Abstract

High-finesse resonant cavities play an important role in many metrology applications such as gravitational wave detectors. The performance of these cavities can be limited by round-trip losses (RTP) generated by light that is scattered by the mirror surface defects into higher-order modes that are close to resonance. In this paper we develop a detailed model of this effect and we study possible strategies to correct the mirror surface. We show that it is possible to restrict the correction to the combination of a reduced set of surface deformations that can be reproduced on the mirror using projected heating patterns. We show with an optical simulation that by acting on the cavity mirrors it is possible to reduce RTP to the large angle scattering limit. We also show that the optimal correction can be computed without any a priori knowledge of the mirror surface, but based only on measurements of the power stored inside the cavity, thus opening up the possibility of a simple implementation of the proposed algorithm.

© 2014 Optical Society of America

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References

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  1. The Virgo Collaboration, “Advanced Virgo Baseline Design,” VIR-027A-09, https://tds.ego-gw.it/ql/?c=6589 .
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    [CrossRef]
  3. K. Kuroda, “Large-scale gravitational wave telescope (LCGT),” Int. J. Mod. Phys. D 20, 1755–1770 (2011).
  4. P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
    [CrossRef]
  5. R. A. Day, G. Vajente, M. Kasprzack, and J. Marque, “Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction (CHRAC),” Phys. Rev. D 87, 082003 (2013).
    [CrossRef]
  6. A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys. Conf. Ser. 363, 012016 (2012).
    [CrossRef]
  7. E. Siegman, Lasers (University Science, 1986).
  8. W. Winkler, R. Schilling, K. Danzmann, J. Mizuno, A. Ridiger, and K. A. Strain, “Light scattering described in the mode picture,” Appl. Opt. 33, 7547–7550 (1994).
    [CrossRef]
  9. G. Vajente and R. A. Day, “Adaptive optics sensing and control technique to optimize the resonance of the Laguerre–Gauss 33 mode in Fabry–Perot cavities,” Phys. Rev. D 87, 122005 (2013).
    [CrossRef]
  10. T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, “Effects of mirror aberrations on Laguerre–Gaussian beams in interferometric gravitational-wave detectors,” Phys. Rev. D 84, 102001 (2011).
    [CrossRef]
  11. A. Haber, A. Polo, S. Ravensbergen, H. P. Urbach, and M. Verhaegen, “Identification of a dynamical model of a thermally actuated deformable mirror,” Opt. Lett. 38, 3061–3064 (2013).
    [CrossRef]
  12. M. Morari, C. E. Garcia, and D. M. Prett, “Model predictive control: theory and practice a survey,” Automatica 25, 335–348 (1989).
    [CrossRef]
  13. M. Ando, K. Arai, K. Kawabe, and K. Tsubono, “Demonstration of power recycling on a Fabry–Perot-type prototype gravitational wave detector,” Phys. Lett. A 248, 145–150 (1998).
    [CrossRef]

2013

R. A. Day, G. Vajente, M. Kasprzack, and J. Marque, “Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction (CHRAC),” Phys. Rev. D 87, 082003 (2013).
[CrossRef]

G. Vajente and R. A. Day, “Adaptive optics sensing and control technique to optimize the resonance of the Laguerre–Gauss 33 mode in Fabry–Perot cavities,” Phys. Rev. D 87, 122005 (2013).
[CrossRef]

A. Haber, A. Polo, S. Ravensbergen, H. P. Urbach, and M. Verhaegen, “Identification of a dynamical model of a thermally actuated deformable mirror,” Opt. Lett. 38, 3061–3064 (2013).
[CrossRef]

2012

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys. Conf. Ser. 363, 012016 (2012).
[CrossRef]

2011

K. Kuroda, “Large-scale gravitational wave telescope (LCGT),” Int. J. Mod. Phys. D 20, 1755–1770 (2011).

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, “Effects of mirror aberrations on Laguerre–Gaussian beams in interferometric gravitational-wave detectors,” Phys. Rev. D 84, 102001 (2011).
[CrossRef]

2010

G. M. Harry, for the LIGO Scientific Collaboration, “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quant. Grav. 27, 084006 (2010).
[CrossRef]

2002

P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
[CrossRef]

1998

M. Ando, K. Arai, K. Kawabe, and K. Tsubono, “Demonstration of power recycling on a Fabry–Perot-type prototype gravitational wave detector,” Phys. Lett. A 248, 145–150 (1998).
[CrossRef]

1994

1989

M. Morari, C. E. Garcia, and D. M. Prett, “Model predictive control: theory and practice a survey,” Automatica 25, 335–348 (1989).
[CrossRef]

Adhikari, R.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, “Effects of mirror aberrations on Laguerre–Gaussian beams in interferometric gravitational-wave detectors,” Phys. Rev. D 84, 102001 (2011).
[CrossRef]

Ando, M.

M. Ando, K. Arai, K. Kawabe, and K. Tsubono, “Demonstration of power recycling on a Fabry–Perot-type prototype gravitational wave detector,” Phys. Lett. A 248, 145–150 (1998).
[CrossRef]

Arai, K.

M. Ando, K. Arai, K. Kawabe, and K. Tsubono, “Demonstration of power recycling on a Fabry–Perot-type prototype gravitational wave detector,” Phys. Lett. A 248, 145–150 (1998).
[CrossRef]

Chen, Y.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, “Effects of mirror aberrations on Laguerre–Gaussian beams in interferometric gravitational-wave detectors,” Phys. Rev. D 84, 102001 (2011).
[CrossRef]

P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
[CrossRef]

Coccia, E.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys. Conf. Ser. 363, 012016 (2012).
[CrossRef]

Danzmann, K.

Day, R. A.

G. Vajente and R. A. Day, “Adaptive optics sensing and control technique to optimize the resonance of the Laguerre–Gauss 33 mode in Fabry–Perot cavities,” Phys. Rev. D 87, 122005 (2013).
[CrossRef]

R. A. Day, G. Vajente, M. Kasprzack, and J. Marque, “Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction (CHRAC),” Phys. Rev. D 87, 082003 (2013).
[CrossRef]

Fafone, V.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys. Conf. Ser. 363, 012016 (2012).
[CrossRef]

Garcia, C. E.

M. Morari, C. E. Garcia, and D. M. Prett, “Model predictive control: theory and practice a survey,” Automatica 25, 335–348 (1989).
[CrossRef]

Haber, A.

Harry, G. M.

G. M. Harry, for the LIGO Scientific Collaboration, “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quant. Grav. 27, 084006 (2010).
[CrossRef]

Hong, T.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, “Effects of mirror aberrations on Laguerre–Gaussian beams in interferometric gravitational-wave detectors,” Phys. Rev. D 84, 102001 (2011).
[CrossRef]

Kasprzack, M.

R. A. Day, G. Vajente, M. Kasprzack, and J. Marque, “Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction (CHRAC),” Phys. Rev. D 87, 082003 (2013).
[CrossRef]

Kawabe, K.

M. Ando, K. Arai, K. Kawabe, and K. Tsubono, “Demonstration of power recycling on a Fabry–Perot-type prototype gravitational wave detector,” Phys. Lett. A 248, 145–150 (1998).
[CrossRef]

Kuroda, K.

K. Kuroda, “Large-scale gravitational wave telescope (LCGT),” Int. J. Mod. Phys. D 20, 1755–1770 (2011).

Malvezzi, V.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys. Conf. Ser. 363, 012016 (2012).
[CrossRef]

Marque, J.

R. A. Day, G. Vajente, M. Kasprzack, and J. Marque, “Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction (CHRAC),” Phys. Rev. D 87, 082003 (2013).
[CrossRef]

Miller, J.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, “Effects of mirror aberrations on Laguerre–Gaussian beams in interferometric gravitational-wave detectors,” Phys. Rev. D 84, 102001 (2011).
[CrossRef]

Minenkov, Y.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys. Conf. Ser. 363, 012016 (2012).
[CrossRef]

Mizuno, J.

Morari, M.

M. Morari, C. E. Garcia, and D. M. Prett, “Model predictive control: theory and practice a survey,” Automatica 25, 335–348 (1989).
[CrossRef]

Polo, A.

Prett, D. M.

M. Morari, C. E. Garcia, and D. M. Prett, “Model predictive control: theory and practice a survey,” Automatica 25, 335–348 (1989).
[CrossRef]

Purdue, P.

P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
[CrossRef]

Ravensbergen, S.

Ridiger, A.

Rocchi, A.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys. Conf. Ser. 363, 012016 (2012).
[CrossRef]

Schilling, R.

Siegman, E.

E. Siegman, Lasers (University Science, 1986).

Sperandio, L.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys. Conf. Ser. 363, 012016 (2012).
[CrossRef]

Strain, K. A.

Tsubono, K.

M. Ando, K. Arai, K. Kawabe, and K. Tsubono, “Demonstration of power recycling on a Fabry–Perot-type prototype gravitational wave detector,” Phys. Lett. A 248, 145–150 (1998).
[CrossRef]

Urbach, H. P.

Vajente, G.

R. A. Day, G. Vajente, M. Kasprzack, and J. Marque, “Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction (CHRAC),” Phys. Rev. D 87, 082003 (2013).
[CrossRef]

G. Vajente and R. A. Day, “Adaptive optics sensing and control technique to optimize the resonance of the Laguerre–Gauss 33 mode in Fabry–Perot cavities,” Phys. Rev. D 87, 122005 (2013).
[CrossRef]

Verhaegen, M.

Winkler, W.

Yamamoto, H.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, “Effects of mirror aberrations on Laguerre–Gaussian beams in interferometric gravitational-wave detectors,” Phys. Rev. D 84, 102001 (2011).
[CrossRef]

Appl. Opt.

Automatica

M. Morari, C. E. Garcia, and D. M. Prett, “Model predictive control: theory and practice a survey,” Automatica 25, 335–348 (1989).
[CrossRef]

Class. Quant. Grav.

G. M. Harry, for the LIGO Scientific Collaboration, “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quant. Grav. 27, 084006 (2010).
[CrossRef]

Int. J. Mod. Phys. D

K. Kuroda, “Large-scale gravitational wave telescope (LCGT),” Int. J. Mod. Phys. D 20, 1755–1770 (2011).

J. Phys. Conf. Ser.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys. Conf. Ser. 363, 012016 (2012).
[CrossRef]

Opt. Lett.

Phys. Lett. A

M. Ando, K. Arai, K. Kawabe, and K. Tsubono, “Demonstration of power recycling on a Fabry–Perot-type prototype gravitational wave detector,” Phys. Lett. A 248, 145–150 (1998).
[CrossRef]

Phys. Rev. D

G. Vajente and R. A. Day, “Adaptive optics sensing and control technique to optimize the resonance of the Laguerre–Gauss 33 mode in Fabry–Perot cavities,” Phys. Rev. D 87, 122005 (2013).
[CrossRef]

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, “Effects of mirror aberrations on Laguerre–Gaussian beams in interferometric gravitational-wave detectors,” Phys. Rev. D 84, 102001 (2011).
[CrossRef]

P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
[CrossRef]

R. A. Day, G. Vajente, M. Kasprzack, and J. Marque, “Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction (CHRAC),” Phys. Rev. D 87, 082003 (2013).
[CrossRef]

Other

E. Siegman, Lasers (University Science, 1986).

The Virgo Collaboration, “Advanced Virgo Baseline Design,” VIR-027A-09, https://tds.ego-gw.it/ql/?c=6589 .

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Figures (6)

Fig. 1.
Fig. 1.

Surface of any mirror can be described as the variation from a reference spherical surface, which is matched with the wavefront curvature of the incoming beam. The deviation is expressed as the distance along the propagation direction of the real surface from reference surface as a function of the transverse coordinates.

Fig. 2.
Fig. 2.

Scheme of mode coupling in a resonant cavity. Here the input mirror is assumed to be perfect, while the end-mirror surface has some figure errors that couple part of the fundamental mode (in black) into HOMs (in gray).

Fig. 3.
Fig. 3.

Representation of the HOM coupling generated by the base maps described in Eq. (18). Each column corresponds to one of the maps. A Gaussian beam is reflected on the map and the amplitude of each HOM is represented by the color scale. The entire map has been rescaled setting the maximum to one.

Fig. 4.
Fig. 4.

Orthogonal set of maps that generates single eighth- or ninth-order mode when applied to a mirror. In this example the mirror has a diameter of 30 cm and the beam radius corresponds to about 48 mm.

Fig. 5.
Fig. 5.

Top row, randomly generated mirror maps for the input (left) and end (right) mirrors of a Fabry–Perot resonant cavity with Advanced Virgo geometry. Bottom row, correction maps computed as linear combination of the mode generating maps of Fig. 4. The mirror diameter is 30 cm.

Fig. 6.
Fig. 6.

Mirror correction found with the bi-dimensional sequential search algorithm described in the text. These should be compared with those shown in Fig. 5 obtained from the knowledge of the mirror surface maps.

Equations (18)

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

Ψ(x,y)=m,nψmnHGmn(x,y),
Ψr(x,y)=re2ikz(x,y)ψ00HG00(x,y)2ikrmnKmnψ00HGmn(x,y),
Kmn=mn|z(x,y)|00=HGmn*(x,y)z(x,y)HG00(x,y)dxdy.
φ3=α·ir2ψ2,
ψ3=(1α22)·ir2ψ2.
ψ3=α·ir2φ2.
[1r1r2(1α22)e2iΦ]ψ1=t1ψ0+ir1r2αe2iΦ+iΦmnφ1,
[1r1r2(1α22)e2iΦ2iΦmn]φ1=ir1r2αe2iΦ+iΦmnψ1.
ψ1=t1ψ01r1r2e2iΦ[1α22(12+r1r2e2iΦ+2iΦmn1r1r2e2iΦ+2iΦmn)].
δϕ=α22·1r12r221+r12r22·112r1r21r1r2cos(2Φ2Φmn),
L=α2·1r12r221+r12r22·112r1r21r1r2cos(2Φ2Φmn).
φ1=αir1r2e2iΦ+2iΦmn1r1r2e2iΦ+2iΦmn·t11r1r2e2iΦψ0.
φ1=t11r1r2·r1r21r1r2e2iΦmn(α1+eiΦmnα2)ψ0,
φr=r1[(α1*+e2iΦmn1r1r2e2iΦmnt1t1r2r21r1r2α1)+(e2iΦmn1r1r2e2iΦmnt1t1r2r21r1r2eiΦmnα2)]ψ0.
PHOM=P0·4kr12r22(1r1r2)2+4r1r2sin2Φmn·[α12+α22+2α1α2cosΦmn],
α2α2+kHGmn*(x,y)c2(x,y)HG00(x,y)dxdy=α2+β2.
Pmn=P00·4kr12r22(1r1r2)2+4r1r2sin2Φmnα12sin2Φmn.
Mmn=HGmnHG00*.

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