Abstract

A typical application for laser interferometers is a precision measurement of length changes that results in interferometric phase shifts. Such phase changes are typically predicted numerically, due to the complexity of the overlap integral that needs to be solved. In this paper we will derive analytical representations of the interferometric phase and contrast (aka fringe visibility) for two beam interferometers, both homodyne and heterodyne. The fundamental Gaussian beams can be arbitrarily misaligned and mismatched to each other. A limitation of the analytical result is that both beams must be detected completely, which can experimentally be realized by a sufficiently large single-element photodetector.

© 2014 Optical Society of America

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References

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  1. IfoCAD, http://www.lisa.aei-hannover.de/ifocad/ .
  2. E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.
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    [CrossRef]
  5. B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
    [CrossRef]
  6. G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
    [CrossRef]
  7. F. Cassaing, “Optical path difference sensors,” C. R. Acad. Sci. Paris 2, 87–98 (2001).
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    [CrossRef]
  14. K. Dahl, “From design to operation: a suspension platform interferometer for the AEI 10  m prototype,” Ph.D. thesis (Leibniz Universität Hannover, 2013).

2012 (2)

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[CrossRef]

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

2004 (1)

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]

2001 (1)

F. Cassaing, “Optical path difference sensors,” C. R. Acad. Sci. Paris 2, 87–98 (2001).

2000 (1)

Y. Surrel, “Fringe analysis,” Top. Appl. Phys. 77, 55–102 (2000).
[CrossRef]

1997 (1)

1993 (1)

1992 (1)

1990 (1)

1974 (1)

Andreas, B.

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[CrossRef]

Barangaccio, D. J.

Bruning, J. H.

Cassaing, F.

F. Cassaing, “Optical path difference sensors,” C. R. Acad. Sci. Paris 2, 87–98 (2001).

Dahl, K.

K. Dahl, “From design to operation: a suspension platform interferometer for the AEI 10  m prototype,” Ph.D. thesis (Leibniz Universität Hannover, 2013).

Danzmann, K.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[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]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.

Freischlad, K.

Freise, A.

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]

Fujii, K.

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[CrossRef]

Gallagher, J. E.

Heinzel, G.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[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]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.

Herriott, D. R.

Kochkina, E.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.

Koliopoulos, C. L.

Kuramoto, N.

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[CrossRef]

Larkin, K. G.

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]

Mahrdt, C.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.

Mana, G.

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[CrossRef]

Müller, V.

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.

Oreb, B. F.

Rosenfeld, D. P.

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]

Schuster, S.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.

Sheard, B.

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.

Sheard, B. S.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

Surrel, Y.

Wanner, G.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.

White, A. D.

Willke, B.

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]

Appl. Opt. (3)

C. R. Acad. Sci. Paris (1)

F. Cassaing, “Optical path difference sensors,” C. R. Acad. Sci. Paris 2, 87–98 (2001).

Class. Quantum Grav. (1)

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]

J. Opt. Soc. Am. A (2)

Metrologia (1)

B. Andreas, K. Fujii, N. Kuramoto, and G. Mana, “The uncertainty of the phase-correction in sphere-diameter measurements,” Metrologia 49, 479–486 (2012).
[CrossRef]

Opt. Commun. (1)

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. S. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

Top. Appl. Phys. (1)

Y. Surrel, “Fringe analysis,” Top. Appl. Phys. 77, 55–102 (2000).
[CrossRef]

Other (4)

IfoCAD, http://www.lisa.aei-hannover.de/ifocad/ .

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in ASP Conference Series (Astronomical Society of the Pacific, 2012), Vol. 467.

R. Schilling, OptoCad, http://www.rzg.mpg.de/ros/optocad.html .

K. Dahl, “From design to operation: a suspension platform interferometer for the AEI 10  m prototype,” Ph.D. thesis (Leibniz Universität Hannover, 2013).

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Figures (2)

Fig. 1.
Fig. 1.

Computation of the electric field E(rb,zb) with rb=xb2+yb2 at a point R on a detector surface (xy plane, represented by a dotted line). Shown is the xz plane where the incident beam—indicated by its coordinate system (e^xB,e^zB)—is shifted with respect to the photodiode center by ri and tilted about a pivot P around the y axis. The beam centroid and therefore the origin of the beam coordinate system is located at M(rirp)+rp. Any point R on the detector surface can be represented in photodiode coordinates by r=xe^x+ye^y+ze^z or equivalently in beam coordinates by rbB=xbe^xB+ybe^yB+zbe^zB, with initially unknown values xb,yb,zb. These can be computed using the vector rb=xbe^x+ybe^y+zbe^z, which is defined as rb=M1(rrp)+rpri. Remark, vectors of the same line type can be mapped by rotation via M or M1 around P.

Fig. 2.
Fig. 2.

Differential phase converted to length (longitudinal path length signal LPS=(ϕϕ|αm=0)/k, top) and fringe visibility (contrast, bottom). The solid lines were generated with Eqs. (23) and (28), respectively, the matching dots show numerical results generated with IfoCAD.

Tables (2)

Tables Icon

Table 1. List of Physical Parametersa

Tables Icon

Table 2. List of Substitutions in the Overlap Integrand IOv [Eq. (15)] for the Special Case (nx,ny,nz)m=(0,1,0) and αr=0a

Equations (35)

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

E(rb,zb,t)=A(rb,zb)exp(iωtiΦ(rb,zb)),
A(rb,zb)=2Z2Pπw2(zb)exp(rb2w2(zb)),
Φ(rb,zb)=krb22R(zb)η(zb)+kzb,
P=0drb2πrbI12Z0drb2πrb|A(rb,zb)|2.
PS=dS|Em+Er|2
=dS12Z|Arexp(iωrtiΦr)+Amexp(iωmtiΦm)|2
=dS12Z(Ar2+Am2)[1+2AmArAm2+Ar2cos(ΔωtΔΦ)]
=Pm+Pr[1+1Pm+PrdS12Z(2AmAr)cos(ΔωtΔΦ)]
P¯[1+1P¯dS2PmPrπwmwrexp(rm2wm2rr2wr2)2cos(ΔωtΔΦ)]
=P¯[1+2PmPrP¯dS2πwmwrexp(rm2wm2rr2wr2)cos(ΔωtΔΦ)]
P¯[1+APdSIOv],
PS=P¯[1+ccos(Δωt+ϕ)],
(xbybzb)m,rMm,r1[(xy0)(xpypzp)m,r]+(xpypzp)m,r(xiyi0)m,r,
Mm,r(nx2(1cos(α))+cos(α)nxny(1cos(α))nzsin(α)nxnz(1cos(α))+nysin(α)nynx(1cos(α))+nzsin(α)ny2(1cos(α))+cos(α)nynz(1cos(α))nxsin(α)nznx(1cos(α))nysin(α)nzny(1cos(α))+nxsin(α)nz2(1cos(α))+cos(α))m,r.
IOv=A0exp[(A1x2+A2xy+A3x+A4y2+A5y+A6)]cos[B1x2+B2xy+B3x+B4y2+B5y+B6]=A0exp[(xy1)T(A1A2A30A4A500A6)(xy1)]cos[(xy1)T(B1B2B30B4B500B6)(xy1)].
IOvR(IOvc)
IOvc=A0exp[(xy1)T(A1A2A30A4A500A6)(xy1)]exp[i(xy1)T(B1B2B30B4B500B6)(xy1)]
=A0exp[(xy1)T(C1C2C30C4C500C6)(xy1)],
OcdSIOvc
=2πexp[C6+C1C52+C32C4C2C3C54C1C4C22]C14C4C22C1,
R[4C4C22C1]>0.
ϕ=arg(Oc|t=0)=arctan(I(Oc|t=0),R(Oc|t=0))
=arctan(tan(D1))=mod(D1,π)π/2,
D1B6+12arg(D2)+I(D2D3*)|D2|2
D2C224C1C4
D3C1C52+C32C4C2C3C5.
c=APR(Oc)cos(Δωt+ϕ)
=2πA0APexp[A6R(D2D3*)|D2|2]|D2|.
LPS1k(ϕϕ|αm=0).
c=AP2zR,mzR,r(zR,m+zR,r)2+(zmzr)2
=AP2wmwr(k2Rmk2Rr)2+(1wm2+1wr2)2,
c=APexp[k(xi,mxi,r)2+(yi,myi,r)24zR]
=APexp[(4R2+k2w4)(xi,mxi,r)2+(yi,myi,r)28R2w2].
LPS(α,ΔzR,Δz)α2k(z4zR+ΔzR2kzR(zp+z)z8zR2+Δzk((z+zp)2zR2)+zR8zR2)
LPS(α,Δw,ΔR)α2(w28R+ΔR2R2(2zp2+w2)+k2w4(zp+R)232R4+Δww2+4zp(zp+R)8Rw).

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