrossRef]

Thorne, K. S.

Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses," Phys. Rev. D 62, 122002 (2000).
[CrossRef]

E. D'Ambrosio, R. O'Shaughnessy, S. Strigin, K. S. Thorne, and S. Vyatchanin, "Reducing thermoelastic noise in gravitational-wave interferometers by flattening the light beams," arXiv.org e-Print archive, gr-qc/0409075, 20 September 2004, http://arxiv.org/abs/gr-qc/0409075.

Vyatchanin, S.

E. D'Ambrosio, R. O'Shaughnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin, "Advanced LIGO: non-Gaussian beams," Class. Quantum Grav. 21, S867-S873 (2004).
[CrossRef]

E. D'Ambrosio, R. O'Shaughnessy, S. Strigin, K. S. Thorne, and S. Vyatchanin, "Reducing thermoelastic noise in gravitational-wave interferometers by flattening the light beams," arXiv.org e-Print archive, gr-qc/0409075, 20 September 2004, http://arxiv.org/abs/gr-qc/0409075.

Willems, P.

E. D'Ambrosio, R. O'Shaughnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin, "Advanced LIGO: non-Gaussian beams," Class. Quantum Grav. 21, S867-S873 (2004).
[CrossRef]

AIP Conf. Proc. (1)

P. Fritschel, "The second generation LIGO interferometers," AIP Conf. Proc. 575, 15-23 (2001).
[CrossRef]

Class. Quantum Grav. (1)

E. D'Ambrosio, R. O'Shaughnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin, "Advanced LIGO: non-Gaussian beams," Class. Quantum Grav. 21, S867-S873 (2004).
[CrossRef]

Phys. Rev. D (1)

Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses," Phys. Rev. D 62, 122002 (2000).
[CrossRef]

Other (1)

E. D'Ambrosio, R. O'Shaughnessy, S. Strigin, K. S. Thorne, and S. Vyatchanin, "Reducing thermoelastic noise in gravitational-wave interferometers by flattening the light beams," arXiv.org e-Print archive, gr-qc/0409075, 20 September 2004, http://arxiv.org/abs/gr-qc/0409075.

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Figures (10)

Fig. 1
Fig. 1

Ideal mirror shape as calculated (dashed curve) and the shape that can be approximated by an electrostatically actuated flat membrane (solid curve).

Fig. 2
Fig. 2

(Color online) Deformable mirror. The circular region in the center of the square is the active region. This central region is a thin membrane that can be distorted by electrostatic actuation from the electrodes underneath. This particular membrane is broken; the bottom half has shattered into many pieces but the top half remains intact.

Fig. 3
Fig. 3

(Color online) Left, electrode pattern and wire layout on the silicon substrate. Right, closeup of the electrode pattern. Electrodes 9–13 are used to create the desired mirror surface with rotational symmetry. Electrodes 3–8 and 14–19 are used to compensate for saddlelike membrane surface shapes that are typical for as-fabricated mirrors. Electrodes 1 and 2 provide a connection with the membrane through the bond pad. Electrodes 20 and 21 are used to keep the remaining surface area at a defined voltage (usually ground). The nominal silicon nitride membrane diameter is 10 mm and covers all the segmented electrodes and the inner concentric ring and circular electrodes. The area defined by the dotted rectangle was measured by a white-light interferometer to determine the surface profile while the mirror was being deformed.

Fig. 4
Fig. 4

Left, surface profile of the center of the actuated mirror. Right, the surface height along a cord through the center (solid diamonds) shows that the mirror surface is within 8 nm of the intended surface profile (solid circles) in the central region of the mirror.

Fig. 5
Fig. 5

Calculated eigenmode shapes for a cavity with a spherical mirror (dashed curve) and our deformable mirror (solid curve).

Fig. 6
Fig. 6

(Color online) Experimental setup of the optical cavity. A white-light interferometer (not shown) is used to monitor the shape of the deformable mirror (DM).

Fig. 7
Fig. 7

(Color online) Top left, flattop mode that resonates in the test cavity. The profiles of a chord taken through the middle of the flattop and Gaussian beams are shown in the top right and bottom left plots.

Fig. 8
Fig. 8

(Color online) Relative power buildup in the cavity as a function of pitch misalignment of the end mirror. The solid curve fits the measured data for the flattop cavity. The dashed curve is calculated for a conventional cavity. We suspect that the asymmetry of the curve for the flattop cavity is due to transverse displacement of the flattop beam from tilting the cavity end mirror coupling to the misalignment of the monitor photodiode.

Fig. 9
Fig. 9

(Color online) Estimated relative amount of thermoelastic noise that couples to the beam for a flattop beam and a Gaussian beam when the end mirror of the cavity is misaligned.

Fig. 10
Fig. 10

Calculated cavity mode shapes for a cavity with a deformable mirror (top row) and a spherical mirror (bottom row) for several values of end mirror tilt. The observed mode shapes obey similar behavior; however, the simulated mode-shape images are much cleaner allowing the effect of misalignment to be more easily seen.

Equations (4)

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U ( r > 0 ) = exp [ ( r w c ) 2 ] [ 1 H ( r 4 w c ) ] ,
C p m p m = u p m ( r , θ , L ) u p m * ( r , θ , L ) × exp { i 2 k [ S ( r , θ ) S 0 ( r , θ ) ] } r d r d θ .
u p m ( r , θ , z ) = 2 p ! ( 1 + δ 0 m ) π ( m + p ) ! × exp { i ( 2 p + m + 1 ) [ ψ ( z ) ψ 0 ] } w ( z ) × [ 2 r w ( z ) ] m L p m [ 2 r 2 w 2 ( z ) ] × exp [ i k r 2 2 q ˜ ( z ) + i m θ ] ,
1 q ˜ ( z ) = 1 R ( z ) i λ π w 2 ( z ) .

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