Abstract

Mirror misalignment or the tilt angle of the Michelson interferometer can be estimated from the modulation depth measured with collimated monochromatic light. The intensity of the light beam is usually assumed to be uniform, but, for example, with gas lasers it generally has a Gaussian distribution, which makes the modulation depth less sensitive to the tilt angle. With this assumption, the tilt angle may be underestimated by about 50%. We have derived a mathematical model for modulation depth with a circular aperture and Gaussian beam. The model reduces the error of the tilt angle estimate to below 1%. The results of the model have been verified experimentally.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. S. Williams, “Mirror misalignment in Fourier spectroscopy using a Michelson interferometer with circular aperture,” Appl. Opt. 5, 1084–1085 (1966).
    [CrossRef]
  2. J. Chamberlain, “The generation and recording of the interferogram in a practical interferometer,” in The Principles of Interferometric Spectroscopy, G. W. Chantry and N. W. B. Stone, eds. (Wiley, 1979), pp. 237–238.
  3. G. A. Vanasse and H. Sakai, “Fourier spectroscopy” in Progress in Optics, E. Wolf, ed. (Elsevier, 1967), Vol. 6, pp. 258–330.
  4. W. H. Steel, “Interferometers without collimation for Fourier spectroscopy,” J. Opt. Soc. Am. 54, 151–156 (1964).
    [CrossRef]
  5. G. W. Stroke, “Photoelectric fringe signal information and range in interferometers with moving mirrors,” J. Opt. Soc. Am. 47, 1097–1103 (1957).
    [CrossRef]
  6. L. W. Kunz and D. Goorvitch, “Combined effects of a converging beam of light and mirror misalignment in Michelson interferometry,” Appl. Opt. 13, 1077–1079 (1974).
    [CrossRef]
  7. D. L. Cohen, “Performance degradation of a Michelson interferometer when its misalignment angle is a rapidly varying, random time series,” Appl. Opt. 36, 4034–4042 (1997).
    [CrossRef]
  8. M. V. R. K. Murty, “Some more aspects of the Michelson interferometer with cube corners,” J. Opt. Soc. Am. 50, 7–10 (1960).
    [CrossRef]
  9. E. R. Peck, “Integrated flux from a Michelson or corner-cube interferometer,” J. Opt. Soc. Am. 45, 931–934 (1955).
    [CrossRef]

1997 (1)

1974 (1)

1966 (1)

1964 (1)

1960 (1)

1957 (1)

1955 (1)

Chamberlain, J.

J. Chamberlain, “The generation and recording of the interferogram in a practical interferometer,” in The Principles of Interferometric Spectroscopy, G. W. Chantry and N. W. B. Stone, eds. (Wiley, 1979), pp. 237–238.

Cohen, D. L.

Goorvitch, D.

Kunz, L. W.

Murty, M. V. R. K.

Peck, E. R.

Sakai, H.

G. A. Vanasse and H. Sakai, “Fourier spectroscopy” in Progress in Optics, E. Wolf, ed. (Elsevier, 1967), Vol. 6, pp. 258–330.

Steel, W. H.

Stroke, G. W.

Vanasse, G. A.

G. A. Vanasse and H. Sakai, “Fourier spectroscopy” in Progress in Optics, E. Wolf, ed. (Elsevier, 1967), Vol. 6, pp. 258–330.

Williams, C. S.

Appl. Opt. (3)

J. Opt. Soc. Am. (4)

Other (2)

J. Chamberlain, “The generation and recording of the interferogram in a practical interferometer,” in The Principles of Interferometric Spectroscopy, G. W. Chantry and N. W. B. Stone, eds. (Wiley, 1979), pp. 237–238.

G. A. Vanasse and H. Sakai, “Fourier spectroscopy” in Progress in Optics, E. Wolf, ed. (Elsevier, 1967), Vol. 6, pp. 258–330.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Coordinates of the circular aperture. Coordinate axes t and u lie on the cross section of the other beam (or the aperture). The optical path difference is x, and y is the extra optical path difference from mirror tilt. When the modulation depth mG is integrated across dy in Eq. 9, the dt is as if it is sliding over the aperture; hence, y2αt. Note that the tilt angle α causes the deviation angle 2α between the beams.

Fig. 2.
Fig. 2.

Finding the correction coefficient c, which was used to modify the standard deviation σ to enhance the convolution approximation mG*. The coefficient c is a logistic function, which was fitted to the standard deviation σF/σ data. Values of σF resulted from fitting the convolution approximation mG* to the exact solution mG with respect to the standard deviation σ.

Fig. 3.
Fig. 3.

Comparison of the modulation depth calculated from mB for the uniform beam together with the exact solution mG and the convolution model mGc for the Gaussian beam. The ratio of the standard deviation of the Gaussian beam to the aperture diameter was 0.16. Tilt angles are valid with the 10 mm aperture diameter and 632.8 nm wavelength.

Fig. 4.
Fig. 4.

The maximum relative error of the mirror tilt angle estimate Δα/α when using the convolution approximation mG* (Eq. 12) and the convolution model mGc (Eq. 15) instead of the exact solution mG (Eq. 9). See Subsection 2.C. for the description of model mGc and Subsection 2.D. for the exact definition of the error.

Fig. 5.
Fig. 5.

Schematic figure of the measurement setup: L1, L2, red and green He–Ne lasers with their own removable 10× expanders; S1, S2, beam stops; BS2, beam splitter; EXPN, second 10× expander; S3, beam stop with a millimeter scale attached to it; L, condensing lens; D, photodiode detector. The interferometer is a Michelson type with beam splitter, BS, mirror with adjustable alignment, M1, and driven mirror, M2.

Fig. 6.
Fig. 6.

Comparison of the modulation depth (with error bars) from measurements and the convolution model mGc (Eq. 15) for a Gaussian beam and the function mB (Eq. 5) for a uniform beam. Clearly, the function mB fits the data inaccurately. The standard deviation of the beam intensity was 1.48 mm, and the aperture diameter was 8.0 mm (ratio 1.48/8=0.19).

Fig. 7.
Fig. 7.

Comparison of the modulation depth (with error bars) from measurements and the convolution model mGc (Eq. 15) for a Gaussian beam and the function mB (Eq. 5) for a uniform beam. The function mB gives a useful fit at this time. The standard deviation of the beam intensity was 13.8 mm, and the aperture diameter was 14.0 mm (ratio 13.8/14=0.99).

Equations (15)

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

IB(x)=2αR2αRI(x+y)2(2αR)2y2[π(2αR)2]1dy.
B(y)={2(2αR)2y2[π(2αR)2]1|y|2αR0elsewhere.
I(x)=E(ν)exp(i2πνx)dν=F{E(ν)},
IB(x)=E(ν)exp(i2πνx)[B(y)exp(i2πνy)dy]dν.
mB=2J1(4πνRα)4πνRα=2J1(πk)πk,
g(t,u)=g0exp(at2au2),a=(2σ2)1,
G(y)dy2α=exp[ay2(2α)2][q(y)q(y)exp(au2)du]dy2α,
IG(x)=E(ν)exp(i2πνx)×[G0G(y)exp(i2πνy)dy]dν.
mG=G02αR2αRexp[ay2(2α)2]×[q(y)q(y)exp(au2)du]exp(i2πνy)dy.
q(y)q(y)du=2q(y)=2παR2B(y),
mGmG*=M0B(y)exp[ay2(2α)2]exp(i2πνy)dy,
mG*=M0(F{B(y)}*F{exp[y22(2ασ)2]}),
F{exp[y22(2ασ)2]}=22πσαexp[2(2πσαν)2].
c(σ2R)=1.230.231+[4.3σ/(2R)]2.8.
mGc(k,σ)=mG*(k,σc)=mG*(k,σc).

Metrics