Abstract

The theoretical investigation of the off-axis z-scan technique for the measurement of nonlinear optical refraction in materials is presented. The normalized transmittance is calculated for different aperture radii and positions. The dependence of both the normalized transmittance amplitude (ΔTpv) and the distance between maximum and minimum (Δzpv) on the aperture radius is analyzed. A condition for the applicability of the pinhole approximation is given.

© 2006 Optical Society of America

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  1. S. R. Friberg and P. W. Smith, " Nonlinear optical glasses for ultrafast optical switches," IEEE. J. Quantum Electron. 23, 2089- 2094 ( 1978).
    [CrossRef]
  2. M. J. Moran, C. Y. She, and R. L. Carman, " Interferometric measurements of nonlinear refractive index coefficient relative to CS2 in laser-system-related materials," IEEE J. Quantum Electron. 11, 259- 263 ( 1975).
    [CrossRef]
  3. 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]
  4. S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
    [CrossRef]
  5. R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
    [CrossRef]
  6. D. H. Osborne, R. F. Haglund, F. Gonella, and F. Garrido, " Laser-induced sign reversal of the nonlinear refractive index of Ag nanoclusters in soda-lime glass," Appl. Phys. B 66, 517- 521 ( 1998).
    [CrossRef]
  7. T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, " Eclipsing Z-scan measurement of Lambda/104 wave-front distortion," Opt. Lett. 19, 317- 319 ( 1994).
    [CrossRef] [PubMed]
  8. A. Marcano, F. E. Hermandes, and A. D. Sena, " Two-color near-field eclipsing Z-scan technique for the determination of nonlinear refraction," J. Opt. Soc. Am. B 14, 3363- 3367 ( 1997).
    [CrossRef]
  9. S. Bian and J. Frejlich, " Z-scan measurements of photorefractive nonlinearities for SNB:Ce crystal," Appl. Phys. B 64, 539- 546 ( 1997).
    [CrossRef]
  10. J. -G. Tian, W. -P. Zang, and G. Zhang, " Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sencitivity," Opt. Commun. 107, 415- 419 ( 1994).
    [CrossRef]
  11. P. B. Chapple, J. Staromlynska and J. A. Hermann, " Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251- 293 ( 1997).
    [CrossRef]
  12. S. C. Mehendale, S. R. Mishra, K. S. Bindra, M. Laghate, T. S. Dhami, and K. S. Rustagi, " Nonlinear refraction in aqueous colloidal gold," Opt. Commun. 133, 273- 276 ( 1997).
    [CrossRef]
  13. H. B. Liao, R. F. Xiao, J. S. Fu, and G. K. L. Wong, " Large third-order nonlinear susceptibility of Au:Al2O3 composite films near the resonant frequency," Appl. Phys. B 65, 673- 676 ( 1997).
    [CrossRef]

2000 (1)

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
[CrossRef]

1998 (2)

R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
[CrossRef]

D. H. Osborne, R. F. Haglund, F. Gonella, and F. Garrido, " Laser-induced sign reversal of the nonlinear refractive index of Ag nanoclusters in soda-lime glass," Appl. Phys. B 66, 517- 521 ( 1998).
[CrossRef]

1997 (5)

A. Marcano, F. E. Hermandes, and A. D. Sena, " Two-color near-field eclipsing Z-scan technique for the determination of nonlinear refraction," J. Opt. Soc. Am. B 14, 3363- 3367 ( 1997).
[CrossRef]

S. Bian and J. Frejlich, " Z-scan measurements of photorefractive nonlinearities for SNB:Ce crystal," Appl. Phys. B 64, 539- 546 ( 1997).
[CrossRef]

P. B. Chapple, J. Staromlynska and J. A. Hermann, " Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251- 293 ( 1997).
[CrossRef]

S. C. Mehendale, S. R. Mishra, K. S. Bindra, M. Laghate, T. S. Dhami, and K. S. Rustagi, " Nonlinear refraction in aqueous colloidal gold," Opt. Commun. 133, 273- 276 ( 1997).
[CrossRef]

H. B. Liao, R. F. Xiao, J. S. Fu, and G. K. L. Wong, " Large third-order nonlinear susceptibility of Au:Al2O3 composite films near the resonant frequency," Appl. Phys. B 65, 673- 676 ( 1997).
[CrossRef]

1994 (2)

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

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

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]

1978 (1)

S. R. Friberg and P. W. Smith, " Nonlinear optical glasses for ultrafast optical switches," IEEE. J. Quantum Electron. 23, 2089- 2094 ( 1978).
[CrossRef]

1975 (1)

M. J. Moran, C. Y. She, and R. L. Carman, " Interferometric measurements of nonlinear refractive index coefficient relative to CS2 in laser-system-related materials," IEEE J. Quantum Electron. 11, 259- 263 ( 1975).
[CrossRef]

Afonso, C. N.

R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
[CrossRef]

Ballesteros, J. M.

R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
[CrossRef]

Bian, S.

S. Bian and J. Frejlich, " Z-scan measurements of photorefractive nonlinearities for SNB:Ce crystal," Appl. Phys. B 64, 539- 546 ( 1997).
[CrossRef]

Bindra, K. S.

S. C. Mehendale, S. R. Mishra, K. S. Bindra, M. Laghate, T. S. Dhami, and K. S. Rustagi, " Nonlinear refraction in aqueous colloidal gold," Opt. Commun. 133, 273- 276 ( 1997).
[CrossRef]

Carman, R. L.

M. J. Moran, C. Y. She, and R. L. Carman, " Interferometric measurements of nonlinear refractive index coefficient relative to CS2 in laser-system-related materials," IEEE J. Quantum Electron. 11, 259- 263 ( 1975).
[CrossRef]

Chapple, P. B.

P. B. Chapple, J. Staromlynska and J. A. Hermann, " Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251- 293 ( 1997).
[CrossRef]

Debrus, S.

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
[CrossRef]

Dhami, T. S.

S. C. Mehendale, S. R. Mishra, K. S. Bindra, M. Laghate, T. S. Dhami, and K. S. Rustagi, " Nonlinear refraction in aqueous colloidal gold," Opt. Commun. 133, 273- 276 ( 1997).
[CrossRef]

Frejlich, J.

S. Bian and J. Frejlich, " Z-scan measurements of photorefractive nonlinearities for SNB:Ce crystal," Appl. Phys. B 64, 539- 546 ( 1997).
[CrossRef]

Friberg, S. R.

S. R. Friberg and P. W. Smith, " Nonlinear optical glasses for ultrafast optical switches," IEEE. J. Quantum Electron. 23, 2089- 2094 ( 1978).
[CrossRef]

Fu, J. S.

H. B. Liao, R. F. Xiao, J. S. Fu, and G. K. L. Wong, " Large third-order nonlinear susceptibility of Au:Al2O3 composite films near the resonant frequency," Appl. Phys. B 65, 673- 676 ( 1997).
[CrossRef]

Garrido, F.

D. H. Osborne, R. F. Haglund, F. Gonella, and F. Garrido, " Laser-induced sign reversal of the nonlinear refractive index of Ag nanoclusters in soda-lime glass," Appl. Phys. B 66, 517- 521 ( 1998).
[CrossRef]

Gonella, F.

D. H. Osborne, R. F. Haglund, F. Gonella, and F. Garrido, " Laser-induced sign reversal of the nonlinear refractive index of Ag nanoclusters in soda-lime glass," Appl. Phys. B 66, 517- 521 ( 1998).
[CrossRef]

Hagan, D. J.

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

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]

Haglund, R. F.

D. H. Osborne, R. F. Haglund, F. Gonella, and F. Garrido, " Laser-induced sign reversal of the nonlinear refractive index of Ag nanoclusters in soda-lime glass," Appl. Phys. B 66, 517- 521 ( 1998).
[CrossRef]

Haglung, R. F.

R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
[CrossRef]

Hermandes, F. E.

Hermann, J. A.

P. B. Chapple, J. Staromlynska and J. A. Hermann, " Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251- 293 ( 1997).
[CrossRef]

Lafait, J.

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
[CrossRef]

Laghate, M.

S. C. Mehendale, S. R. Mishra, K. S. Bindra, M. Laghate, T. S. Dhami, and K. S. Rustagi, " Nonlinear refraction in aqueous colloidal gold," Opt. Commun. 133, 273- 276 ( 1997).
[CrossRef]

Liao, H. B.

H. B. Liao, R. F. Xiao, J. S. Fu, and G. K. L. Wong, " Large third-order nonlinear susceptibility of Au:Al2O3 composite films near the resonant frequency," Appl. Phys. B 65, 673- 676 ( 1997).
[CrossRef]

Marcano, A.

May, M.

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
[CrossRef]

Mehendale, S. C.

S. C. Mehendale, S. R. Mishra, K. S. Bindra, M. Laghate, T. S. Dhami, and K. S. Rustagi, " Nonlinear refraction in aqueous colloidal gold," Opt. Commun. 133, 273- 276 ( 1997).
[CrossRef]

Mishra, S. R.

S. C. Mehendale, S. R. Mishra, K. S. Bindra, M. Laghate, T. S. Dhami, and K. S. Rustagi, " Nonlinear refraction in aqueous colloidal gold," Opt. Commun. 133, 273- 276 ( 1997).
[CrossRef]

Moran, M. J.

M. J. Moran, C. Y. She, and R. L. Carman, " Interferometric measurements of nonlinear refractive index coefficient relative to CS2 in laser-system-related materials," IEEE J. Quantum Electron. 11, 259- 263 ( 1975).
[CrossRef]

Osborne, D. H.

R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
[CrossRef]

D. H. Osborne, R. F. Haglund, F. Gonella, and F. Garrido, " Laser-induced sign reversal of the nonlinear refractive index of Ag nanoclusters in soda-lime glass," Appl. Phys. B 66, 517- 521 ( 1998).
[CrossRef]

Petford-Long, A. K.

R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
[CrossRef]

Pinçon, N.

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
[CrossRef]

Prot, D.

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
[CrossRef]

Rustagi, K. S.

S. C. Mehendale, S. R. Mishra, K. S. Bindra, M. Laghate, T. S. Dhami, and K. S. Rustagi, " Nonlinear refraction in aqueous colloidal gold," Opt. Commun. 133, 273- 276 ( 1997).
[CrossRef]

Said, A. A.

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]

Sella, C.

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
[CrossRef]

Sena, A. D.

Serna, R.

R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
[CrossRef]

She, C. Y.

M. J. Moran, C. Y. She, and R. L. Carman, " Interferometric measurements of nonlinear refractive index coefficient relative to CS2 in laser-system-related materials," IEEE J. Quantum Electron. 11, 259- 263 ( 1975).
[CrossRef]

Sheik-Bahae, M.

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

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]

Smith, P. W.

S. R. Friberg and P. W. Smith, " Nonlinear optical glasses for ultrafast optical switches," IEEE. J. Quantum Electron. 23, 2089- 2094 ( 1978).
[CrossRef]

Solis, J.

R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
[CrossRef]

Staromlynska, J.

P. B. Chapple, J. Staromlynska and J. A. Hermann, " Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251- 293 ( 1997).
[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 sencitivity," Opt. Commun. 107, 415- 419 ( 1994).
[CrossRef]

Van Stryland, E. W.

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

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]

Venturini, J.

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
[CrossRef]

Wei, T. -H.

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]

Wong, G. K. L.

H. B. Liao, R. F. Xiao, J. S. Fu, and G. K. L. Wong, " Large third-order nonlinear susceptibility of Au:Al2O3 composite films near the resonant frequency," Appl. Phys. B 65, 673- 676 ( 1997).
[CrossRef]

Xia, T.

Xiao, R. F.

H. B. Liao, R. F. Xiao, J. S. Fu, and G. K. L. Wong, " Large third-order nonlinear susceptibility of Au:Al2O3 composite films near the resonant frequency," Appl. Phys. B 65, 673- 676 ( 1997).
[CrossRef]

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 sencitivity," 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 sencitivity," Opt. Commun. 107, 415- 419 ( 1994).
[CrossRef]

Appl. Phys. B (3)

D. H. Osborne, R. F. Haglund, F. Gonella, and F. Garrido, " Laser-induced sign reversal of the nonlinear refractive index of Ag nanoclusters in soda-lime glass," Appl. Phys. B 66, 517- 521 ( 1998).
[CrossRef]

S. Bian and J. Frejlich, " Z-scan measurements of photorefractive nonlinearities for SNB:Ce crystal," Appl. Phys. B 64, 539- 546 ( 1997).
[CrossRef]

H. B. Liao, R. F. Xiao, J. S. Fu, and G. K. L. Wong, " Large third-order nonlinear susceptibility of Au:Al2O3 composite films near the resonant frequency," Appl. Phys. B 65, 673- 676 ( 1997).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. J. Moran, C. Y. She, and R. L. Carman, " Interferometric measurements of nonlinear refractive index coefficient relative to CS2 in laser-system-related materials," IEEE J. Quantum Electron. 11, 259- 263 ( 1975).
[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]

IEEE. J. Quantum Electron. (1)

S. R. Friberg and P. W. Smith, " Nonlinear optical glasses for ultrafast optical switches," IEEE. J. Quantum Electron. 23, 2089- 2094 ( 1978).
[CrossRef]

J. Appl. Phys. (1)

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, " Z-scan determination of the third-order optical nonlinearity of gold:silica nanocomposites," J. Appl. Phys. 88, 4469- 4475 ( 2000).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

P. B. Chapple, J. Staromlynska and J. A. Hermann, " Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251- 293 ( 1997).
[CrossRef]

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

Opt. Commun. (2)

S. C. Mehendale, S. R. Mishra, K. S. Bindra, M. Laghate, T. S. Dhami, and K. S. Rustagi, " Nonlinear refraction in aqueous colloidal gold," Opt. Commun. 133, 273- 276 ( 1997).
[CrossRef]

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

Opt. Lett. (1)

Thin Solid Films (1)

R. Serna, J. M. Ballesteros, J. Solis, C. N. Afonso, D. H. Osborne, R. F. Haglung, Jr., and A. K. Petford-Long, " Laser-induced modification of the nonlinear optical response of laser-deposited Cu:Al2O3 nanocomposite films," Thin Solid Films 318, 96- 99 ( 1998).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

z dependence of the normalized transmittance calculated for different aperture radii and displacement: (a) d = 0.1 cm, (b) d = 0.25 cm, and (c) d = 0.5 cm. The experimental parameters are w 0 = 50 μm, l = 90 cm, and z 0 = 1.48 cm.

Fig. 2
Fig. 2

Variation of ΔTpv with aperture displacement d calculated for different values of the aperture radius, ra.

Fig. 3
Fig. 3

Variation of Δzpv with aperture displacement d calculated for different values of the aperture radius, ra.

Fig. 4
Fig. 4

Effective displacement as a function of aperture radius for d = 0.5 cm.

Equations (1)

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T ( z , ρ ) = 1 + 2 exp [ 2 ρ 2 ( 3 x 2 x 2 + 9 ) ] Δ Φ 0 ( x 2 + 3 ) sin ( 8 ρ 2 x x 2 + 9 ) + 2 x cos ( - 8 ρ 2 x x 2 + 9 ) ( x 2 + 9 ) ( x 2 + 1 ) .

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