Abstract

We generalize the concept of diffraction free beams to parabolic scaling beams (PSBs), whose normalized intensity scales parabolically during propagation. These beams are nondiffracting in the circular parabolic coordinate systems, and all the diffraction free beams of Durnin’s type have counterparts as PSBs. Parabolic scaling Bessel beams with Gaussian apodization are investigated in detail, their nonparaxial extrapolations are derived, and experimental results agree well with theoretical predictions.

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  1. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef]
  3. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493 (2000).
    [CrossRef]
  4. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, Opt. Lett. 29, 44 (2004).
    [CrossRef]
  5. M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
    [CrossRef]
  6. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef]
  7. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  8. E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
    [CrossRef]
  9. N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
    [CrossRef]
  10. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, Opt. Express 20, 18955 (2012).
    [CrossRef]
  11. A. E. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
    [CrossRef]
  12. N. Gao, Y. Zhang, and C. Xie, Appl. Opt. 50, G142 (2011).
    [CrossRef]
  13. N. Gao, C. Xie, C. Li, C. Jin, and M. Liu, Appl. Phys. Lett. 98, 151106 (2011).
    [CrossRef]
  14. Generally, to define a scaling beam with scaling factor s(z), the s−1(z) factor must appear outside the scaling function ψ[r/s(z)], so as to make sure the conservation of total beam power, i.e., (∂/∂z)∫|E(r,z)|2d2r=0. Though we do not require the beam to be of finite power at first, this ansatz still proves to be useful. Thus in Figs. 2(a), 3(a), 4, and 5(a), for a better clarity, we neglect the s−1(z) factors, which is equivalent to normalizing the xy profile at each z plane independently.
  15. C. F. R. Caron and R. M. Potvliege, Opt. Commun. 164, 83 (1999).
    [CrossRef]
  16. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Lett. 33, 177 (2008).
    [CrossRef]
  17. Here we use the integral formulas ∫02πdφ exp(imφ)exp[iς cos(φ−θ)]=2πim exp(imθ)Jm(ς) for m∈Z, and ∫0∞dx exp(−βx)J2v(2ax)Jv(bx)=exp[−a2β/(β2+b2)]Jv[a2b/(β2+b2)]/(β2+b2)1/2 for   Re β>0,b>0 and Re v>−0.5. See I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, 7th ed., Series and Products (Academic, 2007).
  18. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
    [CrossRef]
  19. D. Deng, Y. Gao, J. Zhao, P. Zhang, and Z. Chen, Opt. Lett. 38, 3934 (2013).
    [CrossRef]
  20. M. V. Berry, J. Phys. A 27, L391 (1994).
    [CrossRef]
  21. K. Lombardo and M. A. Alonso, Am. J. Phys. 80, 82 (2012).
    [CrossRef]

2013 (1)

2012 (3)

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, Opt. Express 20, 18955 (2012).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

K. Lombardo and M. A. Alonso, Am. J. Phys. 80, 82 (2012).
[CrossRef]

2011 (3)

N. Gao, Y. Zhang, and C. Xie, Appl. Opt. 50, G142 (2011).
[CrossRef]

N. Gao, C. Xie, C. Li, C. Jin, and M. Liu, Appl. Phys. Lett. 98, 151106 (2011).
[CrossRef]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

2010 (1)

2008 (1)

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

2004 (1)

2000 (1)

1999 (1)

C. F. R. Caron and R. M. Potvliege, Opt. Commun. 164, 83 (1999).
[CrossRef]

1994 (1)

M. V. Berry, J. Phys. A 27, L391 (1994).
[CrossRef]

1991 (1)

A. E. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
[CrossRef]

1987 (2)

J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

1979 (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Alonso, M. A.

K. Lombardo and M. A. Alonso, Am. J. Phys. 80, 82 (2012).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Bandres, M. A.

Bekenstein, R.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Berry, M. V.

M. V. Berry, J. Phys. A 27, L391 (1994).
[CrossRef]

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Caron, C. F. R.

C. F. R. Caron and R. M. Potvliege, Opt. Commun. 164, 83 (1999).
[CrossRef]

Chávez-Cerda, S.

Chen, Z.

Christodoulides, D. N.

Deng, D.

Dennis, M. R.

Dholakia, K.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Durnin, J.

J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Efremidis, N. K.

Gao, N.

N. Gao, Y. Zhang, and C. Xie, Appl. Opt. 50, G142 (2011).
[CrossRef]

N. Gao, C. Xie, C. Li, C. Jin, and M. Liu, Appl. Phys. Lett. 98, 151106 (2011).
[CrossRef]

Gao, Y.

Gradshteyn, I. S.

Here we use the integral formulas ∫02πdφ exp(imφ)exp[iς cos(φ−θ)]=2πim exp(imθ)Jm(ς) for m∈Z, and ∫0∞dx exp(−βx)J2v(2ax)Jv(bx)=exp[−a2β/(β2+b2)]Jv[a2b/(β2+b2)]/(β2+b2)1/2 for   Re β>0,b>0 and Re v>−0.5. See I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, 7th ed., Series and Products (Academic, 2007).

Greenfield, E.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Gutiérrez-Vega, J. C.

Iturbe-Castillo, M. D.

Jin, C.

N. Gao, C. Xie, C. Li, C. Jin, and M. Liu, Appl. Phys. Lett. 98, 151106 (2011).
[CrossRef]

Kaminer, I.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Li, C.

N. Gao, C. Xie, C. Li, C. Jin, and M. Liu, Appl. Phys. Lett. 98, 151106 (2011).
[CrossRef]

Lindberg, J.

Liu, M.

N. Gao, C. Xie, C. Li, C. Jin, and M. Liu, Appl. Phys. Lett. 98, 151106 (2011).
[CrossRef]

Lombardo, K.

K. Lombardo and M. A. Alonso, Am. J. Phys. 80, 82 (2012).
[CrossRef]

Mazilu, M.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Mourka, A.

Nemirovsky, J.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Potvliege, R. M.

C. F. R. Caron and R. M. Potvliege, Opt. Commun. 164, 83 (1999).
[CrossRef]

Raz, O.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Ring, J. D.

Ryzhik, I. M.

Here we use the integral formulas ∫02πdφ exp(imφ)exp[iς cos(φ−θ)]=2πim exp(imθ)Jm(ς) for m∈Z, and ∫0∞dx exp(−βx)J2v(2ax)Jv(bx)=exp[−a2β/(β2+b2)]Jv[a2b/(β2+b2)]/(β2+b2)1/2 for   Re β>0,b>0 and Re v>−0.5. See I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, 7th ed., Series and Products (Academic, 2007).

Segev, M.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Siegman, A. E.

A. E. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
[CrossRef]

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Xie, C.

N. Gao, Y. Zhang, and C. Xie, Appl. Opt. 50, G142 (2011).
[CrossRef]

N. Gao, C. Xie, C. Li, C. Jin, and M. Liu, Appl. Phys. Lett. 98, 151106 (2011).
[CrossRef]

Zhang, P.

Zhang, Y.

Zhao, J.

Am. J. Phys. (2)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

K. Lombardo and M. A. Alonso, Am. J. Phys. 80, 82 (2012).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

N. Gao, C. Xie, C. Li, C. Jin, and M. Liu, Appl. Phys. Lett. 98, 151106 (2011).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. E. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
[CrossRef]

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

J. Phys. A (1)

M. V. Berry, J. Phys. A 27, L391 (1994).
[CrossRef]

Opt. Commun. (1)

C. F. R. Caron and R. M. Potvliege, Opt. Commun. 164, 83 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. Lett. (4)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Other (2)

Here we use the integral formulas ∫02πdφ exp(imφ)exp[iς cos(φ−θ)]=2πim exp(imθ)Jm(ς) for m∈Z, and ∫0∞dx exp(−βx)J2v(2ax)Jv(bx)=exp[−a2β/(β2+b2)]Jv[a2b/(β2+b2)]/(β2+b2)1/2 for   Re β>0,b>0 and Re v>−0.5. See I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, 7th ed., Series and Products (Academic, 2007).

Generally, to define a scaling beam with scaling factor s(z), the s−1(z) factor must appear outside the scaling function ψ[r/s(z)], so as to make sure the conservation of total beam power, i.e., (∂/∂z)∫|E(r,z)|2d2r=0. Though we do not require the beam to be of finite power at first, this ansatz still proves to be useful. Thus in Figs. 2(a), 3(a), 4, and 5(a), for a better clarity, we neglect the s−1(z) factors, which is equivalent to normalizing the xy profile at each z plane independently.

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Figures (5)

Fig. 1.
Fig. 1.

(a) Cylindrical wave with the focal line direction represented by ϕ / 2 . (b) Superposing these cylindrical waves with different focal line directions together would produce the PSBs.

Fig. 2.
Fig. 2.

Intensity profile of a PSBB with m = 0 in (a) the circular cylindrical coordinate system and (b) the circular parabolic coordinate system. z 0 is an arbitrarily chosen value, and r 0 = ( 9.6 z 0 / k ) 1 / 2 is the radial location of the first zero points of the wavefront in the z 0 plane. For convenience of exposition, we symmetrically extend the r / r 0 and ξ axis, and neglect the z 1 / 2 factor in Eq. (7) [14]. The same treatment is made for Figs. 3(a), 4, and 5(a).

Fig. 3.
Fig. 3.

(a) Intensity profile of a Gaussian-apodized PSBB with m = 0 and w 0 = 5 r 0 . (b) The radial profile at the focal plane.

Fig. 4.
Fig. 4.

Intensity profile of a Gaussian-apodized PSBB with m = 0 and w 0 = 5 r 0 in the (a) semi-far field and (b) far field, plotted using Eq. (10).

Fig. 5.
Fig. 5.

(a) Experimental setup. (b) Microscopic image of the mask. (c) Normalized x y intensity profile before (left) and 35 mm after (right) the mask. The image area is 4   mm × 4 mm . (d) Normalized x z intensity profile after the aperture.

Equations (15)

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2 i k U ( r , θ , z ) z = [ 1 r r ( r r ) + 1 r 2 2 θ 2 ] U ( r , θ , z ) ,
ξ = k r 2 4 z , φ = 2 θ , z ˜ = z ,
[ 1 ξ ξ ( ξ ξ ) + 1 ξ 2 2 φ 2 + 1 ] ψ = 0 ,
E ( r , θ , z ) = U ( r , θ , z ) exp ( i k z ) = z 1 / 2 ψ ( k r 2 4 z , 2 θ ) exp ( i k r 2 4 z ) exp ( i k z ) .
ψ ( ξ , φ ) = 1 2 π 0 2 π A ( ϕ ) exp [ i ξ cos ( ϕ φ ) ] d ϕ ,
E ( r , θ , z ) = 1 2 π z 1 / 2 exp ( i k z ) 0 2 π A ( ϕ ) exp [ i k r 2 2 z cos 2 ( θ ϕ 2 ) ] d ϕ .
E ( r , θ , z ) = z 1 / 2 J m / 2 ( k r 2 4 z ) exp ( i k r 2 4 z ) exp ( i m θ ) exp ( i k z ) .
E ( r , θ , z ) 1 w ( z + z S ) J | m | / 2 [ μ r 2 w 2 ( z + z S ) ] · exp [ 1 + i ( μ 2 + 1 ) ( z + z S ) / z R w 2 ( z + z S ) r 2 ] exp ( i m θ ) exp [ i k ( z + z S ) ] ,
U ( r , θ , z 0 ) = z 0 1 / 2 J m / 2 ( k r 2 4 z 0 ) exp ( i k r 2 4 z 0 ) exp ( r 2 w 0 2 ) exp ( i m θ ) .
U ( r , θ , z ) = 1 i λ Δ z exp ( i k r 2 2 Δ z ) 0 ρ d ρ exp ( i k ρ 2 2 Δ z ) · 0 2 π d φ U ( ρ , φ , z 0 ) exp [ i k ρ r Δ z cos ( φ θ ) ] = ( α z ) 1 / 2 J m / 2 ( k r 2 4 α z ) exp ( i k r 2 4 β z ) exp ( r 2 γ w 0 2 ) exp ( i m θ ) ,
α = 1 s 2 z 0 z + i s ( 1 + z 0 z ) , β = 1 + s 2 [ z 0 / z + Δ z 2 / ( z 2 + s 2 z 0 z ) ] 1 + 2 s 2 [ z 0 / Δ z + Δ z / ( z + s 2 z 0 ) ] , γ = 2 [ ( z / z 0 + s 2 ) 2 + s 2 Δ z 2 / z 0 2 ] 1 + z 2 / z 0 2 + 2 s 2 .
| E ( r , θ , 0 ) | 2 | J m / 2 ( k r 2 4 z 0 ( s 2 + i s ) ) | 2 exp ( r 2 w 0 2 1 s 2 1 + 2 s 2 1 + s 2 ) .
E ( r , θ , z ) = z 1 / 2 J m / 2 ( k r 2 4 z ) exp ( i k r 2 4 z ) exp ( r 2 2 w 0 2 ) exp ( i m θ ) exp ( i k z ) ,
| E ( r , θ , z ) | 2 | J m / 2 ( i k 2 w 0 2 r 2 8 z 2 ) | 2 exp ( k 2 w 0 2 r 2 4 z 2 ) ,
E ( r , θ , z ) = 1 2 π 0 2 π J 0 { k [ r 2 cos 2 ( θ ϕ 2 ) + ( z i ε ) 2 ] 1 / 2 } A ( ϕ ) d ϕ .

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