Abstract

The properties of TEM01* doughnut modes have been examined by frequency analysis and by two-beam interference. Interpretation in terms of an evolving helix, made up to two orthogonal TEM01 modes of different frequency, is supported by computer simulation of fringe patterns. These patterns are shown to correspond closely with photographic recordings; the implications of the phenomena for the crossed-beam technique in laser velocimetry are outlined. Finally, the possibility of developing a beam of pure helical cophasal surface is discussed and its interference patterns analyzed.

© 1983 Optical Society of America

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References

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  1. J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
    [Crossref]
  2. D. V. Willetts and J. M. Vaughan, “Properties of a laser mode with a helical cophasal surface,” in Laser Advances and Applications, B. S. Wherrett, ed. (Wiley, New York, 1980), pp. 51–56.
  3. P. N. Pusey, J. M. Vaughan, and D. V. Willetts, “Effect of spatial incoherence of the laser in photon-correlation spectroscopy,” J. Opt. Soc. Am. 73, 1012–1017 (1983).
    [Crossref]
  4. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966).
    [Crossref] [PubMed]
  5. M. Bertolotti, B. Daino, F. Gori, and D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
    [Crossref]
  6. M. Carnevale and B. Daino, “Spatial coherence analysis by interferometric methods,” Opt. Acta 24, 1099–1104 (1977).
    [Crossref]
  7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Sec. 4.4.

1983 (1)

1979 (1)

J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[Crossref]

1977 (1)

M. Carnevale and B. Daino, “Spatial coherence analysis by interferometric methods,” Opt. Acta 24, 1099–1104 (1977).
[Crossref]

1966 (1)

1965 (1)

M. Bertolotti, B. Daino, F. Gori, and D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[Crossref]

Bertolotti, M.

M. Bertolotti, B. Daino, F. Gori, and D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[Crossref]

Carnevale, M.

M. Carnevale and B. Daino, “Spatial coherence analysis by interferometric methods,” Opt. Acta 24, 1099–1104 (1977).
[Crossref]

Daino, B.

M. Carnevale and B. Daino, “Spatial coherence analysis by interferometric methods,” Opt. Acta 24, 1099–1104 (1977).
[Crossref]

M. Bertolotti, B. Daino, F. Gori, and D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[Crossref]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Sec. 4.4.

Gori, F.

M. Bertolotti, B. Daino, F. Gori, and D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[Crossref]

Kogelnik, H.

Li, T.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Sec. 4.4.

Pusey, P. N.

Sette, D.

M. Bertolotti, B. Daino, F. Gori, and D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[Crossref]

Vaughan, J. M.

P. N. Pusey, J. M. Vaughan, and D. V. Willetts, “Effect of spatial incoherence of the laser in photon-correlation spectroscopy,” J. Opt. Soc. Am. 73, 1012–1017 (1983).
[Crossref]

J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[Crossref]

D. V. Willetts and J. M. Vaughan, “Properties of a laser mode with a helical cophasal surface,” in Laser Advances and Applications, B. S. Wherrett, ed. (Wiley, New York, 1980), pp. 51–56.

Willetts, D. V.

P. N. Pusey, J. M. Vaughan, and D. V. Willetts, “Effect of spatial incoherence of the laser in photon-correlation spectroscopy,” J. Opt. Soc. Am. 73, 1012–1017 (1983).
[Crossref]

J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[Crossref]

D. V. Willetts and J. M. Vaughan, “Properties of a laser mode with a helical cophasal surface,” in Laser Advances and Applications, B. S. Wherrett, ed. (Wiley, New York, 1980), pp. 51–56.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Nuovo Cimento (1)

M. Bertolotti, B. Daino, F. Gori, and D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[Crossref]

Opt. Acta (1)

M. Carnevale and B. Daino, “Spatial coherence analysis by interferometric methods,” Opt. Acta 24, 1099–1104 (1977).
[Crossref]

Opt. Commun. (1)

J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[Crossref]

Other (2)

D. V. Willetts and J. M. Vaughan, “Properties of a laser mode with a helical cophasal surface,” in Laser Advances and Applications, B. S. Wherrett, ed. (Wiley, New York, 1980), pp. 51–56.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Sec. 4.4.

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Figures (5)

Fig. 1
Fig. 1

(a) Coordinate system used for fringe intensity calculations. The angles θ1 and θ2 are constrained within the limits −(π/2) < θ1 ≤ (3π/2) and −(3π/2) < θ2 ≤ (π/2) chosen to simplify the analysis. (b) Uninverted beams, (c) inverted beams, both shown displaced side-ways.

Fig. 2
Fig. 2

Computer reconstruction of two-beam interferograms with a TEM01* laser mode: (a) beams inverted (i.e., Fig. 1c) but not displaced, (b) beams inverted and displaced horizontally, (c) beams uninverted and displaced horizontally (i.e., Fig. 1b); (d) like (c) but with the addition of 14% TEM00 mode. Note the half-cycle displacements in the fringes in different regions; see text for details of the perspective. The centers of the separated beams are marked by circles.

Fig. 3
Fig. 3

Two-beam interference fringes. The top row shows the beams overlapped exactly without displacement: (a) Gaussian TEM00 mode, (b) doughnut TEM01* mode uninverted (i.e., Fig. 1b), (c) doughnut TEM01* mode inverted (i.e., Fig. 1c). The lower row shows the effect of lateral displacement of the beams: (d) doughnut TEM01* mode uninverted and displaced horizontally, (e) doughnut TEM01* mode uninverted and displaced vertically, (f) doughnut TEM01* mode inverted and displaced horizontally. Note in particular the half-cycle changes on the vertical fringes and compare them with Fig. 2.

Fig. 4
Fig. 4

Optical arrangement to produce a beam of pure helical cophasal form from a beam initially in a two-spot TEM01 mode of single frequency. The two paths between the beam splitters BS1 and BS2 should be arranged to be approximately equal. The axis of the two-spot mode lies in the plane of the paper when the modes are shown normal to the beam and is normal to the plane of the paper when the mode axis lies along the beam. The optical path and phase between the two beams may be varied by the piezoadjustable mirror at PZM. As the phase is changed the beam varies between a left-hand and a right-hand helix, as shown.

Fig. 5
Fig. 5

Phase-difference contour maps (thin lines) and fringe patterns (thick lines, continuous and dashed) for two-beam interference of a true helical TEM01* laser mode: (a) inverted and separated beams, (b) uninverted and separated beams. The contours are labeled in phase difference.

Equations (6)

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E ( ρ , θ , t ) = C [ exp ( - ρ 1 2 / 2 σ 2 ) sin ( ω 0 t - ϕ 1 ) + exp ( - ρ 2 2 / 2 σ 2 ) sin ( ω 0 t + ϕ 2 ) ] + ( ρ 1 / σ ) exp ( - ρ 1 2 / 2 σ 2 ) sin ( ω 1 t ± θ 1 - ϕ 1 ) + ( ρ 2 / σ ) exp ( - ρ 2 2 / 2 σ 2 ) sin ( ω 1 t + θ 2 + ϕ 2 ) ,
I ½ i = 1 2 ( a i 2 + b i 2 ) + a 1 a 2 cos ( ϕ 1 + ϕ 2 ) + b 1 b 2 cos ( ϕ 1 + ϕ 2 θ 1 - θ 2 ) ,
a i = C exp ( - ρ i 2 / 2 σ 2 ) i = 1 , 2 , b i = ( ρ i / σ ) exp ( - ρ i 2 / 2 σ 2 ) i = 1 , 2.
½ b 1 b 2 [ cos ( ϕ 1 + ϕ 2 θ 2 - θ 2 ) + cos ( ϕ 1 + ϕ 2 ± θ 1 + θ 2 ) ] .
I ½ i = 1 2 ( a i 2 + b i 2 ) + [ a 1 a 2 + b 1 b 2 cos ( θ 2 ± θ 1 ) ] cos ( ϕ 1 + ϕ 2 ) .
x = ( - 1 ) i + 1 R - ρ i sin θ i , i = 1 , 2 , y = ρ i cos θ i , i = 1 , 2.