Abstract

The general analytical formula for the propagation of the power-exponent-phase vortex (PEPV) beam through a paraxial ABCD optical system is derived. On that basis the evolution of the intensity distribution of such a beam in free space and the focusing system is investigated. In addition, some experiments are carried out, which verify the theoretical predictions. Both of the theoretical and experimental results show that the beam’s profile can be modulated by the topological charge and the power order of the PEPV beam.

© 2016 Optical Society of America

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References

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    [Crossref]
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  4. C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
    [Crossref]
  5. A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
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  23. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17(12), 9818–9827 (2009).
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2015 (1)

Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54(11), 111306 (2015).
[Crossref]

2014 (2)

P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014).
[Crossref] [PubMed]

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

2013 (1)

2011 (1)

2009 (2)

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17(12), 9818–9827 (2009).
[Crossref] [PubMed]

2008 (2)

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008).
[Crossref] [PubMed]

2005 (2)

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005).
[Crossref] [PubMed]

S. Tao, X. C. Yuan, J. Lin, X. Peng, and H. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005).
[Crossref] [PubMed]

2004 (2)

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

2003 (1)

2001 (1)

2000 (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

1998 (1)

1996 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1989 (1)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

1970 (1)

Alexander, T. J.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Anzolin, G.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Arlt, J.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

Barbieri, C.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Barnett, S. M.

Basistiy, I. V.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Baumann, S. M.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bekshaev, A. Y.

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Bianchini, A.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Cheng, S.

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

Collins, S. A.

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Curtis, J. E.

Dholakia, K.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

Dowling, J. P.

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005).
[Crossref] [PubMed]

Flossmann, F.

Franke-Arnold, S.

Gahagan, K. T.

Galvez, E. J.

Gan, X.

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Götte, J. B.

Grier, D. G.

Hitomi, T.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

Kalb, D. M.

Kapale, K. T.

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005).
[Crossref] [PubMed]

Karamoch, A. I.

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Kivshar, Y. S.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

Leanhardt, A. E.

Li, H.

Li, P.

Lin, J.

Liu, S.

Luo, W.

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

MacMillan, L. H.

Mari, E.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Molina-Terriza, G.

Niu, H.

O’Holleran, K.

Padgett, M. J.

Pas ko, V. A.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Peng, T.

Peng, X.

Prasciolu, M.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Preece, D.

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Romanato, F.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Rozas, D.

Rumala, Y. S.

Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54(11), 111306 (2015).
[Crossref]

Y. S. Rumala and A. E. Leanhardt, “Multiple-beam interference in a spiral phase plate,” J. Opt. Soc. Am. B 30(3), 615–621 (2013).
[Crossref]

Sacks, Z. S.

Slyusar, V. V.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Soskin, M. S.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Sponselli, A.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Sukhorukov, A. A.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

Swartzlander, G. A.

Tamburini, F.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Tao, S.

Torner, L.

Umbriaco, G.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Vasnetsov, M. V.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Villoresi, P.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Wright, E. M.

Xie, G.

Yin, J.

Yuan, X. C.

Yuan, Z.

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

Zhao, J.

Acta Opt. Sin. (1)

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

Appl. Phys. B (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

Earth Moon Planets (1)

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Opt. Eng. (1)

Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54(11), 111306 (2015).
[Crossref]

Opt. Express (4)

Opt. Lett. (4)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005).
[Crossref] [PubMed]

Other (3)

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, Beijing, 2000).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 1980).

J. Alda, “Laser and Gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering (Dekker, 2003), 999–1013.

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Figures (8)

Fig. 1
Fig. 1 Experimental setup for generating a PEPV beam and measuring its intensity properties in free space and focusing system. BE, beam expander; P, polarizer; SLM, spatial light modulator; L, thin lens; PC1, PC2, personal computers.
Fig. 2
Fig. 2 The holographs for the generation of the PEPV beam with different power orders n and topological charges m . (a) n = 2 , m = 1 ; (b) n = 2 , m = 2 ; (c) n = 2 , m = 6 ; (d) n = 4 , m = 2 .
Fig. 3
Fig. 3 Theoretical (a-d) and experimental (e-h) results of the intensity distributions of the PEPV beam with m = 1 at the propagation distance z = 0.7 m for different values of power order n . (a, e) n = 2 ; (b, f) n = 4 ; (c, g) n = 6 ; (d, h) n = 10 .
Fig. 4
Fig. 4 Theoretical results of the optical intensity distribution of the PEPV beam with large power order. (a) n = 20 ; (b) n = 50 ; (c) n = 100 ; (d) n = 200 . The other parameters are the same as in Fig. 3.
Fig. 5
Fig. 5 Theoretical (a-d) and experimental (e-f) results for the optical intensity distribution and the phase contour (i-l) of the PEPV beam with different topological charges when the power order n = 2 and the propagation distance z = 0.7 m . (a, e, i) m = 2 ; (b, f, j) m = 4 ; (c, g, k) m = 6 ; (d, h, l) m = 10 . The locations of singularities are labeled by white circles.
Fig. 6
Fig. 6 The theoretically (a-c) and experimentally (d-f) focused intensity distributions of the PEPV beam. (a, d) m = 2 , n = 3 ; (b, e) m = 3 , n = 4 ; (c, f) m = 4 , n = 3 .
Fig. 7
Fig. 7 The relationship between estimated value of ln | M l | and l . Because of the error of Eq. (17), the curve becomes nearly flat when | l | > 30 .
Fig. 8
Fig. 8 Theoretical (a-d) results for the optical intensity distribution and the phase contour (e-h) of the PEPV beam at different propagation distances z with m = 4 , n = 3 . (a, e) z = 1 m ; (b, f) z = 2 m ; (c, g) z = 4 m ; (d, h) z = 8 m . The locations of singularities are labeled by white circles.

Equations (28)

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E ( 0 ) ( r , φ ) = A 0 exp ( r 2 w 2 ) exp [ i 2 m π ( φ 2 π ) n ] ,
E ( ρ , θ , z ) = i k 2 π B 0 0 2 π E ( 0 ) ( r , φ ) exp { i k 2 B [ A r 2 2 r ρ cos ( θ φ ) + D ρ 2 ] } r d r d φ ,
E ( ρ , θ , z ) = i k 2 π B exp ( i k D ρ 2 2 B ) 0 0 2 π exp ( r 2 w 2 ) exp ( i k A r 2 2 B ) × exp [ i k ρ r B cos ( θ φ ) ] exp [ i 2 m π ( φ 2 π ) n ] r d r d φ .
exp [ i k r ρ B cos ( φ θ ) ] = h = i h J h ( k r ρ B ) exp [ i h ( φ θ ) ] , J l ( x ) = ( 1 ) l J l ( x ) , J l ( x ) = p = 0 ( 1 ) p 1 p ! Γ ( l + p + 1 ) ( x 2 ) l + 2 p , Γ ( x ) = 0 exp ( t ) t x 1 d t , exp ( s x n ) = j = 0 s j x n j j ! ,
E ( ρ , θ , z ) = ( i 2 λ B R ) exp ( i k D 2 B ρ 2 ) { exp ( k 2 ρ 2 4 B 2 R ) M 0 + l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] } ,
M l = 0 2 π exp { i [ 2 π m ( φ 2 π ) n + l φ ] } d φ = { j = 0 h = 0 i j + h m j l h ( 2 π ) j + h + 1 j ! h ! ( n j + h + 1 ) , l 0 , j = 0 i j m j ( 2 π ) j + 1 j ! ( n j + 1 ) , l = 0.
I ( ρ , θ , z ) = E * ( ρ , θ , z ) E ( ρ , θ , z ) .
( A B C D ) = ( 1 z 0 1 ) .
E ( ρ , θ , z ) = ( i E 0 2 λ z R ) exp ( i k 2 z ρ 2 ) { exp ( k 2 ρ 2 4 z 2 R ) M 0 + l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 z 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] } ,
( A B C D ) = ( 1 f 0 1 ) ( 1 0 1 / f 1 ) ( 1 s 0 1 ) = ( 0 f 1 / f 1 s / f ) .
E ( ρ , θ , z ) = ( i E 0 w 2 2 λ f ) exp [ i k ( f s ) 2 f 2 ρ 2 ] { exp [ ( k ρ w 2 f ) 2 ] M 0 + l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( i k ρ w 2 f ) 2 p + l ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] } .
l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] .
Γ ( p + l / 2 + 1 ) Γ ( p + 1 ) ~ Γ ( p + l + 1 ) Γ ( p + l / 2 + 1 ) , Γ ( x ) ~ 2 π x ( x e ) x ,
p = 0 S p = p = 0 p max S p + p = p max + 1 S p ,
Δ p = p = p max + 1 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ~ p = p max + 1 1 Γ ( p + l / 2 + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ~ p = p max + 1 1 2 π ( p + l / 2 + 1 ) [ k 2 ρ 2 e 4 B 2 R ( p + l / 2 + 1 ) ] p + l / 2 .
Δ p ~ p = p max + 1 1 2 π ( p max ) ( k 2 ρ 2 e 4 B 2 R p max ) p ~ 10 10 .
M l = 0 2 π exp { i [ 2 π m ( φ 2 π ) n + l φ ] } d φ q = 0 s exp { 2 π i [ m ( q s ) n + l q s ] } 2 π s ,
j = 0 h = 0 i j + h m j l h ( 2 π ) j + h + 1 j ! h ! ( n j + h + 1 ) = j = 0 h = 0 M l , j , h ,
j = 0 h = 0 M l , j , h = j = 0 j max h = 0 h max M l , j , h + j = j max + 1 h = 0 h max M l , j , h + j = 0 j max h = h max + 1 M l , j , h + j = j max + 1 h = h max + 1 M l , j , h ,
Δ M = j = j max + 1 h = 0 h max M l , j , h + j = 0 j max h = h max + 1 M l , j , h + j = j max + 1 h = h max + 1 M l , j , h .
M l , j , h = i j + h m j l h ( 2 π ) j + h + 1 j ! h ! ( n j + h + 1 ) ~ e 2 ( i 2 m π e j + 1 ) j ( i 2 l π e h + 1 ) h ( n j + h + 1 ) ( j + 1 ) ( h + 1 ) ( j + 1 ) ( h + 1 ) .
M l , j , h = 2 π ( 2 i m π ) j ( 2 i l π ) h j ! h ! ( n j max + h max + 1 ) ,
j = j max + 1 h = h max + 1 2 π ( 2 i m π ) j ( 2 i l π ) h j ! h ! ( n j max + h max + 1 ) ~ 2 π ( 2 × 10 -12 8 × 10 -12 i ) ( 4 × 10 15 8 × 10 15 ) = 5 × 10 25 10 25 i .
M l , j , h ~ 2 π ( 2 i m π ) j j ! ( 2 i l π ) h ( n j max + 1 ) h ! ,
j = j max + 1 h = 0 h max M l , j , h ~ h = 0 h max 2 π ( 2 i l π ) h ( n j max + 1 ) h ! [ exp ( 2 i m π ) j = 0 j max ( 2 i m π ) j j ! ] ~ 2 π ( n j max + 1 ) × ( 2 × 10 -12 8 × 10 -12 i ) = 8 × 10 -14 3 × 10 -13 i .
M l , j , h ~ 2 π ( 2 i m π ) j ( 2 i l π ) h ( h max + 1 ) j ! h ! ,
j = 0 j max h = h max + 1 M l , j , h ~ j = 0 j max h = h max + 1 2 π ( 2 i m π ) j ( 2 i l π ) h ( h max + 1 ) j ! h ! = 8 × 10 17 2 × 10 16 i .
Δ l ~ l = l max + 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] ~ l = l max + 1 ( k 2 ρ 2 4 B 2 R ) l / 2 [ p = 0 1 p ! ( k 2 ρ 2 4 B 2 R ) p ] × | M l max | ~ exp ( k 2 ρ 2 4 B 2 R ) | M l max | ( k 2 ρ 2 4 B 2 R ) ( l max + 1 ) / 2 1 ( k 2 ρ 2 4 B 2 R ) 1 / 2 ~ 2 × 10 3 .

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