Abstract

A detailed analysis of the plane-wave diffraction by a finite-radius circular spiral phase plate (SPP) with integer and fractional topological charge and with variable transmission coefficients inside and outside of the plate edge is presented. We characterize the effect of varying the transmission coefficients and the parameters of the SPP on the propagated field. The vortex structure for integer and fractional phase step of the SPPs with and without phase apodization at the plate edge is also analyzed. The consideration of the interference between the light crossing the SPP and the light that undergoes no phase alteration at the aperture plane reveals new and interesting phenomena associated to this classical problem.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  19. J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt. 10, 015009 (2008).
    [CrossRef]

2008

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt. 10, 015009 (2008).
[CrossRef]

2007

2006

2005

2004

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

1999

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

1994

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

Q. Haider and L. C. Liu, “Fourier or Bessel transformations of highly oscillatory functions,” J. Phys. A 25, 6755-6760 (1992).
[CrossRef]

C. J. R. Sheppard and M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274-281 (1992).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985-990 (1992).
[CrossRef]

1987

1985

1964

1962

M. H. Sussman, “Fresnel diffraction with phase objects,” Am. J. Phys. 30, 44-48 (1962).
[CrossRef]

Almazov, A. A.

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985-990 (1992).
[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]

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]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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]

Condell, W. J.

Dholakia, K.

Dubra, A.

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

Elfstrom, H.

English, R. E.

Ferrari, J. A.

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

Fischer, P.

George, N.

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt. 10, 015009 (2008).
[CrossRef]

Haider, Q.

Q. Haider and L. C. Liu, “Fourier or Bessel transformations of highly oscillatory functions,” J. Phys. A 25, 6755-6760 (1992).
[CrossRef]

Hrynevych, M.

C. J. R. Sheppard and M. Hrynevych, “Structure of the axial intensity minima in the Fresnel diffraction on a circular opening and superluminous effects,” Opt. Commun. 271, 316-322 (2007).
[CrossRef]

C. J. R. Sheppard and M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274-281 (1992).
[CrossRef]

Khonina, S. N.

Kotlyar, V. V.

Kovalev, A. A.

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 (2004).
[CrossRef]

Leburn, C. G.

Liu, L. C.

Q. Haider and L. C. Liu, “Fourier or Bessel transformations of highly oscillatory functions,” J. Phys. A 25, 6755-6760 (1992).
[CrossRef]

López-Mariscal, C.

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt. 10, 015009 (2008).
[CrossRef]

Marchand, W.

Moiseev, O. Y.

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 (2004).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and M. Hrynevych, “Structure of the axial intensity minima in the Fresnel diffraction on a circular opening and superluminous effects,” Opt. Commun. 271, 316-322 (2007).
[CrossRef]

C. J. R. Sheppard and M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274-281 (1992).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Skelton, S. E.

Skidanov, R. V.

Soifer, V. A.

Soskin, M. S.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985-990 (1992).
[CrossRef]

Streuber, C. T.

Sussman, M. H.

M. H. Sussman, “Fresnel diffraction with phase objects,” Am. J. Phys. 30, 44-48 (1962).
[CrossRef]

Turunen, J.

Vasnetsov, M. V.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985-990 (1992).
[CrossRef]

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]

Wolf, E.

Wright, E. M.

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 (2004).
[CrossRef]

Am. J. Phys.

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

M. H. Sussman, “Fresnel diffraction with phase objects,” Am. J. Phys. 30, 44-48 (1962).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985-990 (1992).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

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. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt. 10, 015009 (2008).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

Q. Haider and L. C. Liu, “Fourier or Bessel transformations of highly oscillatory functions,” J. Phys. A 25, 6755-6760 (1992).
[CrossRef]

New J. Phys.

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

Opt. Commun.

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]

C. J. R. Sheppard and M. Hrynevych, “Structure of the axial intensity minima in the Fresnel diffraction on a circular opening and superluminous effects,” Opt. Commun. 271, 316-322 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Other

A. E. Siegman, Lasers (University Science Books, 1986).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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Figures (15)

Fig. 1
Fig. 1

Geometry of the SPP aperture function.

Fig. 2
Fig. 2

(a) Intensity and (b) phase of diffracted plane waves by uniform-phase disks of β = π 5 with A 1 = 1 and A 2 = 1 (solid curve), with A 1 = 0.8 and A 2 = 0.4 (dashed curve), and with A 1 = 0.4 and A 2 = 0.8 (dashed-dotted curve), observed at z L = 2 .

Fig. 3
Fig. 3

Radial phase distributions of the diffraction patterns at z L = 2 caused by disks with (a) phase shift β = π 6 , inner transmission coefficient A 1 = 1 , and outer transmission coefficients A 2 = 0.2 (solid curve), 0.5 (dashed curve), and 0.8 (dashed-dotted curve); (b) phase shift β = π 3 , outer transmission coefficient A 2 = 0.9 , and inner transmission coefficients A 1 = 0.2 (solid curve), 0.5 (dashed curve), and 0.8 (dashed-dotted curve).

Fig. 4
Fig. 4

Comparison between (a) intensity and (b) phase of ( A 1 + A 2 ) U 1 , 1 (solid curve) and U A 1 , A 2 + U A 2 , A 1 (dashed curve) for A 1 = 0.8 , A 2 = 0.4 and β = π 5 , observed at z L = 2 .

Fig. 5
Fig. 5

Intensity and phase distributions of the diffraction pattern caused by a n = 1 finite-radius SPP, observed on 3 a × 3 a windows at z L = ( a ) 1.0, (b) 2.5, and (c) 4.0.

Fig. 6
Fig. 6

Intensity and phase distributions of the diffraction pattern caused by a n = 3 finite-radius SPP, observed at z L = 1.0 for A 1 = 1 and A 2 = ( a ) 0, (b) 0.25, (c) 0.5, (d) 0.75, and (e) 1.

Fig. 7
Fig. 7

Location of vortices in the phase distribution of the diffraction pattern caused by a transparent finite-radius SPP, observed at z L = 1.0 for n = ( a ) 1 and (b) 2. Positive and negative vortices are represented by white and black small circles, respectively.

Fig. 8
Fig. 8

Intensity and phase distribution of the diffraction pattern caused by a transparent n = 1 finite-radius SPP observed at z L = ( a ) 1.0, (b) 15, (c) 25, and (d) 110 for A 2 = 1 . Positive and negative vortices are represented by white and black small circles, respectively. A zoom of the central region of the phase in (a) is included in Fig. 7a.

Fig. 9
Fig. 9

Intensity and phase distribution of the diffraction pattern caused by a n = 2 finite-radius SPP observed at z L = 1.0 , for apodization distances of ϵ = ( a ) 0, (b) 0.25 a , and (c) 0.5 a .

Fig. 10
Fig. 10

Phase distribution of the diffraction pattern caused by a n = 2 finite-radius SPP observed at z L = 1.0 , for apodization distances of ϵ = ( a ) 0, (b) 0.25 a , and (c) 0.5 a .

Fig. 11
Fig. 11

Propagation distance before the last vortex-pair annihilation event for different topological charges n and with apodization distances ϵ = 0 (squares), 0.2 a (circles), and 0.5 a (triangles).

Fig. 12
Fig. 12

Intensity and phase of the diffraction pattern observed at z L = 1.0 , caused by a finite-radius SPP with fractional charges α = ( a ) 1.25, (b) 1.5, and (c) 1.75.

Fig. 13
Fig. 13

Phase distribution of the diffraction pattern observed at z L = 2.0 , caused by a finite-radius SPP with fractional charges α = ( a ) 2.3 and (b) 2.7.

Fig. 14
Fig. 14

Phase distribution of the diffraction pattern observed at z L = 2.0 , caused by a finite-radius SPP with fractional topological charge α = 2.5 . We can see the chain of unit strength vortices on the + x axis.

Fig. 15
Fig. 15

Intensity and phase of the diffraction pattern observed at z L = 1.0 , caused by a finite-radius SPP with apodization distance ϵ = 0.5 a and fractional charges α = ( a ) 1.25, (b) 1.5, and (c) 1.75.

Equations (22)

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U ( r 0 , φ 0 ) = { A 1 exp ( i α φ 0 + i β ) r 0 a A 2 r 0 > a } ,
U ( r ) = i k 2 π z exp ( i k r 2 2 z ) 0 r 0 d r 0 0 2 π d φ 0 U ( r 0 , φ 0 ) × exp { i k 2 z [ r 0 2 2 r r 0 cos ( φ 0 φ ) ] } ,
z L = 2 a ( k a π ) 1 3 ,
U far field ( r ) = k i 2 π z exp ( i k r 2 2 z ) 0 r 0 d r 0 0 2 π d φ 0 U ( r 0 , φ 0 ) × exp [ i k r r 0 z cos ( φ 0 φ ) ] .
U far field ( r ) = k a 2 i z exp ( i k r 2 2 z ) { A 1 ( i ) n exp ( i n φ ) ( n + 2 ) Γ ( n + 1 ) ( ρ 2 ) n F 2 1 ( 1 + n 2 , 2 + n 2 , 1 + n ; ρ 2 4 ) + A 2 [ δ ( ρ ) J 1 ( ρ ) ρ ] } ,
F 2 1 ( α , β , γ ; x ) = m = 0 ( α ) m ( β ) m ( γ ) m m ! x m ,
0 2 π d φ 0 exp [ i k r z r 0 cos ( φ 0 φ ) + i n φ 0 ] = i n 2 π exp ( i n φ ) J n ( k r z r 0 ) ,
U n ( r , φ ; z ) = i k a 2 z exp ( i k r 2 2 z ) × [ A 1 ( i ) n exp ( i n φ + i β ) G n ( r , z ) A 2 G 0 ( r , z ) ] + A 2 ,
G n ( r , z ) = 0 1 w exp ( i k a 2 2 z w 2 ) J n ( k a r z w ) d w ,
G n ( r , z ) = 1 n ! ( i k a r 2 z ) n m = 0 1 ( 2 m + n + 2 ) m ! × ( i k a 2 2 z ) m F 2 1 [ 2 + 2 m + n 2 , 4 + 2 m + n 2 , 1 + n ; ( k a r 2 z ) 2 ] .
U A 1 , A 2 + U 1 A 1 , 1 A 2 = U 1 , 1 .
U A 1 , A 2 + U A 2 , A 1 = ( A 1 + A 2 ) U 1 , 1 .
S n = 1 2 π C d φ Φ ( r ) φ ,
U ( r 0 , φ 0 ) = exp [ i n φ 0 f ( r 0 ) ] .
f ( r 0 ) = { 1 r 0 a cos [ π 2 ϵ ( r 0 a ) ] a < r 0 a + ϵ , 0 r 0 > a + ϵ }
U ( r 0 , φ 0 ) = { A 1 exp ( i α φ 0 ) r 0 a A 2 r 0 > a } ,
exp ( i α φ 0 ) = ( 1 ) α sin ( α π ) π m = exp ( i m φ 0 ) α m .
U ( r , φ ; z ) = A 2 i k a 2 z exp ( i k 2 z r 2 ) × m = C m exp ( i m φ ) G m ( r , z ) ,
C m = A 1 ( 1 ) α sin ( α π ) π ( i ) m α m A 2 δ m , 0 ,
lim α n [ ( 1 ) α sin ( α π ) π ( α m ) ] = { 1 for m = n 0 for m n } ,
C m α = n = A 1 ( i ) m δ m , n A 2 δ m , 0 .
U ( r 0 , φ 0 ) = exp [ i α φ 0 f ( r 0 ) ] ,

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