Abstract

We report an investigation of the popular Z-scan technique for elliptical Gaussian beams. A thin lens wave optics theory is developed for such a beam by generalizing the existing theory for circular Gaussian beams. We find that the Z-scan signature is greatly influenced by two parameters, namely, the ellipticity of the input beam and the waist separation. Depending on these parameters, an additional peak and valley may appear in a closed-aperture Z scan, whereas a small asymmetry may be observed in an open-aperture Z scan. We demonstrate what the qualitative features of both open and closed Z scans would be for a beam with any ellipticity.

© 1996 Optical Society of America

Full Article  |  PDF Article

Corrections

S. M. Mian, B. Taheri, and J. P. Wicksted, "Effects of beam ellipticity on Z-scan measurements: erratum," J. Opt. Soc. Am. B 13, 2671-2671 (1996)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-13-11-2671

References

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  1. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
    [CrossRef] [PubMed]
  2. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  3. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
    [CrossRef]
  4. M. Sheik-Bahae, J. Wang, R. Desalvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two color Z Scan,” Opt. Lett. 17, 258–260 (1992).
    [CrossRef] [PubMed]
  5. J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009–1017 (1994).
    [CrossRef]
  6. W. D. St. John, B. Taheri, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Time-dependent thermal lensing in a lead oxide-modified silicate glass,” J. Opt. Soc. Am. B 9, 610–616 (1992).
    [CrossRef]
  7. B. Taheri, A. Munoz, F. W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. 71, 3693–3700 (1992).
    [CrossRef]
  8. H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Infrared nonlinearity of commercial Cd (S, Se) glass composites,” Opt. Commun. 87, 19–22 (1992).
    [CrossRef]
  9. W. Zhao and P. Palffy-Muhoray, “Z-scan measurement of χ(3) using top-hat beams,” Appl. Phys. Lett. 65, 673–675 (1994).
    [CrossRef]
  10. Y. M. Cheung and S. K. Gayen, “Optical nonlinearities of tea studied by Z-scan and four-wave mixing techniques,” J. Opt. Soc. Am. B 11, 636–643 (1994).
    [CrossRef]
  11. B. A. Rockwell, W. P. Roach, M. E. Rogers, M. W. Mayo, C. A. Toth, C. P. Cain, and G. D. Noojin, “Nonlinear refraction in vitreous humor,” Opt. Lett. 18, 1792–1794 (1993).
    [CrossRef] [PubMed]
  12. H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
    [CrossRef]
  13. J. Castillo, V. P. Kozich, and A. Marcano, “Thermal lensing resulting from one- and two-photon absorption studied with a two-color time-resolved Z scan,” Opt. Lett. 19, 171–173 (1994).
    [CrossRef] [PubMed]
  14. T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wavefront distortion,” Opt. Lett. 19, 317–319 (1994).
    [CrossRef] [PubMed]
  15. J. G. Tian, W. P. Zang, and G. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
    [CrossRef]
  16. D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, “Reflection Z-scan technique for measurement of optical properties of surfaces,” Appl. Phys. Lett. 65, 1067–1069 (1994).
    [CrossRef]
  17. R. L. Sutherland, “Effects of multiple internal sample reflections on the nonlinear refractive Z-scan measurements,” Appl. Opt. 33, 5576–5584 (1994).
    [CrossRef] [PubMed]
  18. M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
    [CrossRef]
  19. J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
    [CrossRef]
  20. P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and the thick-sample limit,” J. Opt. Soc. Am. B 11, 975–982 (1994).
    [CrossRef]
  21. W. Zhao and P. Palffy-Muhoray “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
    [CrossRef]
  22. S. M. Mian and J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
    [CrossRef]
  23. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.
  24. P. P. Banerjee, R. G. Lindquist, J. M. Lee, and R. M. Misra, “Z-scan measurement of optical nonlinearities in nematic– liquid crystals,” in Digest of 1994 OSA Annual Meeting (Optical Society of America, Washington, D.C., 1994), paper WII5, p. 129.

1995 (1)

S. M. Mian and J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

1994 (9)

P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and the thick-sample limit,” J. Opt. Soc. Am. B 11, 975–982 (1994).
[CrossRef]

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009–1017 (1994).
[CrossRef]

W. Zhao and P. Palffy-Muhoray, “Z-scan measurement of χ(3) using top-hat beams,” Appl. Phys. Lett. 65, 673–675 (1994).
[CrossRef]

Y. M. Cheung and S. K. Gayen, “Optical nonlinearities of tea studied by Z-scan and four-wave mixing techniques,” J. Opt. Soc. Am. B 11, 636–643 (1994).
[CrossRef]

J. Castillo, V. P. Kozich, and A. Marcano, “Thermal lensing resulting from one- and two-photon absorption studied with a two-color time-resolved Z scan,” Opt. Lett. 19, 171–173 (1994).
[CrossRef] [PubMed]

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wavefront distortion,” Opt. Lett. 19, 317–319 (1994).
[CrossRef] [PubMed]

J. G. Tian, W. P. Zang, and G. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, “Reflection Z-scan technique for measurement of optical properties of surfaces,” Appl. Phys. Lett. 65, 1067–1069 (1994).
[CrossRef]

R. L. Sutherland, “Effects of multiple internal sample reflections on the nonlinear refractive Z-scan measurements,” Appl. Opt. 33, 5576–5584 (1994).
[CrossRef] [PubMed]

1993 (3)

1992 (5)

1991 (2)

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

1989 (1)

Banerjee, P. P.

P. P. Banerjee, R. G. Lindquist, J. M. Lee, and R. M. Misra, “Z-scan measurement of optical nonlinearities in nematic– liquid crystals,” in Digest of 1994 OSA Annual Meeting (Optical Society of America, Washington, D.C., 1994), paper WII5, p. 129.

Blackburn, D. H.

W. D. St. John, B. Taheri, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Time-dependent thermal lensing in a lead oxide-modified silicate glass,” J. Opt. Soc. Am. B 9, 610–616 (1992).
[CrossRef]

B. Taheri, A. Munoz, F. W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. 71, 3693–3700 (1992).
[CrossRef]

Cain, C. P.

Castillo, J.

Chapple, P. B.

Cheung, Y. M.

Cranmer, D. C.

B. Taheri, A. Munoz, F. W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. 71, 3693–3700 (1992).
[CrossRef]

W. D. St. John, B. Taheri, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Time-dependent thermal lensing in a lead oxide-modified silicate glass,” J. Opt. Soc. Am. B 9, 610–616 (1992).
[CrossRef]

de Araujo, C. B.

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, “Reflection Z-scan technique for measurement of optical properties of surfaces,” Appl. Phys. Lett. 65, 1067–1069 (1994).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Infrared nonlinearity of commercial Cd (S, Se) glass composites,” Opt. Commun. 87, 19–22 (1992).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

Desalvo, R.

Gayen, S. K.

Gomes, A. S. L.

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, “Reflection Z-scan technique for measurement of optical properties of surfaces,” Appl. Phys. Lett. 65, 1067–1069 (1994).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Infrared nonlinearity of commercial Cd (S, Se) glass composites,” Opt. Commun. 87, 19–22 (1992).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

Hagan, D. J.

Hermann, J. A.

Kozich, V. P.

Lee, J. M.

P. P. Banerjee, R. G. Lindquist, J. M. Lee, and R. M. Misra, “Z-scan measurement of optical nonlinearities in nematic– liquid crystals,” in Digest of 1994 OSA Annual Meeting (Optical Society of America, Washington, D.C., 1994), paper WII5, p. 129.

Lindquist, R. G.

P. P. Banerjee, R. G. Lindquist, J. M. Lee, and R. M. Misra, “Z-scan measurement of optical nonlinearities in nematic– liquid crystals,” in Digest of 1994 OSA Annual Meeting (Optical Society of America, Washington, D.C., 1994), paper WII5, p. 129.

Ma, H.

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Infrared nonlinearity of commercial Cd (S, Se) glass composites,” Opt. Commun. 87, 19–22 (1992).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

Marcano, A.

Mayo, M. W.

McDuff, R. G.

Mian, S. M.

S. M. Mian and J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

Misra, R. M.

P. P. Banerjee, R. G. Lindquist, J. M. Lee, and R. M. Misra, “Z-scan measurement of optical nonlinearities in nematic– liquid crystals,” in Digest of 1994 OSA Annual Meeting (Optical Society of America, Washington, D.C., 1994), paper WII5, p. 129.

Munoz, A.

B. Taheri, A. Munoz, F. W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. 71, 3693–3700 (1992).
[CrossRef]

Noojin, G. D.

Palffy-Muhoray, P.

W. Zhao and P. Palffy-Muhoray, “Z-scan measurement of χ(3) using top-hat beams,” Appl. Phys. Lett. 65, 673–675 (1994).
[CrossRef]

W. Zhao and P. Palffy-Muhoray “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

Petrov, D. V.

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, “Reflection Z-scan technique for measurement of optical properties of surfaces,” Appl. Phys. Lett. 65, 1067–1069 (1994).
[CrossRef]

Powell, R. C.

B. Taheri, A. Munoz, F. W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. 71, 3693–3700 (1992).
[CrossRef]

W. D. St. John, B. Taheri, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Time-dependent thermal lensing in a lead oxide-modified silicate glass,” J. Opt. Soc. Am. B 9, 610–616 (1992).
[CrossRef]

Roach, W. P.

Rockwell, B. A.

Rogers, M. E.

Said, A. A.

Sheik-Bahae, M.

Soileau, M. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

St. John, F. W. D.

B. Taheri, A. Munoz, F. W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. 71, 3693–3700 (1992).
[CrossRef]

St. John, W. D.

Staromlynska, J.

Sutherland, R. L.

Taheri, B.

W. D. St. John, B. Taheri, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Time-dependent thermal lensing in a lead oxide-modified silicate glass,” J. Opt. Soc. Am. B 9, 610–616 (1992).
[CrossRef]

B. Taheri, A. Munoz, F. W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. 71, 3693–3700 (1992).
[CrossRef]

Tian, J. G.

J. G. Tian, W. P. Zang, and G. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

Toth, C. A.

Van Stryland, E. W.

Wang, J.

Wei, T. H.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Wicksted, J. P.

S. M. Mian and J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

B. Taheri, A. Munoz, F. W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. 71, 3693–3700 (1992).
[CrossRef]

W. D. St. John, B. Taheri, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Time-dependent thermal lensing in a lead oxide-modified silicate glass,” J. Opt. Soc. Am. B 9, 610–616 (1992).
[CrossRef]

Xia, T.

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.

Young, J.

Zang, W. P.

J. G. Tian, W. P. Zang, and G. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

Zhang, G.

J. G. Tian, W. P. Zang, and G. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

Zhao, W.

W. Zhao and P. Palffy-Muhoray, “Z-scan measurement of χ(3) using top-hat beams,” Appl. Phys. Lett. 65, 673–675 (1994).
[CrossRef]

W. Zhao and P. Palffy-Muhoray “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991).
[CrossRef]

W. Zhao and P. Palffy-Muhoray, “Z-scan measurement of χ(3) using top-hat beams,” Appl. Phys. Lett. 65, 673–675 (1994).
[CrossRef]

W. Zhao and P. Palffy-Muhoray “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, “Reflection Z-scan technique for measurement of optical properties of surfaces,” Appl. Phys. Lett. 65, 1067–1069 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

J. Appl. Phys. (2)

B. Taheri, A. Munoz, F. W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. 71, 3693–3700 (1992).
[CrossRef]

S. M. Mian and J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

J. Opt. Soc. Am. B (6)

Opt. Commun. (2)

J. G. Tian, W. P. Zang, and G. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Infrared nonlinearity of commercial Cd (S, Se) glass composites,” Opt. Commun. 87, 19–22 (1992).
[CrossRef]

Opt. Eng. (1)

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Opt. Lett. (5)

Other (2)

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.

P. P. Banerjee, R. G. Lindquist, J. M. Lee, and R. M. Misra, “Z-scan measurement of optical nonlinearities in nematic– liquid crystals,” in Digest of 1994 OSA Annual Meeting (Optical Society of America, Washington, D.C., 1994), paper WII5, p. 129.

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Figures (7)

Fig. 1
Fig. 1

Contour plot of the closed-aperture transmission. The circular beam lies along y = x. One can envisage all possible closed Z-scan signatures from this plot by using Eq. (47). The solid and the dotted curves represent major and minor contour lines, respectively.

Fig. 2
Fig. 2

Specific examples of the effects of beam ellipticity on the closed-aperture Z-scan signature. A small nonlinearity of ΔΦ0 = 0.1 and a 250-mm focal-length lens are used. The beams are characterized by their ellipticity. (a) The beams have waists wy = 2 mm, wx = 2, 1.5, 1 mm. (b) The beams have waists wy = 2 mm, wx = 0.4, 0.3, 0.2 mm.

Fig. 3
Fig. 3

Beam with an ellipticity of e = 5.5 (wy = 1.1 mm, wx = 0.2 mm) and a sample nonlinearity of ΔΦ0 = 0.1 in a closed-aperture Z-scan, where an extra peak is observed. This curve corresponds to y = 16.48x − 15.05, as given by Eq. (47).

Fig. 4
Fig. 4

Nonlinear decrease of the ratio ΔTp−v/ΔΦ0 as a function of ellipticity. The large waist wy is kept at 2 mm while wx is changed. Two focusing lenses, 50 and 500 mm, are used to separate the waists. The arrow denotes the direction of increasing waist separation.

Fig. 5
Fig. 5

Contour plot of the open-aperture transmission. The circular beam lies along y = x. One can envisage all possible open-aperture Z-scan signatures from this plot by using Eq. (47). The solid and the dotted curves represent major and minor contour lines, respectively.

Fig. 6
Fig. 6

Specific examples of the effects of beam ellipticity on an open-aperture Z-scan signature. A dimensionless irradiance of Q0 = 0.9 and a 250-mm focal-length lens are used. The beams have waists wy = 2 mm, wx = 2, 1.0, 0.5 mm and are characterized in terms of their ellipticity.

Fig. 7
Fig. 7

Open-aperture Z scan for a beam with waists wy = 0.8 mm, wx = 0.2 mm and a focusing lens of 250 mm. A dimensionless irradiance of Q0 = 0.9 is used, and a small degree of asymmetry is observed.

Equations (47)

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

E ( x , y , z , t ) = E 0 ( t ) [ w 0 x w x ( z ) w 0 y w y ( z ) ] 1 / 2 exp { i [ k z η ( z ) ] } × exp { x 2 [ 1 w x 2 ( z ) + i k 2 R x ( z ) ] y 2 [ 1 w y 2 ( z ) + i k 2 R y ( z ) ] } ,
w x , y 2 ( z ) = w 0 x , 0 y 2 [ 1 + ( z z x , y ) 2 z 0 x , 0 y 2 ] ,
R x , y ( z ) = ( z z x , y ) [ 1 + z 0 x , 0 y 2 ( z z x , y ) 2 ] ,
η ( z ) = 1 2 tan 1 ( z z x z 0 x ) + 1 2 tan ( z z y z 0 y ) .
d Δ ϕ d z = Δ n ( I ) k ,
d I d z = α ( I ) I ,
Δ ϕ e ( z , t ) = Δ ϕ 0 ( z , t ) exp [ 2 x 2 w x 2 ( z ) 2 y 2 w y 2 ( z ) ] ,
Δ ϕ 0 ( z , t ) = Δ Φ 0 ( t ) { [ 1 + ( z z x ) 2 / z 0 x 2 ] [ 1 + ( z z y ) 2 / z 0 y 2 ] } 1 / 2 ,
Δ Φ 0 ( t ) = γ k L eff 2 P ( t ) / π w 0 x w 0 y .
I ( x , y , z , t ) = 2 P ( t ) π w x ( z ) w y ( z ) exp [ 2 x 2 w x 2 ( z ) 2 y 2 w y 2 ( z ) ] .
E e ( x , y , z , t ) = E ( x , y , z , t ) exp ( α L / 2 ) × exp [ i Δ ϕ e ( x , y , z , t ) ] = E ( 0 , 0 , z , t ) exp ( α L / 2 ) × m = 0 [ i Δ ϕ 0 ( z , t ) ] m m ! × exp [ i k x 2 2 q m 0 x ( z ) i k y 2 2 q m 0 y ( z ) ] ,
1 q m 0 x , m 0 y ( z ) = 1 R x , y ( z ) i λ π w m 0 x , m 0 y 2 ( z ) ,
w m 0 x , m 0 y 2 ( z ) = w x , y 2 ( z ) / ( 2 m + 1 ) .
g x , y ( z ) = 1 + d / R x , y ( z ) ,
R m x , m y ( z ) = d [ 1 g x , y ( z ) g x , y 2 ( z ) + d 2 / d m 0 x , m 0 y 2 ( z ) ] 1 ,
w m x , m y 2 ( z ) = w m 0 x , m 0 y 2 ( z ) [ g x , y 2 ( z ) + d 2 d m 0 x , m 0 y 2 ( z ) ] ,
d m 0 x , m 0 y ( z ) = k w m 0 x , m 0 y 2 ( z ) / 2 ,
[ w m 0 x ( z ) w m x ( z ) w m 0 y ( z ) w m y ( z ) ] 1 / 2 exp [ i η m ( z ) ] = [ g x ( z ) + i d d m 0 x ( z ) ] 1 / 2 [ g y ( z ) + i d d m 0 y ( z ) ] 1 / 2 ,
E a ( x , y , z , t ) = E ( 0 , 0 , z , t ) exp ( α L / 2 ) × m = 0 [ w m 0 x ( z ) w m x ( z ) w m 0 y ( z ) w m y ( z ) ] 1 / 2 exp [ i η m ( z ) ] × exp [ i k x 2 2 q m x ( z ) i k y 2 2 q m y ( z ) ] × [ i Δ ϕ 0 ( z , t ) ] m m ! .
1 q m x , m y ( z ) = 1 R m x , m y ( z ) i λ π w m x , m y 2 ( z ) .
I e ( x , y , z , t ) = I ( x , y , z , t ) exp ( α L ) 1 + β I ( x , y , z , t ) L eff ,
Δ ϕ e ( x , y , z , t ) = k γ β ln [ 1 + β I ( x , y , z , t ) L eff ] .
E e ( x , y , z , t ) = E ( x , y , z , t ) exp ( α L / 2 ) × [ 1 + β L eff I ( x , y , z , t ) ] i k γ / β 1 / 2 .
n = 0 m [ 1 + ( 2 n 1 ) i β k γ ]
T ( z ) = c 0 n 0 2 0 r a 0 2 π | E a ( r , θ , z , t ) | 2 r d r d θ d t S P ( t ) d t ,
T ( z ) = | E a ( 0 , 0 , z , Δ Φ 0 ) | 2 | E a ( 0 , 0 , z , Δ Φ 0 = 0 ) | 2 = | 1 + i Δ Φ 0 ( g x + i d d 00 x ) 1 / 2 ( g y + i d d 00 y ) 1 / 2 × ( g x + i d d 10 x ) 1 / 2 ( g y + i d d 10 y ) 1 / 2 | 2 ,
T ( z ) = | 1 + i Δ Φ 0 a i b h | 2 ,
h 2 = ( d 00 x 2 + 9 R x 2 ) ( d 00 y 2 + 9 R y 2 ) ,
a = ( d 00 x 2 + 3 R x 2 ) ( d 00 y 2 + 3 R y 2 ) 4 R x R y d 00 x d 00 y ,
b = 2 [ R x d 00 x ( d 00 x 2 + 3 R x 2 ) + R y d 00 y ( d 00 y 2 + 3 R y 2 ) ] .
T ( z ) = 1 + 2 Δ Φ 0 B / h = 1 + 2 Δ Φ 0 p [ a 2 b 2 a 2 h 2 ] 1 / 2 ,
x ( z ) = ( z z x ) / z 0 x ,
y ( z ) = ( z z y ) / z 0 y ,
T ( z ) = 1 + 2 Δ Φ 0 p { [ ( x 2 + 9 ) ( x 2 + 1 ) ( y 2 + 9 ) ( y 2 + 1 ) ] 1 / 2 ( x 2 + 3 ) ( y 2 + 3 ) + 4 x y 2 ( x 2 + 9 ) ( x 2 + 1 ) ( y 2 + 9 ) ( y 2 + 1 ) } 1 / 2 .
T ( u ) = 1 + 4 Δ Φ 0 u ( u 2 + 9 ) ( u 2 + 1 ) ,
q ( r , θ , z , t ) β L eff I ( r , θ , z , t ) = q 0 ( z , t ) exp [ 2 r 2 f ( θ , z ) ] ,
q 0 ( z , t ) = β L eff 2 P ( t ) / π w x ( z ) w y ( z ) ,
f ( θ , z ) = cos 2 θ w x 2 ( z ) + sin 2 θ w y 2 ( z ) .
P e ( z , t ) = 0 0 2 π q 0 ( z , t ) exp [ 2 r 2 f ( θ , z ) ] exp ( α L ) r d r d θ β L eff { 1 + q 0 ( z , t ) exp [ 2 r 2 f ( θ , z ) ] } = π w x ( z ) w y ( z ) 2 β L eff ln [ 1 + q 0 ( z , t ) ] exp ( α L ) .
T ( z ) = k = 0 [ q 0 ( z , 0 ) ] k ( k + 1 ) 3 / 2 ,
q 0 ( z , 0 ) = 2 P 0 β L eff / π w x ( z ) w y ( z ) .
q 0 ( z , 0 ) = Q 0 [ ( 1 + x 2 ) ( 1 + y 2 ) ] 1 / 2 ,
Q 0 = 2 P 0 β L eff / π w 0 x w 0 y .
x ( y 2 + 3 ) + y ( x 2 + 3 ) = 0 .
z null = ( z x z 0 y + z y z 0 x ) / ( z 0 x + z 0 y ) .
z null = 1 2 { ( z y + z x ) ± [ ( z y z x ) 2 12 z 0 x z 0 y ] 1 / 2 } ,
y = ( z 0 x / z 0 y ) x ( Δ z x y / z 0 y ) ,

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