Abstract

The sounding rocket principle of equivalence measurement uses a set of four laser gauges operating in Fabry–Perot cavities to determine the relative acceleration of two test masses that are chemically different. One end of each cavity is a flat mirror on a test mass. Because the test masses are unconstrained and thus expected to rotate slightly during measurement, and because the distance measurements are made at the sub-picometer level, it is essential to have real-time alignment of the beam entering the cavity. However, the cavity must be used in reflection and space is limited. We show that Anderson’s alignment method can be used in reflection, but that it requires that the Fabry–Perot cavity have mirrors with significantly unequal reflectivities.

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References

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  1. R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
    [CrossRef]
  2. R. Thapa, J. D. Phillips, E. Rocco, and R. D. Reasenberg, “Subpicometer length measurement using semiconductor laser tracking frequency gauge,” Opt. Lett. 36, 3759–3761 (2011).
    [CrossRef]
  3. J. D. Phillips and R. D. Reasenberg, “Semiconductor laser tracking frequency distance gauge,” Proc. SPIE 7436, 74360T (2009).
    [CrossRef]
  4. D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23, 2944–2949 (1984).
    [CrossRef]
  5. N. M. Sampas and D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. 29, 394–403 (1990).
    [CrossRef]
  6. E. Morrison, B. J. Meers, D. I. Robertson, and H. Ward, “Automatic alignment of optical interferometers,” Appl. Opt. 33, 5041–5049 (1994).
    [CrossRef]
  7. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  8. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1976).

2011 (2)

R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
[CrossRef]

R. Thapa, J. D. Phillips, E. Rocco, and R. D. Reasenberg, “Subpicometer length measurement using semiconductor laser tracking frequency gauge,” Opt. Lett. 36, 3759–3761 (2011).
[CrossRef]

2009 (1)

J. D. Phillips and R. D. Reasenberg, “Semiconductor laser tracking frequency distance gauge,” Proc. SPIE 7436, 74360T (2009).
[CrossRef]

1994 (1)

1990 (1)

1984 (1)

1983 (1)

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

Anderson, D. Z.

Drever, R. W. P.

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

Ford, G. M.

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

Hall, J. L.

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

Hough, J.

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

Kowalski, F. V.

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

Lorenzini, E. C.

R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
[CrossRef]

Meers, B. J.

Morrison, E.

Munley, A. J.

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

Patla, B. R.

R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
[CrossRef]

Phillips, J. D.

R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
[CrossRef]

R. Thapa, J. D. Phillips, E. Rocco, and R. D. Reasenberg, “Subpicometer length measurement using semiconductor laser tracking frequency gauge,” Opt. Lett. 36, 3759–3761 (2011).
[CrossRef]

J. D. Phillips and R. D. Reasenberg, “Semiconductor laser tracking frequency distance gauge,” Proc. SPIE 7436, 74360T (2009).
[CrossRef]

Popescu, E. E.

R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
[CrossRef]

Reasenberg, R. D.

R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
[CrossRef]

R. Thapa, J. D. Phillips, E. Rocco, and R. D. Reasenberg, “Subpicometer length measurement using semiconductor laser tracking frequency gauge,” Opt. Lett. 36, 3759–3761 (2011).
[CrossRef]

J. D. Phillips and R. D. Reasenberg, “Semiconductor laser tracking frequency distance gauge,” Proc. SPIE 7436, 74360T (2009).
[CrossRef]

Robertson, D. I.

Rocco, E.

R. Thapa, J. D. Phillips, E. Rocco, and R. D. Reasenberg, “Subpicometer length measurement using semiconductor laser tracking frequency gauge,” Opt. Lett. 36, 3759–3761 (2011).
[CrossRef]

R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
[CrossRef]

Sampas, N. M.

Thapa, R.

R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
[CrossRef]

R. Thapa, J. D. Phillips, E. Rocco, and R. D. Reasenberg, “Subpicometer length measurement using semiconductor laser tracking frequency gauge,” Opt. Lett. 36, 3759–3761 (2011).
[CrossRef]

Ward, H.

E. Morrison, B. J. Meers, D. I. Robertson, and H. Ward, “Automatic alignment of optical interferometers,” Appl. Opt. 33, 5041–5049 (1994).
[CrossRef]

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

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1976).

Appl. Opt. (3)

Appl. Phys. B (1)

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

Class. Quantum Grav. (1)

R. D. Reasenberg, E. C. Lorenzini, B. R. Patla, J. D. Phillips, E. E. Popescu, E. Rocco, and R. Thapa, “A quick test of the WEP enabled by a sounding rocket,” Class. Quantum Grav. 28, 094014 (2011).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

J. D. Phillips and R. D. Reasenberg, “Semiconductor laser tracking frequency distance gauge,” Proc. SPIE 7436, 74360T (2009).
[CrossRef]

Other (1)

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1976).

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Figures (1)

Fig. 1.
Fig. 1.

γ(η) over the allowed range of η from Rc1/2 to Rc1/2 and shown for the case of Rc=0.85. Solid for a reflection cavity; dashed for a transmission cavity.

Tables (1)

Tables Icon

Table 1. Effect of ρ and γ on the Observability of the Alignment Parameters p (Displacement Measure) and q (Rotation Measure) for η=1, Except That Values Given in Square Brackets Have η1 Such That γ=0.5a

Equations (24)

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Um,n=Vm(x)Vn(y),
V0(x)=(2πwx2)1/4e(x/wx)2eix2π/λRV1(x)=(2πwx2)1/42xwxe(x/wx)2eix2π/λR.
ViVj*dx=δi,j.
U0=U0,0,U1=U1,0.
Ψm=E0eiωtU0(J0(m)+J1(m)(eiΩte1Ωt)+),
Ψdec=E0eiωt(C0U0+(p+iq)U1),
Ψin=E0eiωt(C0U0(J0+J1(eiΩteiΩt))+(p+iq)U1(J0+J1(eiΩteiΩt))).
ΓR=(1eiδ)Rc1Rceiδ,
ΓR=(ηη1eiδ)Rc1Rceiδ,
QR(δ)=ΓRΓR*=η2+η22+4sin2(δ/2)(1Rc)2+4Rcsin2(δ/2)Rc,
ΓT=(1Rc)2Rc(η2+η22)1Rceiδ,
QT(δ)=ΓTΓT*=(1Rc)2Rc(η2+η22)(1Rc)2+4Rcsin2(δ/2).
Ψout=E0eiωt(C0U0(γ0eiϕ0J0+ρJ1(eiΩteiΩt))+(p+iq)U1(J0ρ+J1(γ1eiϕ1eiΩtρeiΩt))),
Sout=ΨoutΨout*/E02S0+pSp+qSq.
S0=U02(γ02J02+4γ0ρJ0J1sinφosin(Ωt)+4ρ2J12sin2(Ωt)),
Sq=2U0U1[γ0ρJ02cosφ0+γ0J0J1(ρcos(Ωt+φ1)+γ1cos(Ωt+φ1φ0))+J12(ρ2(1cos(2Ωt))+γ1ρ(cosφ1cos(2Ωt+φ1)))]
Sp=2U0U1[γ0ρJ02sinφ0+J0J1(2ρ2sin(Ωt)γ0(ρsin(Ωt+φ0)+γ1sin(Ωt+φ1φ0)))+J12(ρ2sin(2Ωt)+γ1ρ(sin(2Ωt+φ1)sinφ1))].
0V0(x)V1*(x)dx=0V0(x)V1*(x)dx=12π.
γ0=(η2+η22)Rc1Rc1+Rc(1Rc)2/41Rc(|1η|(1η)22),
S˜0=U02(γ02J02+2ρ2J12(1cos(2Ωt))),
S˜p=2U0U1(J02ργ0J0J1γ0(ργ1)cos(Ωt)+J12ρ(ρ+γ1)(1cos(2Ωt))),
S˜q=2U0U1(J0J1[2ρ2γ0(ρ+γ1)]sin(Ωt)J12ρ(ργ1)sin(2Ωt)).
1Rb=(1Ra)41γ2
1Rb=(1Ra)(1γ2)4.

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