Abstract

We present a new model to predict diffraction patterns of femtosecond pulses through complex optical systems. The model is based on the extension of an ABCD matrix formalism combined with generalized Huygens–Fresnel transforms (already used in the CW regime) to the femtosecond regime. The model is tested to describe femtosecond digital in-line holography experiments realized in situ through a cylindrical Plexiglas pipe. The model allows us to establish analytical relations that link the holographic reconstruction process to the experimental parameters of the pipe and of the incident beam itself. Simulations and experimental results are in good concordance. Femtosecond digital in-line holography is shown to allow significant coherent noise reduction, and this model will be particularly efficient to describe a wide range of optical geometries. More generally, the model developed can be easily used in any experiment where the knowledge of the precise evolution of femtosecond transverse patterns is required.

© 2012 Optical Society of America

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2010 (2)

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[CrossRef]

N. Verrier, C. Rémacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

2009 (2)

2008 (3)

2007 (4)

Q. Zou and B. Lu, “Propagation properties of ultrashort pulsed beams with constant waist width in free space,” Opt. Laser Technol. 39, 619–625 (2007).
[CrossRef]

X. Du and D. Zhao, “Propagation of elliptical Gaussian beams modulated by an elliptical annular aperture,” J. Opt. Soc. Am. A 24, 444–450 (2007).
[CrossRef]

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Suppression of the Moiré effect in subpicosecond digital in-line holography,” Opt. Express 15, 887–895 (2007).
[CrossRef]

2006 (5)

J. Garcia-Sucerquia, W. B. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef]

O. Aydin and M. Avci, “Heat and fluid flow characteristics of gases in micropipes,” Int. J. Heat Mass Trans. 49, 1723–1730 (2006).

C. Zheng, D. Zhao, and X. Du, “Analytical expression of elliptical Gaussian beams through nonsymmetric systems with an elliptical aperture,” Optik 117, 296–298 (2006).
[CrossRef]

X. Du and D. Zhao, “Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems,” J. Opt. Soc. Am. A 23, 625–631 (2006).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[CrossRef]

2005 (4)

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

C. H. Amon, S. C. Yao, C. F. Wu, and C. C. Hsieh, “Microelectromechanical system-based evaporative thermal management of high heat flux electronics,” J. Heat Transfer 127, 66–75 (2005).
[CrossRef]

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time averaging in digital Fresnel holography,” Appl. Opt. 44, 337–343 (2005).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. E. M Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

2003 (3)

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258(2003).
[CrossRef]

G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827–833 (2003).
[CrossRef]

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

2002 (2)

S. Coëtmellec, D. Lebrun, and C. Ozkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

2001 (3)

2000 (3)

K. Konig, “Multiphoton microscopy in life sciences,” J. Microsc. 200, 83–104 (2000).
[CrossRef]

T. Q. Nguyen, I. B. Martini, J. Liu, and B. J. Schwartz, “Controlling interchain interactions in conjugated polymers: the effects of chain morphology on exciton-exciton annihilation and aggregation in MEH-PPV films,” J. Phys. Chem. B 104, 237–255 (2000).
[CrossRef]

T. A. Birks, W. J. Wadsworth, and P. S. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000).
[CrossRef]

1999 (2)

S. Link and M. A. El-Sayed, “Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods,” J. Phys. Chem. B 103, 8410–8426 (1999).
[CrossRef]

F. Dubois, L. Joannes, and J. C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38, 7085–7094 (1999).
[CrossRef]

1998 (2)

1997 (2)

C. Palma and V. Bagini, “Extension of the Fresnel transform to ABCD systems,” J. Opt. Soc. Am. A 14, 1774–1779 (1997).
[CrossRef]

X. Liu, D. Du, and G. Mourou, “Laser ablation and micromachining with ultrashort laser pulses,” IEEE J. Quantum Electron. 33, 1706–1716 (1997).
[CrossRef]

1993 (2)

L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2180–2186 (1993).
[CrossRef]

1988 (1)

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1987 (3)

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

L. Onural and P. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[CrossRef]

1985 (1)

IP Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25, 241–265 (1980).
[CrossRef]

1871 (1)

W. Sellmeier, “Zur Erkla¨rung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. und Chemie 219, 272–282 (1871).
[CrossRef]

Allano, D.

Ameur, K. A.

Amon, C. H.

C. H. Amon, S. C. Yao, C. F. Wu, and C. C. Hsieh, “Microelectromechanical system-based evaporative thermal management of high heat flux electronics,” J. Heat Transfer 127, 66–75 (2005).
[CrossRef]

Andre, Y. B.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Avci, M.

O. Aydin and M. Avci, “Heat and fluid flow characteristics of gases in micropipes,” Int. J. Heat Mass Trans. 49, 1723–1730 (2006).

Aydin, O.

O. Aydin and M. Avci, “Heat and fluid flow characteristics of gases in micropipes,” Int. J. Heat Mass Trans. 49, 1723–1730 (2006).

Bagini, V.

Birks, T. A.

Blaisot, J. B.

Bor, Z.

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

Bourayou, R.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Breazeale, M.

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. J. Wen and M. Breazeale, “Gaussian beam functions as a base function set for acoustical field calculations,” Proceedings of IEEE Ultrasonics Symposium (IEEE, 1987), pp. 1137–1140.

Brunel, M.

N. Verrier, C. Rémacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

M. Brunel, S. Coëtmellec, D. Lebrun, and K. A. Ameur, “Digital phase contrast with the fractional Fourier transform,” Appl. Opt. 48, 579–583 (2009).
[CrossRef]

N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A 25, 1459–1466 (2008).
[CrossRef]

N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt. 47, 4147–4157 (2008).
[CrossRef]

M. Brunel, D. Mgharaz, and S. Coëtmellec, “Generation of femtosecond diffraction-compensated beam through an opaque disk,” Opt. Express 16, 10390–10397 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Suppression of the Moiré effect in subpicosecond digital in-line holography,” Opt. Express 15, 887–895 (2007).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. E. M Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

Christov, IP

IP Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Coëtmellec, S.

N. Verrier, C. Rémacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

M. Brunel, S. Coëtmellec, D. Lebrun, and K. A. Ameur, “Digital phase contrast with the fractional Fourier transform,” Appl. Opt. 48, 579–583 (2009).
[CrossRef]

M. Brunel, D. Mgharaz, and S. Coëtmellec, “Generation of femtosecond diffraction-compensated beam through an opaque disk,” Opt. Express 16, 10390–10397 (2008).
[CrossRef]

N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt. 47, 4147–4157 (2008).
[CrossRef]

N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A 25, 1459–1466 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Suppression of the Moiré effect in subpicosecond digital in-line holography,” Opt. Express 15, 887–895 (2007).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. E. M Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

S. Coëtmellec, D. Lebrun, and C. Ozkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

De Nicola, S.

Du, D.

X. Liu, D. Du, and G. Mourou, “Laser ablation and micromachining with ultrashort laser pulses,” IEEE J. Quantum Electron. 33, 1706–1716 (1997).
[CrossRef]

Du, X.

Dubois, F.

Duh, Y. J.

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

El-Sayed, M. A.

S. Link and M. A. El-Sayed, “Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods,” J. Phys. Chem. B 103, 8410–8426 (1999).
[CrossRef]

Ferraro, P.

Finizio, A.

Fraser, D.

Frey, S.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Garcia-Sucerquia, J.

Garufis, C.

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

Girasole, T.

Gougeon, S.

Grilli, S.

Hanson, S. G.

Hernández, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258(2003).
[CrossRef]

Hsieh, C. C.

C. H. Amon, S. C. Yao, C. F. Wu, and C. C. Hsieh, “Microelectromechanical system-based evaporative thermal management of high heat flux electronics,” J. Heat Transfer 127, 66–75 (2005).
[CrossRef]

Idlahcen, S.

Illueca, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258(2003).
[CrossRef]

Ivanov, C. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[CrossRef]

Janssen, A. J. E. M

Janssen, A. J. E. M.

Jericho, M. H.

Jericho, S. K.

Joannes, L.

Juhasz, T.

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

Kasarova, S. N.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[CrossRef]

Kasparian, J.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Katz, J.

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[CrossRef]

Kerr, F. H.

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Klages, P.

Konig, K.

Krauss, O.

Kreuzer, H. J.

Kurtz, R. M.

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: With Applications in Optics and Signal Processing (Wiley, 2001).

Lambert, A. J.

Lebrun, D.

N. Verrier, C. Rémacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

M. Brunel, S. Coëtmellec, D. Lebrun, and K. A. Ameur, “Digital phase contrast with the fractional Fourier transform,” Appl. Opt. 48, 579–583 (2009).
[CrossRef]

N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A 25, 1459–1466 (2008).
[CrossRef]

N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt. 47, 4147–4157 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Suppression of the Moiré effect in subpicosecond digital in-line holography,” Opt. Express 15, 887–895 (2007).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. E. M Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

S. Coëtmellec, D. Lebrun, and C. Ozkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

Legros, J. C.

Leval, J.

Link, S.

S. Link and M. A. El-Sayed, “Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods,” J. Phys. Chem. B 103, 8410–8426 (1999).
[CrossRef]

Liu, J.

T. Q. Nguyen, I. B. Martini, J. Liu, and B. J. Schwartz, “Controlling interchain interactions in conjugated polymers: the effects of chain morphology on exciton-exciton annihilation and aggregation in MEH-PPV films,” J. Phys. Chem. B 104, 237–255 (2000).
[CrossRef]

Liu, X.

X. Liu, D. Du, and G. Mourou, “Laser ablation and micromachining with ultrashort laser pulses,” IEEE J. Quantum Electron. 33, 1706–1716 (1997).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2180–2186 (1993).
[CrossRef]

Lu, B.

Q. Zou and B. Lu, “Propagation properties of ultrashort pulsed beams with constant waist width in free space,” Opt. Laser Technol. 39, 619–625 (2007).
[CrossRef]

Martini, I. B.

T. Q. Nguyen, I. B. Martini, J. Liu, and B. J. Schwartz, “Controlling interchain interactions in conjugated polymers: the effects of chain morphology on exciton-exciton annihilation and aggregation in MEH-PPV films,” J. Phys. Chem. B 104, 237–255 (2000).
[CrossRef]

Mas, D.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258(2003).
[CrossRef]

McBride, A. C.

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mees, L.

Mejean, G.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Meng, H.

Meucci, R.

Mgharaz, D.

Miret, J. J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258(2003).
[CrossRef]

Mounier, D.

Mourou, G.

X. Liu, D. Du, and G. Mourou, “Laser ablation and micromachining with ultrashort laser pulses,” IEEE J. Quantum Electron. 33, 1706–1716 (1997).
[CrossRef]

Mysyrowicz, A.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25, 241–265 (1980).
[CrossRef]

Nguyen, T. Q.

T. Q. Nguyen, I. B. Martini, J. Liu, and B. J. Schwartz, “Controlling interchain interactions in conjugated polymers: the effects of chain morphology on exciton-exciton annihilation and aggregation in MEH-PPV films,” J. Phys. Chem. B 104, 237–255 (2000).
[CrossRef]

Nicolas, F.

Nikolov, I. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[CrossRef]

Onural, L.

L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
[CrossRef]

L. Onural and P. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: With Applications in Optics and Signal Processing (Wiley, 2001).

Ozkul, C.

Palma, C.

Pan, G.

Pérez, J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258(2003).
[CrossRef]

Picart, P.

Pierattini, G.

Porras, M. A.

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Rémacha, C.

Riemann, I.

Rodriguez, M.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Roze, C.

Russell, P. S.

Salmon, E.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Sarayba, M. B.

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

Sauerbrey, R.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Schwartz, B. J.

T. Q. Nguyen, I. B. Martini, J. Liu, and B. J. Schwartz, “Controlling interchain interactions in conjugated polymers: the effects of chain morphology on exciton-exciton annihilation and aggregation in MEH-PPV films,” J. Phys. Chem. B 104, 237–255 (2000).
[CrossRef]

Scott, P.

L. Onural and P. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

Sellmeier, W.

W. Sellmeier, “Zur Erkla¨rung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. und Chemie 219, 272–282 (1871).
[CrossRef]

Sheng, J.

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[CrossRef]

Soltes, C. R.

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

Sultanova, N. G.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[CrossRef]

Tran, D. B.

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

Vázquez, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258(2003).
[CrossRef]

Verrier, N.

Vikram, C. S.

C. S. Vikram, Particle Field Holography, Cambridge Studies in Modern Optics (Cambridge University, 1992).

Wadsworth, W. J.

Wen, J. J.

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. J. Wen and M. Breazeale, “Gaussian beam functions as a base function set for acoustical field calculations,” Proceedings of IEEE Ultrasonics Symposium (IEEE, 1987), pp. 1137–1140.

Wille, H.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Wolf, J. P.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Woste, L.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Wu, C. F.

C. H. Amon, S. C. Yao, C. F. Wu, and C. C. Hsieh, “Microelectromechanical system-based evaporative thermal management of high heat flux electronics,” J. Heat Transfer 127, 66–75 (2005).
[CrossRef]

Xu, W. B.

Yao, S. C.

C. H. Amon, S. C. Yao, C. F. Wu, and C. C. Hsieh, “Microelectromechanical system-based evaporative thermal management of high heat flux electronics,” J. Heat Transfer 127, 66–75 (2005).
[CrossRef]

Yu, J.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Yura, H. T.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: With Applications in Optics and Signal Processing (Wiley, 2001).

Zhao, D.

Zheng, C.

C. Zheng, D. Zhao, and X. Du, “Analytical expression of elliptical Gaussian beams through nonsymmetric systems with an elliptical aperture,” Optik 117, 296–298 (2006).
[CrossRef]

Zou, Q.

Q. Zou and B. Lu, “Propagation properties of ultrashort pulsed beams with constant waist width in free space,” Opt. Laser Technol. 39, 619–625 (2007).
[CrossRef]

Ann. Phys. und Chemie (1)

W. Sellmeier, “Zur Erkla¨rung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. und Chemie 219, 272–282 (1871).
[CrossRef]

Annu. Rev. Fluid Mech. (1)

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[CrossRef]

Appl. Opt. (7)

IEEE J. Quantum Electron. (1)

X. Liu, D. Du, and G. Mourou, “Laser ablation and micromachining with ultrashort laser pulses,” IEEE J. Quantum Electron. 33, 1706–1716 (1997).
[CrossRef]

IMA J. Appl. Math. (2)

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25, 241–265 (1980).
[CrossRef]

Int. J. Heat Mass Trans. (1)

O. Aydin and M. Avci, “Heat and fluid flow characteristics of gases in micropipes,” Int. J. Heat Mass Trans. 49, 1723–1730 (2006).

J. Acoust. Soc. Am. (1)

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Cataract Refract. Surg. (1)

D. B. Tran, M. B. Sarayba, Z. Bor, C. Garufis, Y. J. Duh, C. R. Soltes, T. Juhasz, and R. M. Kurtz, “Randomized prospective clinical study comparing induced aberrations with IntraLase and Hansatome flap creation in fellow eyes: potential impact on wavefront-guided laser in situ keratomileusis,” J. Cataract Refract. Surg. 31, 97–105 (2005).
[CrossRef]

J. Heat Transfer (1)

C. H. Amon, S. C. Yao, C. F. Wu, and C. C. Hsieh, “Microelectromechanical system-based evaporative thermal management of high heat flux electronics,” J. Heat Transfer 127, 66–75 (2005).
[CrossRef]

J. Microsc. (1)

K. Konig, “Multiphoton microscopy in life sciences,” J. Microsc. 200, 83–104 (2000).
[CrossRef]

J. Opt. Soc. Am. A (9)

X. Du and D. Zhao, “Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems,” J. Opt. Soc. Am. A 23, 625–631 (2006).
[CrossRef]

X. Du and D. Zhao, “Propagation of elliptical Gaussian beams modulated by an elliptical annular aperture,” J. Opt. Soc. Am. A 24, 444–450 (2007).
[CrossRef]

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[CrossRef]

S. Idlahcen, L. Mees, C. Roze, T. Girasole, and J. B. Blaisot, “Time gate, optical layout, and wavelength effects on ballistic imaging,” J. Opt. Soc. Am. A 26, 1995–2004 (2009).
[CrossRef]

C. Palma and V. Bagini, “Extension of the Fresnel transform to ABCD systems,” J. Opt. Soc. Am. A 14, 1774–1779 (1997).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2180–2186 (1993).
[CrossRef]

S. Coëtmellec, D. Lebrun, and C. Ozkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. E. M Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A 25, 1459–1466 (2008).
[CrossRef]

J. Phys. Chem. B (2)

T. Q. Nguyen, I. B. Martini, J. Liu, and B. J. Schwartz, “Controlling interchain interactions in conjugated polymers: the effects of chain morphology on exciton-exciton annihilation and aggregation in MEH-PPV films,” J. Phys. Chem. B 104, 237–255 (2000).
[CrossRef]

S. Link and M. A. El-Sayed, “Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods,” J. Phys. Chem. B 103, 8410–8426 (1999).
[CrossRef]

Opt. Commun. (3)

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[CrossRef]

IP Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258(2003).
[CrossRef]

Opt. Eng. (1)

L. Onural and P. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

Opt. Express (5)

Opt. Laser Technol. (1)

Q. Zou and B. Lu, “Propagation properties of ultrashort pulsed beams with constant waist width in free space,” Opt. Laser Technol. 39, 619–625 (2007).
[CrossRef]

Opt. Lasers Eng. (1)

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Opt. Lett. (3)

Opt. Mater. (1)

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29, 1481–1490 (2007).
[CrossRef]

Optik (1)

C. Zheng, D. Zhao, and X. Du, “Analytical expression of elliptical Gaussian beams through nonsymmetric systems with an elliptical aperture,” Optik 117, 296–298 (2006).
[CrossRef]

Phys. Rev. E (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Science (1)

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[CrossRef]

Other (3)

J. J. Wen and M. Breazeale, “Gaussian beam functions as a base function set for acoustical field calculations,” Proceedings of IEEE Ultrasonics Symposium (IEEE, 1987), pp. 1137–1140.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: With Applications in Optics and Signal Processing (Wiley, 2001).

C. S. Vikram, Particle Field Holography, Cambridge Studies in Modern Optics (Cambridge University, 1992).

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Figures (12)

Fig. 1.
Fig. 1.

System under consideration.

Fig. 2.
Fig. 2.

Theoretical hologram for e1=32.5cm, z1=1.2cm, z2=1.9cm, z3=1.7cm, z4=1.2cm, and e2=3.4cm, in the case of a 1 mm opaque disk present into the pipe, under femtosecond illumination.

Fig. 3.
Fig. 3.

Experimental hologram recorded for e1=32.5cm, z1=1.2cm, z2=1.9cm, z3=1.7cm, z4=1.2cm, and e2=3.4cm under femtosecond illumination.

Fig. 4.
Fig. 4.

Theoretical and experimental profiles along the x-and y-axes in the case of a 1 mm opaque disk present into the pipe.

Fig. 5.
Fig. 5.

Influence of dispersion: index variations of the pipe made in Plexiglas (a) or of a pipe made with a virtual material (b), a detail of the intensity pattern distribution obtained with the Plexiglas pipe (c) or with the virtual material’s pipe (d). The pattern that would be obtained if dispersion was not taken into account is reported in red. There is only a difference for the virtual material.

Fig. 6.
Fig. 6.

FRFT reconstruction of the simulated hologram of Fig. 2 with αx=0.723π/2 and αy=0.272π/2.

Fig. 7.
Fig. 7.

FRFT reconstruction of the experimental hologram of Fig. 3 with αx=0.903π/2 and αy=0.675π/2.

Fig. 8.
Fig. 8.

Experimental hologram recorded with e1=12cm, z1=1.2cm, z2=2.05cm, z3=1.55cm, z4=1.2cm, and e2=3.5cm under femtosecond illumination.

Fig. 9.
Fig. 9.

FRFT reconstruction of the opaque object from Fig. 8 with αx=0.90π/2 and αy=0.674π/2.

Fig. 10.
Fig. 10.

FRFT reconstruction of the pipe’s defect from Fig. 8 with αx=0.779π/2 and αy=0.584π/2.

Fig. 11.
Fig. 11.

Experimental hologram recorded with e1=32.5cm, z1=1.2cm, z2=1.9cm, z3=1.7cm, z4=1.2cm, and e2=3.4cm under CW illumination.

Fig. 12.
Fig. 12.

FRFT reconstruction of Fig. 11 with αx=0.903π/2 and αy=0.675π/2.

Equations (38)

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

E˜(r,ω)=G˜(r,ω)U˜(ω),
I(r)=C0+|G˜(r,ω)|2|U˜(ω)|2dω,
G1˜(ξ,η,e1+z1+z2,ω)=ωexp(iωcE1)i2πcB1xB1y×R2G0˜(x,y,0,ω)exp[iω2cB1x(A1xx22ξx+D1xξ2)]×exp[iω2cB1y(A1yy22ηy+D1yη2)]dxdy,
M1x,y=(A1x,yB1x,yC1x,yD1x,y)=Mz2×MD2x,y×Mz1×MD1x,y×Me1.
G1˜(ξ,η,e1+z1+z2,ω)=ωexp(iωcE1)i2πcB1xB1yK1xK1yexp[(ξ2ω1x2+η2ω1y2)]exp[iω2c(ξ2R1x+η2R1y)],
K1x,y=(πω021iA1x,yωw022cB1x,y)1/2.
w1x,y=(2cB1x,yωw0)[1+(A1x,yωw022cB1x,y)2]1/2,R1x,y=B1x,yD1x,yA1x,y(ωw022cB1x,y)21+(A1x,yωw022cB1x,y)2.
T(ξ,η)=k=1NAkexp[Bkb2(ξ2+Rell2η2)].
G˜(x,y,z,ω)=ωexp(iωcE2)i2πcB2xB2yR2G1˜(ξ,η,e1+z1+z2,ω)[1T(ξ,η)]×exp[iω2cB2x(A2xξ22xξ+D2xx2)]×exp[iω2cB2y(A2yη22yη+D2yy2)]dξdη,
G˜(x,y,z,ω)=ωexp(iωcE2)i2πcB2xB2y×(R˜(x,y,z,ω)O˜(x,y,z,ω)).
R˜(x,y,z,ω)=ωexp(iωcE1)i2πcB1xB1yK1xK1yK2xK2y×exp[ω2c(NxB2xx2+NyB2yy2)]×exp[iω2c(MxB2xx2+MyB2yy2)],
Mx,y=D2x,y+(ωw1x,y22cB2x,y)2(B2x,yR1x,yA2x,y)1+(ωw1x,y22cB2x,y)2(B2x,yR1x,yA2x,y)2,Nx,y=ωw1x,y22cB2x,y1+(ωw1x,y22cB2x,y)2(B2x,yR1x,yA2x,y)2,
K2x,y=[πw1x,y21+iωw1x,y22cB2x,y(B2x,yR1x,yA2x,y)]1/2.
1w1xeq2=1w1x2+{Bk}b2,1w1yeq2=1w1y2+Rell2{Bk}b2,
1R1xeq=1R1x+c{Bk}2ωb2,1R1yeq=1R1y+Rell2c{Bk}2ωb2
O˜(x,y,z,ω)=ωexp(iωcE1)i2cB1xB1yK1xK1y×exp[iω2c(D2xB2xx2+D2yB2yy2)]×k=1NAkK2xeqK2yeq×exp[ω2c(NxeqB2xx2+NyeqB2yy2)]×exp[iω2c(MxeqB2xx2+MyeqB2yy2)],
Mx,yeq=(ωw1x,yeq22cB2x,y)2(B2x,yR1x,yeqA2x,y)1+(ωw1x,yeq22cB2x,y)2(B2x,yR1x,yeqA2x,y)2,Nx,yeq=ωw1x,yeq22cB2x,y1+(ωw1x,yeq22cB2x,y)2(B2x,yR1x,yeqA2x,y)2,
K2x,yeq=[πw1x,yeq21+iωw1x,yeq22cB2x,y(B2x,yR1x,yeqA2x,y)]1/2.
M2x,y=Me2×MD4x,y×Mz4×MD3x,y×Mz3=(A2x,yB2x,yC2x,yD2x,y).
n2(λ)=1+i=0kBiλ2λ2λ0i2,
n2(λ)=A1+A2λ2+A3λ2+A4λ4+A5λ6+A6λ8+,
Fαx,αy[I(x,y)](xα,yα)=R2Nαx(x,xα)Nαy(y,yα)I(x,y)dxdy,
Nαp(x,xα)=C(αp)exp(iπx2+xα2sp2tanαp)exp(i2πxαxsp2sinαp),
C(αp)=exp(i(π4sign(sinαp)αp2))|sp2sinαp|1/2.
αxopt=arctan[B2xλ0sx2(MxD2x)],αyopt=arctan[B2yλ0sy2(MyD2y)].
gx,y=wtotx,y/w1x,y,
w1x,y=(λ0B1x,yπw0)[1+(A1x,yπw02λ0B1x,y)2]1/2
wtotx,y=(λ0Btotx,yπw0)[1+(Atotx,yπw02λ0Btotx,y)2]1/2,
Me1=(1e101).
MRx1=(10n0n1Rx11),MR1y=(10n0n1Ry11).
Mz1=(1z1n101).
MRx2=(10n1n2Rx21),MRy2=(10n1n2Ry21).
Mz2=(1z2n201).
Mz3=(1z3n201).
MRx3=(10n2n1Rx31),MRy3=(10n2n1Ry31),
Mz4=(1z4n101),
MRx4=(10n1n0Rx41),MRy4=(10n1n0Ry41),
Me4=(1e401),

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