Abstract

An angle-measuring technique based on an optical interferometer is reported. The technique exploits a Michelson interferometric configuration in which a right-angle prism and a glass strip are introduced into a probe beam. Simultaneous rotation of both components along an axis results in an optical path difference between the reference and the probe beams. In a second arrangement two right-angle prisms and glass strips are introduced into two beams of a Michelson interferometer. The prisms and the strips are rotated simultaneously to introduce an optical path difference between the two beams. In our arrangement, optimization of various parameters makes the net optical path difference between the two beams approximately linear for a rotation as great as ±20°. Results are simulated that show an improvement of 2–3 orders of magnitude in error and nonlinearity compared with a previously reported technique.

© 1999 Optical Society of America

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References

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  1. M. Ben-Levy, S. G. Braun, J. Shamir, “Angular velocity measuring interferometer,” Appl. Opt. 18, 4265–4266 (1989).
    [CrossRef]
  2. H. K. Chiang, R. P. Kenan, N. F. Hartman, C. J. Summers, “Optical alignment and tilt angle measurement technique based on Lloyd’s mirror arrangement,” Opt. Lett. 17, 1024–1025 (1992).
    [CrossRef] [PubMed]
  3. L. Zeng, H. Mataumoto, K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variations using an acousto-optic deflector,” Opt. Eng. 35, 1662–1667 (1996).
    [CrossRef]
  4. I. Shavirin, O. Strelkov, A. Vetskous, L. Norton-Wayne, R. Harwood, “Internal reflection sensors with high angular resolution,” Appl. Opt. 35, 4133–4141 (1996).
    [CrossRef] [PubMed]
  5. X. Dai, O. Sasaki, J. E. Greivenkamp, T. Suzuki, “Measurement of two-dimensional small rotation angles by using orthogonal parallel interference patterns,” Appl. Opt. 35, 5657–5666 (1996).
    [CrossRef] [PubMed]
  6. D. Malacara, O. Harris, “Interferometric measurement of angles,” Appl. Opt. 9, 1630–1633 (1970).
    [CrossRef] [PubMed]
  7. P. Shi, E. Stijns, “New optical method for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
    [CrossRef] [PubMed]
  8. P. Shi, E. Stijns, “Improving the linearity of the Michelson interferometric angular measurement by a parametric compensation method,” Appl. Opt. 32, 44–51 (1993).
    [CrossRef] [PubMed]
  9. F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, Singapore, 1985), p. 529.

1996

1993

1992

1989

M. Ben-Levy, S. G. Braun, J. Shamir, “Angular velocity measuring interferometer,” Appl. Opt. 18, 4265–4266 (1989).
[CrossRef]

1988

1970

Ben-Levy, M.

M. Ben-Levy, S. G. Braun, J. Shamir, “Angular velocity measuring interferometer,” Appl. Opt. 18, 4265–4266 (1989).
[CrossRef]

Braun, S. G.

M. Ben-Levy, S. G. Braun, J. Shamir, “Angular velocity measuring interferometer,” Appl. Opt. 18, 4265–4266 (1989).
[CrossRef]

Chiang, H. K.

Dai, X.

Greivenkamp, J. E.

Harris, O.

Hartman, N. F.

Harwood, R.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, Singapore, 1985), p. 529.

Kawachi, K.

L. Zeng, H. Mataumoto, K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variations using an acousto-optic deflector,” Opt. Eng. 35, 1662–1667 (1996).
[CrossRef]

Kenan, R. P.

Malacara, D.

Mataumoto, H.

L. Zeng, H. Mataumoto, K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variations using an acousto-optic deflector,” Opt. Eng. 35, 1662–1667 (1996).
[CrossRef]

Norton-Wayne, L.

Sasaki, O.

Shamir, J.

M. Ben-Levy, S. G. Braun, J. Shamir, “Angular velocity measuring interferometer,” Appl. Opt. 18, 4265–4266 (1989).
[CrossRef]

Shavirin, I.

Shi, P.

Stijns, E.

Strelkov, O.

Summers, C. J.

Suzuki, T.

Vetskous, A.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, Singapore, 1985), p. 529.

Zeng, L.

L. Zeng, H. Mataumoto, K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variations using an acousto-optic deflector,” Opt. Eng. 35, 1662–1667 (1996).
[CrossRef]

Appl. Opt.

Opt. Eng.

L. Zeng, H. Mataumoto, K. Kawachi, “Scanning beam collimation method for measuring dynamic angle variations using an acousto-optic deflector,” Opt. Eng. 35, 1662–1667 (1996).
[CrossRef]

Opt. Lett.

Other

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, Singapore, 1985), p. 529.

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Figures (9)

Fig. 1
Fig. 1

Michelson interferometer with a right-angle prism P having refractive index n p in the probe beam. BS, beam splitter; mirrors M1, M2, end reflectors.

Fig. 2
Fig. 2

Michelson interferometer with a plane transparent glass strip S having refractive index n s of thickness d at an initial angle ϕ with the probe beam. R1, R2, beam intercepts on the diagonal face of the prism.

Fig. 3
Fig. 3

Optical path of the laser beam through a glass strip of thickness d. Solid lines, initial position of the glass strip at angle ϕ; dashed lines, glass strip after rotation through angle θ; α, β, angles of refraction in the two position.

Fig. 4
Fig. 4

Michelson interferometer with two prisms and glass strips. Δx, Δy, distances between the apexes of the prisms.

Fig. 5
Fig. 5

Plot of error function [Eq. (23)] against initial angle ϕ of the glass strip to as great as ±20°.

Fig. 6
Fig. 6

Plot of beam intercepts R1 and R2 for the one-prism case.

Fig. 7
Fig. 7

Comparison of error in the present research with the research of Shi and Stijns8 for a rotation range of ±5°.

Fig. 8
Fig. 8

Error in the single- and the double-prism cases for a rotation range of ±10°.

Fig. 9
Fig. 9

Nonlinearity present in the single- and the double-prism case for rotation to as great as ±10°.

Tables (1)

Tables Icon

Table 1 Error and Nonlinearity in One- and Two-Prism Setups for ±20 deg

Equations (36)

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ΔPOθ=2x sin θ+2y tanθ2sin θ+2anp2-sin2 θ1/2-np+sin θ,
δpS0=nsd-d cosϕ-αcos α,
δpSθ=dns-cosϕ-αcos α-ns-cosϕ-θ-βcos β,
Ψ=2 tan-1(np2 sin245-sin-11/npsin θ-1)np cos45-sin-11/npsin θ1/2 -4 tan-1np2 sin245-1np cos451/2 +2 tan-1(np2 sin245+sin-11/npsin θ-1)np cos45+sin-11/npsin θ1/2.
ΔPOSθ=2x sin θ+2y tanθ2sin θ+2anp2-sin2 θ1/2-np+sin θ+2dns-cosϕ-αcos α-ns-cosϕ-θ-βcos β+Ψ λ360.
δplθ=2x1 sin θ-2y1cos θ-1+Ψ λ360,
δprθ=2x2 sin θ-2y2cos θ-1+Ψ λ360.
ΔPTθ=δprθ-δplθ=2x2-x1sin θ-2y2-y1cos θ-1=2Δx sin θ-2Δycos θ-1,
δp2Sθ=dns-cosϕ+θ+γcos γ-ns-cosϕ-θ-βcos β,
ΔPTSθ=2Δx sin θ+2Δycos θ-1+δp2Sθ.
δθ=2/λΔPθ-mθ,
Ln=δθmθ.
i=1θδi2=mini=1θ2λ ΔPi-mi2=mini=1θ4λ2ΔPi-mi λ22.
i=1θδi2=mini=1θ4λ2Aix+Biy+Cid+Di2,
xi=1θδi2=0.
xi=1θAix+Biy+Cid+Di2=0.
x i=1θAi2+y i=1θ AiBi+d i=1θ AiCi+i=1θ AiDi=0,
x i=1θ AiBi+y i=1θBi2+d i=1θ BiCi+i=1θ BiDi=0,
x i=1θ AiCi+y i=1θ CiBi+d i=1θCi2+i=1θ CiDi=0.
x=xaden,
y=yaden,
d=daden,
xa=i=1θ AiBii=1θ BiCii=1θ Ci Di+i=1θ AiCii=1θ BiCii=1θ BiDi-i=1θ AiCii=1θ CiDii=1θBi2+i=1θ AiDii=1θBi2i=1θCi2-i=1θ AiBii=1θCi2i=1θ BiDi-i=1θ AiDii=1θ CiDi2,
ya=i=1θ CiDii=1θ AiBii=1θ AiCi+i=1θ AiCii=1θ BiCii=1θ AiDi-i=1θ BiCii=1θ CiDii=1θAi2+i=1θ BiDii=1θAi2i=1θCi2-i=1θ AiDii=1θCi2i=1θ AiBi-i=1θ BiDii=1θ AiCi2,
da=i=1θ AiCii=1θ AiBii=1θ BiDi+i=1θ AiDii=1θ BiCii=1θ AiBi-i=1θ BiCii=1θ BiDii=1θAi2+i=1θ CiDii=1θBi2i=1θAi2-i=1θ AiDii=1θBi2i=1θ AiCi-i=1θ CiDii=1θ AiBi2,
den=i=1θAi2i=1θ BiCi2+i=1θCi2i=1θ AiBi2-i=1θAi2i=1θCi2i=1θBi2+i=1θ AiCi2i=1θBi2-2 i=1θ AiCii=1θ AiBii=1θ BiCi.
δθ=2x sin θ+2y tanθ2sin θ+2anp2-sin2 θ1/2-np+sin θ2λ+dns-cosϕ-αcos α-ns-cosϕ-θ-βcos β4λ-mθ+Ψ180.
Ai=2 sini,
Bi=2 tani2sini,
Ci=ns-cosϕ-αcos α-ns-cosϕ-i-ββ,
Di=2anp2-sin2i1/2-np+sini-mi λ2+Ψ λ360.
δθ=2Δx sin θ+2Δycos θ-12λ-mθ+2dλns-cosϕ+θ+γcos γ-ns-cosϕ-θ-βcos β.
Ai=2 sini,
Bi=2cosi-1,
Ci=ns-cosϕ+i+γcos γ-ns-cosϕ-i-βcos β,
Di=mi λ2.

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