Abstract

Direct observation of Gouy phase shift on an optical vortex was presented through investigating the intensity profiles of a modified LGpm beam with an asymmetric defect, around at the focal point. In addition, the three-dimensional trajectory of the defect was found to describe a uniform straight line. It was quantitatively found that the rotation profile of a modified LGpm beam manifests the Gouy phase effect where the rotation direction depends on only the sign of topological charge m. This profile measurement method by introducing an asymmetric defect is a simple and useful technique for obtaining the information of the Gouy phase shift, without need of a conventional interference method.

© 2006 Optical Society of America

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2006 (1)

2005 (1)

F. Flossmann, U. T. Schwarz, and M. Maier, "Optical vortices in a Laguerre-Gaussian LG01beam," J. Mod. Opt. 52, 1009-1017 (2005).
[CrossRef]

2004 (1)

2003 (1)

J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).

2001 (4)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholaki, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

S. Feng and H. G. Winful, "Physical origin of the Gouy phase shift," Opt. Lett. 26, 485-487 (2001).
[CrossRef]

L.-M. Duan, J. I. Cirac, and P. Zoller, "Geometric manipulation of trapped ions for quantum computation," Science 292, 1695-1697 (2001).
[CrossRef] [PubMed]

2000 (2)

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, "Geometric quantum computation using nuclear magnetic resonance," Nature 403, 869-871 (2000).
[CrossRef] [PubMed]

R. W. McGown, R. A. Cheville, and D. Grischkowsky, "Direct observation of the Gouy phase shift in THz impulse ranging," Appl. Phys. Lett. 76, 670-672 (2000).
[CrossRef]

1999 (2)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

1997 (2)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

1996 (2)

K. T. Gahagan and G. A. Swartzlander, Jr., "Optical vortex trapping of particles," Opt. Lett. 21, 827-829 (1996).
[CrossRef] [PubMed]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, "Second-harmonic generation and the orbital angular momentum of light," Phys. Rev. A 54, R3742-3745 (1996).
[CrossRef] [PubMed]

1995 (2)

D. Subbarao, "Topological phase in Gaussian beam optics," Opt. Lett. 20, 2162-2164 (1995).
[CrossRef] [PubMed]

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995).
[CrossRef]

1994 (1)

1993 (1)

R. Simon and N. Mukunda, "Bargmann invariant and the geometry of the Gouy effect," Phys. Rev. Lett. 70, 880-883 (1993).
[CrossRef] [PubMed]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A45, 8185-8189 (1992).
[CrossRef] [PubMed]

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

1984 (1)

M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London A 392, 45-57 (1984).
[CrossRef]

1959 (1)

C. R. Carpenter, "Gouy phase advance with microwaves," Am. J. Phys. 27, 98-100 (1959).

1890 (1)

L. G. Gouy, "Sur une propriete nouvelle des ondes lumineuses," Acad. Sci., Paris, C. R.  110, 1251-1253 (1890).

Allen, L.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, "Second-harmonic generation and the orbital angular momentum of light," Phys. Rev. A 54, R3742-3745 (1996).
[CrossRef] [PubMed]

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A45, 8185-8189 (1992).
[CrossRef] [PubMed]

Arlt, J.

J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholaki, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Ashkin, A.

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A45, 8185-8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London A 392, 45-57 (1984).
[CrossRef]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholaki, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Carpenter, C. R.

C. R. Carpenter, "Gouy phase advance with microwaves," Am. J. Phys. 27, 98-100 (1959).

Castagnoli, G.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, "Geometric quantum computation using nuclear magnetic resonance," Nature 403, 869-871 (2000).
[CrossRef] [PubMed]

Cheville, R. A.

R. W. McGown, R. A. Cheville, and D. Grischkowsky, "Direct observation of the Gouy phase shift in THz impulse ranging," Appl. Phys. Lett. 76, 670-672 (2000).
[CrossRef]

Chow, J. H.

Cirac, J. I.

L.-M. Duan, J. I. Cirac, and P. Zoller, "Geometric manipulation of trapped ions for quantum computation," Science 292, 1695-1697 (2001).
[CrossRef] [PubMed]

de Vine, G.

Dennis, M. R.

Dholaki, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholaki, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Dholakia, K.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, "Second-harmonic generation and the orbital angular momentum of light," Phys. Rev. A 54, R3742-3745 (1996).
[CrossRef] [PubMed]

Duan, L.-M.

L.-M. Duan, J. I. Cirac, and P. Zoller, "Geometric manipulation of trapped ions for quantum computation," Science 292, 1695-1697 (2001).
[CrossRef] [PubMed]

Ekert, A.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, "Geometric quantum computation using nuclear magnetic resonance," Nature 403, 869-871 (2000).
[CrossRef] [PubMed]

Feng, S.

S. Feng and H. G. Winful, "Physical origin of the Gouy phase shift," Opt. Lett. 26, 485-487 (2001).
[CrossRef]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

Flossmann, F.

F. Flossmann, U. T. Schwarz, and M. Maier, "Optical vortices in a Laguerre-Gaussian LG01beam," J. Mod. Opt. 52, 1009-1017 (2005).
[CrossRef]

Gahagan, K. T.

Gouy, L. G.

L. G. Gouy, "Sur une propriete nouvelle des ondes lumineuses," Acad. Sci., Paris, C. R.  110, 1251-1253 (1890).

Gray, M. B.

Grischkowsky, D.

R. W. McGown, R. A. Cheville, and D. Grischkowsky, "Direct observation of the Gouy phase shift in THz impulse ranging," Appl. Phys. Lett. 76, 670-672 (2000).
[CrossRef]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Jones, J. A.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, "Geometric quantum computation using nuclear magnetic resonance," Nature 403, 869-871 (2000).
[CrossRef] [PubMed]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Luther-Davies, B.

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholaki, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier, "Optical vortices in a Laguerre-Gaussian LG01beam," J. Mod. Opt. 52, 1009-1017 (2005).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

McClelland, D. E.

McGown, R. W.

R. W. McGown, R. A. Cheville, and D. Grischkowsky, "Direct observation of the Gouy phase shift in THz impulse ranging," Appl. Phys. Lett. 76, 670-672 (2000).
[CrossRef]

Mehta, A. D.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Mukunda, N.

R. Simon and N. Mukunda, "Bargmann invariant and the geometry of the Gouy effect," Phys. Rev. Lett. 70, 880-883 (1993).
[CrossRef] [PubMed]

O’Holleran, K.

Padgett, M. J.

K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four, and five plane waves," Opt. Express 14, 3039-3044 (2006).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, "Second-harmonic generation and the orbital angular momentum of light," Phys. Rev. A 54, R3742-3745 (1996).
[CrossRef] [PubMed]

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995).
[CrossRef]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholaki, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Powels, R.

Rief, M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Rudd, J. V.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

Ruffin, A. B.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, and M. Maier, "Optical vortices in a Laguerre-Gaussian LG01beam," J. Mod. Opt. 52, 1009-1017 (2005).
[CrossRef]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Sibbet, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholaki, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Simmons, R. M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Simon, R.

R. Simon and N. Mukunda, "Bargmann invariant and the geometry of the Gouy effect," Phys. Rev. Lett. 70, 880-883 (1993).
[CrossRef] [PubMed]

Simpson, N. B.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, "Second-harmonic generation and the orbital angular momentum of light," Phys. Rev. A 54, R3742-3745 (1996).
[CrossRef] [PubMed]

Smith, D. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A45, 8185-8189 (1992).
[CrossRef] [PubMed]

Spudich, J. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Subbarao, D.

Swartzlander, G. A.

Tikhonenko, V.

Torii, Y.

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[CrossRef] [PubMed]

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, "Geometric quantum computation using nuclear magnetic resonance," Nature 403, 869-871 (2000).
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V. Garćes-Chávez, D. M. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, "Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle," Phys. Rev. Lett. 91, 093602-1-4 (2003).
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D. P. Caetano, M. P. Almeida, P. H. Souto Ribeiro, J. A. O. Huguenin, B. Coutinho dos Santos, and A. Z. Khoury, "Conservation of orbital angular momentum in stimulated down-conversion," Phys. Rev. A 66, R041801-1-4 (2002).
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A. Vinçotte and L. Bergé, "Femtosecond optical vortices in air," Phys. Rev. Lett. 95, 193901-1-4 (2005).
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D. Neshev, A. Nepomnyashchy, and Y. S. Kivshar, "Nonlinear Aharonov-Bohm scattering by optical vortices," Phys. Rev. Lett. 87, 043901-1-4 (2001).
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F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmarkis, M. Lezius, and F. Karusz, "Gouy phase shift for few-cycle-laser pulses," Phys. Rev. Lett. 92, 113001-1-4 (2004).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Experimental setup for spatial evolution measurement of a modified LG beam with an asymmetric defect, (b) a normal spiral hologram pattern, (c) a pure LG beam, (d) a spiral hologram pattern with an asymmetric defect (defect angle π/3), and (e) a modified LG beam with an asymmetric defect.

Fig. 2.
Fig. 2.

Experimantally-observed spatial evolution of modified LG beams with an aymmetric defect for (a) (m, p) = (+10,0), (b) (m, p) = (-10,0) and (c) (m, p) = (0,0). The intensity profiles at z = -15, 0, and +15 cm are magnified 1.7, 2, and 1.7 times, respectively.

Fig. 3.
Fig. 3.

(a) Schematic drawing for an intensity profile of a modified LG beam with an aymmetric defect with the coordinates x, y, z, and average defect angle φ D when observed from +z-direction on CCD. (b) Dependence of observed defect angle φ D on the propagation distance z for modified LG beams with (m, p) = (+10,0) and (-10,0).

Fig. 4.
Fig. 4.

(a) 3D trajectories of the asymmetric defect for modified LG beams with (m, p) = (+10,0) and (-10,0) projected on x-y plane as a function of propagation distance z. (b) Relationship between coordinates x,y and x′, y′. (x′ ,y′, z) coordinates are obtained from (x,y, z) coordinates by a roation of Φ 0 around the z axis. In our experimental configuration, we put φ D = 0 at z = -65 cm with the initial phase Φ 0.

Fig. 5.
Fig. 5.

Distributions of |Cmp |2 as a function of indices m and p for modified LG beams with (a) (m, p) = (+10,0), (b) (m, p) = (-10,0) (c) (m, p) = (0,0) and (d) (m, p) = (+10,5).

Fig. 6.
Fig. 6.

Calculated spatial evolution of modified LG beams with an aymmetric defect for (a) (m, p) = (+10,0), (b) (m, p) = (-10,0), (c) (m, p) = (0,0) and (d) (m, p) = (+10,+5) at propagation distances of z = -∞,-z R,0,-z R, and +∞. Dimensions of beams are scaled by w(z).

Equations (36)

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E m p ( ρ , φ , z , t ) = u m p ( ρ , φ , z ) exp [ i ( k z ω t ) ] ,
u m p ( ρ , φ , z ) = 2 p ! π ( p + m ) ! [ 2 ρ w ( z ) ] m L p m ( 2 ρ 2 w ( z ) 2 ) w 0 w ( z )
× exp [ ρ 2 w ( z ) 2 i k ρ 2 2 R ( z ) + i m φ i Φ G ( z ) ] ,
L p m ( x ) = r = 0 p ( 1 ) r p + m p r x r r ! .
R ( z ) = ( z R 2 + z 2 ) z , w ( z ) = w 0 1 z 2 z R 2 .
z R = k w 0 2 2 .
Φ G ( z ) = ( 2 p + m + 1 ) Φ ( z ) ( 2 p + m + 1 ) arctan ( z z R ) ,
sgn ( m ) = { + 1 , m > 0 , 1 , m < 0 .
Ψ m p C ( ρ , φ , z ) = u m p ( ρ , φ , z ) { 1 exp [ φ 2 2 ( δ φ D ) 2 ] } ,
Ψ m p C ( ρ , φ , z ) = p = 0 m = C m p u m p ( ρ , φ , z ) .
C m p = 1 w 0 2 0 d ρ ρ π π d φ u m p * ( ρ , φ , z ) Ψ m p C ( ρ , φ , z ) ,
Ψ m p C ( ρ , φ , z ) 2
= p = 0 m = p = 0 m = C m p * C m p u m p * ( ρ , φ , z ) u m p ( ρ , φ , z )
= 2 π w 0 2 w ( z ) 2 exp [ 2 ρ 2 w ( z ) 2 ] p = 0 m = p = 0 m = C m p * C m p p ! p ( p + m ) ! ( p + m ) !
× [ 2 ρ w ( z ) ] m + m L p m ( 2 ρ 2 w ( z ) 2 ) L p m ( 2 ρ 2 w ( z ) 2 )
× exp { i ( m m ) φ i [ 2 ( p p ) + m m ] Φ ( z ) }
4 π w 0 2 w ( z ) 2 exp [ 2 ρ 2 w ( z ) 2 ] ( A + B + C + D + E ) ,
A = p = 0 m = C m p 2 p ! ( p + m ) ! [ 2 ρ w ( z ) ] 2 m [ L p m ( 2 ρ 2 w ( z ) 2 ) ] 2 ,
B = p = 0 m = m < m C m p * C m p p ! ( p + m ) ! ( p + m ) ! [ 2 ρ w ( z ) ] m + m
× L p m ( 2 ρ 2 w ( z ) 2 ) L p m ( 2 ρ 2 w ( z ) 2 ) cos [ ( m m ) φ ( m m ) Φ ( z ) ] ,
C = p = 0 m = p < p C m p * C m p p ! p ! ( p + m ) ! ( p + m ) !
× [ 2 ρ w ( z ) ] 2 m L p m ( 2 ρ 2 w ( z ) 2 ) L p m ( 2 ρ 2 w ( z ) 2 ) cos { 2 ( p p ) Φ ( z ) } ,
D = p = 0 m = p < p m < m C m p * C m p p ! p ! ( p + m ) ! ( p + m ) !
× [ 2 ρ w ( z ) ] m + m L p m ( 2 ρ 2 w ( z ) 2 ) L p m ( 2 ρ 2 w ( z ) 2 )
× cos { ( m m ) φ [ 2 ( p p ) + m m ] Φ ( z ) } ,
E = p = 0 m = p > p m < m C m p * C m p p ! p ! ( p + m ) ! ( p + m ) !
× [ 2 ρ w ( z ) ] m + m L p m ( 2 ρ 2 w ( z ) 2 ) L p m ( 2 ρ 2 w ( z ) 2 )
× cos { ( m m ) φ [ 2 ( p p ) + m m ] Φ ( z ) } .
m = m + δ m ( δ m m ) ,
m = m + δ m ( δ m m ) .
cos Δ = cos { ( δ m δ m ) [ φ sgn ( m ) Φ ( z ) ] } .
Δ = ( δ m δ m ) φ ( δ m δ m ) Φ ( z )
= { ( δ m δ m ) [ φ Φ ( z ) ] , δ m > 0 , δ m > 0 , ( δ m δ m ) φ ( δ m + δ m ) Φ ( z ) , δ m > 0 , δ m < 0 , ( δ m δ m ) [ φ Φ ( z ) ] , δ m < 0 , δ m < 0 , ( δ m δ m ) φ + ( δ m + δ m ) Φ ( z ) , δ m < 0 , δ m > 0 .
x ( z ) = w ( z ) m 2 cos ( Φ G + Φ 0 ) , y ( z ) = w ( z ) m 2 sin ( Φ G + Φ 0 ) .
[ x y ] = [ cos Φ 0 sin Φ 0 sin Φ 0 cos Φ 0 ] [ x y ] ,
x ( z ) = w 0 m 2 , y ( z ) = w 0 m 2 z z R .

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