Abstract

We demonstrate that an optical vortex with fractional topological charge can be approximated as a linear combination of the two adjacent integer topological charges. This theory is experimentally verified utilizing a vortex grating spectrum analyzer, i.e., a computer-generated hologram with a charge 1 vortex grating. By extracting the intensity value of the center portion of the target diffraction orders, both the number and the sign of the topological charge of any input vortex can be identified and analyzed.

© 2010 Optical Society of America

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  1. P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. R. Soc. London Ser. A  133, 60–72 (1931).
    [CrossRef]
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    [CrossRef]
  3. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.  75, 826–829 (1995).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. J. Lin, X. C. Yuan, S. H. Tao, and R. E. Burge, “Multiplexing free-space optical signals using superimposed collinear orbital angular momentum states,” Appl. Opt.  46, 4680–4685 (2007).
    [CrossRef] [PubMed]
  6. K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.  89, 251114–251113 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  14. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett.  94, 233902 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  16. 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, S166–S169 (2004).
    [CrossRef]
  17. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.  6, 71–78 (2004).
    [CrossRef]
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2009 (2)

2007 (1)

2006 (1)

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.  89, 251114–251113 (2006).
[CrossRef]

2005 (3)

2004 (4)

K. Crabtree, J. A. Davis, and I. Moreno, “Image processing with vortex-producing lenses,” Appl. Opt.  43, 1360–1367 (2004).
[CrossRef] [PubMed]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt.  6, 259–268 (2004).
[CrossRef]

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, S166–S169 (2004).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.  6, 71–78 (2004).
[CrossRef]

2000 (1)

1999 (1)

1996 (1)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.  75, 826–829 (1995).
[CrossRef] [PubMed]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.  112, 321–327 (1994).
[CrossRef]

1992 (2)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett.  17, 221–223 (1992).
[CrossRef] [PubMed]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt.  39, 1147–1154 (1992).
[CrossRef]

1989 (1)

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

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A  336, 165–190 (1974).
[CrossRef]

1931 (1)

P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. R. Soc. London Ser. A  133, 60–72 (1931).
[CrossRef]

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, S166–S169 (2004).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.  112, 321–327 (1994).
[CrossRef]

Bernet, S.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett.  94, 233902 (2005).
[CrossRef] [PubMed]

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express  13, 689–694 (2005).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt.  6, 259–268 (2004).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A  336, 165–190 (1974).
[CrossRef]

Bu, J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.  89, 251114–251113 (2006).
[CrossRef]

Burge, R. E.

J. Lin, X. C. Yuan, S. H. Tao, and R. E. Burge, “Multiplexing free-space optical signals using superimposed collinear orbital angular momentum states,” Appl. Opt.  46, 4680–4685 (2007).
[CrossRef] [PubMed]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.  89, 251114–251113 (2006).
[CrossRef]

Campos, J.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.  112, 321–327 (1994).
[CrossRef]

Cottrell, D. M.

Coullet, P.

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

Crabtree, K.

Davis, J. A.

Dirac, P. A. M.

P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. R. Soc. London Ser. A  133, 60–72 (1931).
[CrossRef]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.  75, 826–829 (1995).
[CrossRef] [PubMed]

Fürhapter, S.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express  13, 689–694 (2005).
[CrossRef] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett.  94, 233902 (2005).
[CrossRef] [PubMed]

Gahagan, K. T.

Gil, L.

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

Haist, T.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.  75, 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.  75, 826–829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett.  17, 221–223 (1992).
[CrossRef] [PubMed]

Jesacher, A.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express  13, 689–694 (2005).
[CrossRef] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett.  94, 233902 (2005).
[CrossRef] [PubMed]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt.  39, 1147–1154 (1992).
[CrossRef]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt.  39, 1147–1154 (1992).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.  112, 321–327 (1994).
[CrossRef]

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.  6, 71–78 (2004).
[CrossRef]

Lin, J.

Low, D. K. Y.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.  89, 251114–251113 (2006).
[CrossRef]

McDuff, R.

McNamara, D. E.

Mitry, M. J.

Moh, K. J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.  89, 251114–251113 (2006).
[CrossRef]

Moreno, I.

Niu, H.

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A  336, 165–190 (1974).
[CrossRef]

Osten, W.

Padgett, M. J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.  6, 71–78 (2004).
[CrossRef]

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, S166–S169 (2004).
[CrossRef]

Pascoguin, B. M. L.

Pedrini, G.

Peng, X.

Reicherter, M.

Ritsch-Marte, M.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express  13, 689–694 (2005).
[CrossRef] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett.  94, 233902 (2005).
[CrossRef] [PubMed]

Rocca, F.

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

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.  75, 826–829 (1995).
[CrossRef] [PubMed]

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt.  39, 1147–1154 (1992).
[CrossRef]

Situ, G.

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, S166–S169 (2004).
[CrossRef]

Smith, C. P.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt.  39, 1147–1154 (1992).
[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, S166–S169 (2004).
[CrossRef]

Swartzlander, , G. A.

Tao, S.

Tao, S. H.

Tiziani, H. J.

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt.  39, 1147–1154 (1992).
[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, S166–S169 (2004).
[CrossRef]

Wagemann, E. U.

White, A. G.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.  112, 321–327 (1994).
[CrossRef]

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.  6, 71–78 (2004).
[CrossRef]

Yuan, X. C.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.  89, 251114–251113 (2006).
[CrossRef]

J. Mod. Opt. (1)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt.  39, 1147–1154 (1992).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt.  6, 259–268 (2004).
[CrossRef]

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, S166–S169 (2004).
[CrossRef]

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

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys.  6, 71–78 (2004).
[CrossRef]

Opt. Commun. (2)

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

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.  112, 321–327 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (2)

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett.  94, 233902 (2005).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.  75, 826–829 (1995).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (2)

P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. R. Soc. London Ser. A  133, 60–72 (1931).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A  336, 165–190 (1974).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Calculated intensity components of a vortex beam with fractional charge 0 q 1 : (a)  I 0 versus q in the range [ 1 , + 1 ] , (b)  I 0 and I 1 versus q; (c)  I 0 + I 1 versus q.

Fig. 2
Fig. 2

Intensity of the Fourier spectrum components of fractional vortices with (a)  q = 0.25 , (b)  q = 0.5 , (c)  q = 0.75 , (d)  q = 1.25 , (e)  q = 1.5 , and (f)  q = 1.75 .

Fig. 3
Fig. 3

(a), (b) Simulated phase distribution of the spiral phase plates and intensity patterns at the focal plane produced by optical vortices with topological charges q ranging from 0 to + 1 ; (c)  rotation of nonuniform intensity distribution of optical vortices with q = 0.5 by rotating the spiral phase plate by α = 0 ° , 45 ° , and 90 ° . Here the gray scale of the spiral phase plates indicates a 0 (darkest) to 2 π (brightest) phase change; in the intensity distributions, white denotes the highest intensity.

Fig. 4
Fig. 4

Optical configuration of the grating vortex spectrum analyzer experimental setup. A beam splitter (BS) is used to separate the input and reflected beams.

Fig. 5
Fig. 5

(a) Experimental results generated by the detection grating using input vortices with topological charges from 1 to + 1 (listed at right); the diffraction patterns shown are at orders 1 , 0, and + 1 (listed at bottom). (b) Intensity extracted from the center pixel at diffraction order 0, experimental value ( I e ), and simulated value ( I t ) versus input topological charges.

Fig. 6
Fig. 6

(a) Experimental results generated by the vortex grating spectrum analyzer using input vortices with topological charges from 1 to 3 (listed at right); the diffraction patterns shown are at orders + 1 , + 2 , and + 3 (listed at bottom). (b) Intensity extracted from the center pixel at diffraction order 2, experimental value ( I e ), and simulated value ( I t ) versus input topological charges.

Equations (15)

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

v q ( r , ϕ ) = exp [ i ( q ϕ + α ) ] = n = + c n exp ( i n ϕ ) exp ( i α ) .
c n = 1 2 π π π e i q ϕ e i n ϕ d ϕ = 1 2 π π π e i ( q n ) ϕ d ϕ = sin [ π ( q n ) ] π [ q n ] .
I n = | c n | 2 = | sin ( π ( q n ) ) π ( q n ) | 2 = sinc 2 ( q n ) .
v q ( r , ϕ ) = exp ( i q ϕ ) c N ( q ) exp ( i N ϕ ) + c N + 1 ( q ) exp [ i ( N + 1 ) ϕ ] ,
V q ( ρ , Φ ) = c N ( q ) V N ( ρ , Φ ) + c N + 1 ( q ) V N + 1 ( ρ , Φ ) ,
| V q ( ρ , Φ ) | 2 = | c 0 ( q ) V 0 ( ρ , Φ ) | 2 + | c 1 ( q ) V 1 ( ρ , Φ ) | 2 + ( c 0 * ( q ) c 1 ( q ) V 0 * ( ρ , Φ ) V 1 ( ρ , Φ ) + c 0 ( q ) c 1 * ( q ) V 0 ( ρ , Φ ) V 1 * ( ρ , Φ ) ) ,
g ( x , y ) = m = + g m exp ( i m ϕ ) exp ( i m γ x ) .
G ( ρ , Φ ) = m = + g m V m ( ρ , Φ ) δ ( p m γ ) .
W ( ρ , Φ ) = m = + [ g m V q ( ρ , Φ ) V m ( ρ , Φ ) ] δ ( p m γ ) = m = + E m ( ρ , Φ ) δ ( p m γ ) ,
exp ( i q ϕ ) g ( x , y ) { ( c N ( q ) exp ( i N ϕ ) + c N + 1 ( q ) exp [ i ( N + 1 ) ϕ ] ) } × { m = + g m exp ( i m ϕ ) exp ( i m γ x ) } .
exp ( i q ϕ ) g ( x , y ) g m = N { c N ( q ) + c N + 1 ( q ) exp [ i ϕ ] } exp ( i N γ x ) + g m = N 1 { c N ( q ) exp [ i ϕ ] + c N + 1 ( q ) } exp ( i ( N + 1 ) γ x ) .
W ( ρ , Φ ) E N ( ρ , Φ ) δ ( p + N γ ) + E N 1 ( ρ , Φ ) δ [ p + ( N + 1 ) γ ] ,
E N ( ρ , Φ ) g N [ c N ( q ) V 0 ( ρ , Φ ) + c N + 1 ( q ) V 1 ( ρ , Φ ) ] ;
E N 1 ( ρ , Φ ) g N 1 [ c N ( q ) V 1 ( ρ , Φ ) + c N + 1 ( q ) V 0 ( ρ , Φ ) ] .
| E N ( ρ , Φ ) | 2 = | g m = N | 2 { | c N ( q ) V 0 ( ρ , Φ ) | 2 + | c N + 1 ( q ) V 1 ( ρ , Φ ) | 2 + c N * ( q ) c N + 1 ( q ) V 0 * ( ρ , Φ ) V 1 ( ρ , Φ ) + c N ( q ) c N + 1 * ( q ) V 0 ( ρ , Φ ) V 1 * ( ρ , Φ ) } .

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