Abstract

The Gouy phase anomaly, well established for stigmatic beams, is validated here for astigmatic beams. We simulate the predicted Gouy phase anomaly near astigmatic foci using a beam propagation algorithm integrated within lens design software. We then compare computational results with experimental data acquired using a modified Mertz–Sagnac interferometer. Both in simulation and in experiment, results show that a π/2-phase change occurs as the beam passes through each of the astigmatic foci, experimentally validating results derived in a recent paper by Visser and Wolf [Opt. Commun. 283, 3371–3375 (2010) [CrossRef]  ].

© 2012 Optical Society of America

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  1. L. G. Gouy, “Sur une propriété nouvelle des ondes,” C. R. Acad. Sci. 110, 1251–1253 (1890).
  2. L. G. Gouy, “Sur la propagation anomale des ondes,” C. R. Acad. Sci. 111, 33–40 (1890) [appears under the same author and title in Ann. Chim. Phys. 24, 145 6e series (1891)].
  3. T. D. Visser, E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
    [CrossRef]
  4. D. W. Diehl, T. D. Visser, “Phase singularities of the longitudinal field components in the focal region of a high-aperture optical system,” J. Opt. Soc. Am. A 21, 2103–2108 (2004).
    [CrossRef]
  5. J. Walker, The Analytical Theory of Light (Cambridge University, 1904), pp. 91–93.
  6. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veer, J. P. Woerdman, “Astigmatic laser mode converters and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  7. L. Mertz, “A new demonstration of the anomalous phase change of light at a focus,” J. Opt. Soc. Am. 49, iv (1959).
  8. G. Sagnac, “Intérferomètre à faisceaux lumineux superposés inverses donnant en lumière blanche polarisée une frange central étroite à teinte sensible et des franges colorées étroites à intervalles blancs,” C. R. Acad. Sci. 150, 1676–1680 (1910).
  9. G. Sagnac, “Luminous ether demonstrated by the effect of relative wind of ether in a uniform rotation of an interferometer,” C. R. Acad. Sci. 157, 708–710 (1913).
  10. L. Silberstein, “The propagation of light in rotating systems,” J. Opt. Soc. Am. 5, 291–307 (1921).
    [CrossRef]

2010 (1)

T. D. Visser, E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[CrossRef]

2004 (1)

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veer, J. P. Woerdman, “Astigmatic laser mode converters and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1959 (1)

L. Mertz, “A new demonstration of the anomalous phase change of light at a focus,” J. Opt. Soc. Am. 49, iv (1959).

1921 (1)

1913 (1)

G. Sagnac, “Luminous ether demonstrated by the effect of relative wind of ether in a uniform rotation of an interferometer,” C. R. Acad. Sci. 157, 708–710 (1913).

1910 (1)

G. Sagnac, “Intérferomètre à faisceaux lumineux superposés inverses donnant en lumière blanche polarisée une frange central étroite à teinte sensible et des franges colorées étroites à intervalles blancs,” C. R. Acad. Sci. 150, 1676–1680 (1910).

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes,” C. R. Acad. Sci. 110, 1251–1253 (1890).

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veer, J. P. Woerdman, “Astigmatic laser mode converters and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veer, J. P. Woerdman, “Astigmatic laser mode converters and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Diehl, D. W.

Gouy, L. G.

L. G. Gouy, “Sur la propagation anomale des ondes,” C. R. Acad. Sci. 111, 33–40 (1890) [appears under the same author and title in Ann. Chim. Phys. 24, 145 6e series (1891)].

L. G. Gouy, “Sur une propriété nouvelle des ondes,” C. R. Acad. Sci. 110, 1251–1253 (1890).

Mertz, L.

L. Mertz, “A new demonstration of the anomalous phase change of light at a focus,” J. Opt. Soc. Am. 49, iv (1959).

Sagnac, G.

G. Sagnac, “Luminous ether demonstrated by the effect of relative wind of ether in a uniform rotation of an interferometer,” C. R. Acad. Sci. 157, 708–710 (1913).

G. Sagnac, “Intérferomètre à faisceaux lumineux superposés inverses donnant en lumière blanche polarisée une frange central étroite à teinte sensible et des franges colorées étroites à intervalles blancs,” C. R. Acad. Sci. 150, 1676–1680 (1910).

Silberstein, L.

van der Veer, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veer, J. P. Woerdman, “Astigmatic laser mode converters and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Visser, T. D.

T. D. Visser, E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[CrossRef]

D. W. Diehl, T. D. Visser, “Phase singularities of the longitudinal field components in the focal region of a high-aperture optical system,” J. Opt. Soc. Am. A 21, 2103–2108 (2004).
[CrossRef]

Walker, J.

J. Walker, The Analytical Theory of Light (Cambridge University, 1904), pp. 91–93.

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veer, J. P. Woerdman, “Astigmatic laser mode converters and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Wolf, E.

T. D. Visser, E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[CrossRef]

C. R. Acad. Sci. (4)

L. G. Gouy, “Sur une propriété nouvelle des ondes,” C. R. Acad. Sci. 110, 1251–1253 (1890).

L. G. Gouy, “Sur la propagation anomale des ondes,” C. R. Acad. Sci. 111, 33–40 (1890) [appears under the same author and title in Ann. Chim. Phys. 24, 145 6e series (1891)].

G. Sagnac, “Intérferomètre à faisceaux lumineux superposés inverses donnant en lumière blanche polarisée une frange central étroite à teinte sensible et des franges colorées étroites à intervalles blancs,” C. R. Acad. Sci. 150, 1676–1680 (1910).

G. Sagnac, “Luminous ether demonstrated by the effect of relative wind of ether in a uniform rotation of an interferometer,” C. R. Acad. Sci. 157, 708–710 (1913).

J. Opt. Soc. Am. (2)

L. Silberstein, “The propagation of light in rotating systems,” J. Opt. Soc. Am. 5, 291–307 (1921).
[CrossRef]

L. Mertz, “A new demonstration of the anomalous phase change of light at a focus,” J. Opt. Soc. Am. 49, iv (1959).

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

T. D. Visser, E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veer, J. P. Woerdman, “Astigmatic laser mode converters and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Other (1)

J. Walker, The Analytical Theory of Light (Cambridge University, 1904), pp. 91–93.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic layout of the original Mertz–Sagnac interferometer [6].

Fig. 2.
Fig. 2.

(a) Schematic layout of the stigmatic Mertz–Sagnac interferometer with simulated interferograms computed using beam propagation software. The dark central fringe in the middle simulated interferogram is the manifestation of the Gouy phase anomaly. (b) Linear illustration of the phase change of the clockwise and the counterclockwise rotating beams in the interferometer after they exit the interferometer through focus.

Fig. 3.
Fig. 3.

A geometric ray trace tool used to select the focal lengths for the catalog lenses used in the experiment predicts the size of the beam pattern for each of the counterrotating beams to determine the region over which the phase of the null fringe will be visible. The beam prints are shown at key locations as follows: (a) neither beam has passed through a focus, (b) the location of the first astigmatic focus of the beam that traveled the counterclockwise path, (c) at the location where one beam (counterclockwise) has passed through the first astigmatic focus and the other has not (clockwise) and the beams are the same size, maximizing the size of the central fringe, (d) at the location of the first astigmatic focus of the beam that traveled the clockwise path, (e) somewhat ahead of medial focus for both beams, (f) near the medial astigmatic focus of both beams, (g) somewhat behind the medial focus for both beams, (h) the location of the second astigmatic focus of the beam that traveled the counterclockwise path, (i) at the location where one of the beams has passed through the second astigmatic focus and one has not, (j) the location of the second astigmatic focus of the beam that traveled the clockwise path, and (k) exiting the region of interest, beyond both astigmatic foci for both beams.

Fig. 4.
Fig. 4.

(a) Schematic layout of a Mertz–Sagnac interferometer modified for illustration and measurement of the Gouy phase anomaly for a pair of astigmatic beams. Simulated interferograms computed using beam propagation software are shown in key positions along the axis as described in the text. Only two of the four ray sets of interest are illustrated for clarity. One set of rays is in the plane of cylindrical power (red) and the other set is in the plane of no cylindrical power (blue). (b) Linear graph of the phase regions of interest as the two astigmatic beams interact as they pass through their respective astigmatic foci after exiting the interferometer.

Fig. 5.
Fig. 5.

Schematic layout of the components of the Mertz–Sagnac interferometer, modified for astigmatic beams with key dimensions and a photograph of the experimental setup.

Fig. 6.
Fig. 6.

Measured interferograms recorded at illustrative axial positions along the direction of propagation. These data provide a clear demonstration of the π / 2 -phase shift between each of the pair of astigmatic foci. The beam prints are shown at the following key locations: (a) neither beam has passed through a focus, (b) the location of the first astigmatic focus of the beam that traveled the counterclockwise path, (c) at the location where one beam (counterclockwise) has passed through the first astigmatic focus and one has not (clockwise) and the beams are the same size, maximizing the size of the central fringe, which is gray due to the π / 2 -phase shift between the beams, (d) at the location of the first astigmatic focus of the beam that traveled the clockwise path, (e) near the medial astigmatic focus for both beams, (f) the location of the second astigmatic focus of the beam that traveled the counterclockwise path, (g) at the location where one of the beams has passed through the second astigmatic focus and one has not, noting the central fringe, which is gray due to the π / 2 -phase shift between the beams, and (h) the location of the second astigmatic focus of the beam that traveled the clockwise path. Note that the locations illustrated are identical to those shown in Fig. 3, except that locations (e), (g), and (k) from Fig. 3 are not included here.

Tables (1)

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Table 1. Specific Components Used in the Experiment to Validate the π / 2 Phase Shift as a Focusing Beam Passes Through Astigmatic Foci

Equations (1)

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I = I 1 + I 2 + 2 I 1 I 2 cos ( Δ ϕ ) ,

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