Abstract

The Gouy phase, sometimes called the phase anomaly, is the remarkable effect that in the region of focus a converging wave field undergoes a rapid phase change by an amount of π, compared to the phase of a plane wave of the same frequency. This phenomenon plays a crucial role in any application where fields are focused, such as optical coherence tomography, mode selection in laser resonators, and interference microscopy. However, when the field is spatially partially coherent, as is often the case, its phase is a random quantity. When such a field is focused, the Gouy phase is therefore undefined. The correlation properties of partially coherent fields are described by their so-called spectral degree of coherence. We demonstrate that this coherence function does exhibit a generalized Gouy phase. Its precise behavior in the focal region depends on the transverse coherence length. We show that this effect influences the fringe spacing in interference experiments in a nontrivial manner.

© 2012 Optical Society of America

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References

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  1. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C.R. Acad. Sci. 110, 1251–1253 (1890).
  2. L. G. Gouy, “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145–213 (1891).
  3. E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956).
    [CrossRef]
  4. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).
  5. T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
    [CrossRef]
  6. G. S. Kino and T. R. Korle, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).
  7. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  8. G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
    [CrossRef]
  9. T. Klaassen, A. Hoogeboom, M. P. van Exter, and J. P. Woerdman, “Gouy phase of nonparaxial eigenmodes in a folded resonator,” J. Opt. Soc. Am. A 21, 1689–1693 (2004).
    [CrossRef]
  10. F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
    [CrossRef]
  11. J. F. Federici, R. L. Wample, D. Rodriguez, and S. Mukherjee, “Application of terahertz Gouy phase shift from curved surfaces for estimation of crop yield,” Appl. Opt. 48, 1382–1388 (2009).
    [CrossRef]
  12. M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98, 191114 (2011).
    [CrossRef]
  13. W. Wang, A. T. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel number,” J. Opt. Soc. Am. A 14, 491–496 (1997).
    [CrossRef]
  14. A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
    [CrossRef]
  15. B. Lü, B. Zhang, and B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
    [CrossRef]
  16. T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
    [CrossRef]
  17. G. Gbur and T. D. Visser, “Can coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28, 1627–1629(2003).
    [CrossRef]
  18. T. van Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575–581 (2008).
    [CrossRef]
  19. S. B. Raghunathan, T. van Dijk, E. J. G. Peterman, and T. D. Visser, “Experimental demonstration of an intensity minimum at the focus of a laser beam created by spatial coherence: application to the optical trapping of dielectric particles,” Opt. Lett. 35, 4166–4168 (2010).
    [CrossRef]
  20. G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics, Vol. 55, E. Wolf, ed. (Elsevier, 2010), pp. 285–341.
  21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  22. D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097–2102 (2004).
    [CrossRef]
  23. J. T. Foley and E. Wolf, “Wave-front spacing in the focal region of high-numerical-aperture systems,” Opt. Lett. 30, 1312–1314 (2005).
    [CrossRef]
  24. K. Creath, “Calibration of numerical aperture effects in interferometric microscope objectives,” Appl. Opt. 28, 3333–3338 (1989).
    [CrossRef]

2011 (1)

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98, 191114 (2011).
[CrossRef]

2010 (2)

2009 (1)

2008 (1)

2005 (1)

2004 (3)

2003 (2)

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

G. Gbur and T. D. Visser, “Can coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28, 1627–1629(2003).
[CrossRef]

2002 (1)

T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

2001 (1)

A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

1997 (1)

1995 (1)

B. Lü, B. Zhang, and B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

1993 (1)

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

1989 (1)

1956 (1)

E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956).
[CrossRef]

1891 (1)

L. G. Gouy, “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145–213 (1891).

1890 (1)

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

Allen, L.

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

Beijersbergen, M. W.

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

Cai, B.

B. Lü, B. Zhang, and B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

Creath, K.

Dufour, M. L.

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

Federici, J. F.

Fischer, D. G.

Foley, J. T.

Friberg, A. T.

A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

W. Wang, A. T. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel number,” J. Opt. Soc. Am. A 14, 491–496 (1997).
[CrossRef]

Gauthier, B.

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

Gbur, G.

T. van Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575–581 (2008).
[CrossRef]

G. Gbur and T. D. Visser, “Can coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28, 1627–1629(2003).
[CrossRef]

T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics, Vol. 55, E. Wolf, ed. (Elsevier, 2010), pp. 285–341.

Gouy, L. G.

L. G. Gouy, “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145–213 (1891).

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

Grasbon, F.

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

Hartmann, R.

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

Herzig, H. P.

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98, 191114 (2011).
[CrossRef]

Hoogeboom, A.

Kim, M.-S.

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98, 191114 (2011).
[CrossRef]

Kino, G. S.

G. S. Kino and T. R. Korle, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).

Klaassen, T.

Korle, T. R.

G. S. Kino and T. R. Korle, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).

Lamouche, G.

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

Lindner, F.

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956).
[CrossRef]

Lü, B.

B. Lü, B. Zhang, and B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Monchalin, J.-P.

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

Mühlig, S.

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98, 191114 (2011).
[CrossRef]

Mukherjee, S.

Paulus, G. G.

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

Peterman, E. J. G.

Raghunathan, S. B.

Rockstuhl, C.

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98, 191114 (2011).
[CrossRef]

Rodriguez, D.

Scharf, T.

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98, 191114 (2011).
[CrossRef]

Schätzel, M. G.

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

Stremme, W.

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

Strüder, L.

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

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

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

van Dijk, T.

van Exter, M. P.

Visser, T. D.

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

S. B. Raghunathan, T. van Dijk, E. J. G. Peterman, and T. D. Visser, “Experimental demonstration of an intensity minimum at the focus of a laser beam created by spatial coherence: application to the optical trapping of dielectric particles,” Opt. Lett. 35, 4166–4168 (2010).
[CrossRef]

T. van Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575–581 (2008).
[CrossRef]

D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097–2102 (2004).
[CrossRef]

G. Gbur and T. D. Visser, “Can coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28, 1627–1629(2003).
[CrossRef]

T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics, Vol. 55, E. Wolf, ed. (Elsevier, 2010), pp. 285–341.

Walther, H.

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

Wample, R. L.

Wang, W.

A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

W. Wang, A. T. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel number,” J. Opt. Soc. Am. A 14, 491–496 (1997).
[CrossRef]

Woerdman, J. P.

T. Klaassen, A. Hoogeboom, M. P. van Exter, and J. P. Woerdman, “Gouy phase of nonparaxial eigenmodes in a folded resonator,” J. Opt. Soc. Am. A 21, 1689–1693 (2004).
[CrossRef]

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

Wolf, E.

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

J. T. Foley and E. Wolf, “Wave-front spacing in the focal region of high-numerical-aperture systems,” Opt. Lett. 30, 1312–1314 (2005).
[CrossRef]

T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

W. Wang, A. T. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel number,” J. Opt. Soc. Am. A 14, 491–496 (1997).
[CrossRef]

E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Zhang, B.

B. Lü, B. Zhang, and B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

Ann. Chim. Phys. (1)

L. G. Gouy, “Sur la propagation anomale des ondes,” Ann. Chim. Phys. 24, 145–213 (1891).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. 98, 191114 (2011).
[CrossRef]

C.R. Acad. Sci. (1)

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

J. Mod. Opt. (1)

B. Lü, B. Zhang, and B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (5)

T. D. Visser, G. Gbur, and E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

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

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

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

Opt. Lett. (3)

Phys. Rev. A (1)

F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003).
[CrossRef]

Proc. Phys. Soc. B (1)

E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

G. S. Kino and T. R. Korle, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics, Vol. 55, E. Wolf, ed. (Elsevier, 2010), pp. 285–341.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (5)

Fig. 1.
Fig. 1.

Illustration of the notation.

Fig. 2.
Fig. 2.

Classical Gouy phase δ(z) along the optical axis for a deterministic (i.e., fully coherent) wave field. In this example, a=1cm, f=2cm, and λ=0.6328μm.

Fig. 3.
Fig. 3.

Generalized Gouy phase δμ(0,z2) of a partially coherent field for different values of the transverse coherence length of the field in the aperture, namely, σ=0.5cm 1, 2, and 3 cm. In all examples, the aperture radius a=1cm, the focal length f=2cm, and the wavelength λ=0.6328μm.

Fig. 4.
Fig. 4.

Interferogram for a coherent field (σ/a=50) (dashed red curve) and for a partially coherent field (σ/a=0.5) (solid blue curve). In both cases, a=1cm, f=2cm and λ=0.6328μm.

Fig. 5.
Fig. 5.

Same as Fig. 4, but for larger values of the axial position z2.

Tables (1)

Tables Icon

Table 1. Fringe Spacings for Three Values of the Transverse Coherence Length σa

Equations (21)

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

U(r,ω)=iλsU(0)(r,ω)exp(iks)sd2r,
δ(z)=arg[U(z,ω)]kz.
δ(0)=π/2.
δ(z)+δ(z)=π.
W(0)(r1,r2,ω)=U(0)*(r1,ω)U(0)(r2,ω).
W(r1,r2,ω)=U*(r1,ω)U(r2,ω)
W(r1,r2,ω)=1λ2SW(0)(r,r,ω)×exp[ik(s2s1)]s1s2d2rd2r,
W(0)(r,r)=W(0)(ρ,ρ)
=A2exp[(ρρ)2/2σ2],
μ(r1,r2)=W(r1,r2)S(r1)S(r2),
S(ri)=W(ri,ri),i=1,2.
W(z1,z2)=(2πAλf)20a0aexp[(ρ2+ρ2)/2σ2]×exp{ik[z1(1ρ2/2f2)+z2(1ρ2/2f2)]}×I0(ρρσ2)ρρdρdρ,
δμ(z1,z2)=arg[W(z1,z2)]+kz1kz2.
δμ(0,z2)=arg[0a0aexp[(ρ2+ρ2)/2σ2]×exp{ik[z2ρ2/2f2]}×I0(ρρσ2)ρρdρdρ].
δμ(0,0)=0,
δμ(0,z2)+δμ(0,z2)=0,
W(z1,z2)=U*(z1)U(z2).
δμ(0,z2)=arg[U(z2)]kz2+π/2,
δμ(0,z2)=kz2a2/4f2,
I(z)=|U(0)+U(z)|2,
=S(0)+S(z)+2S(0)S(z)Re[μ(0,z)],

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