OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 6899–6904
« Show journal navigation

Quantitative description of the self-healing ability of a beam

Xiuxiang Chu and Wei Wen  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 6899-6904 (2014)
http://dx.doi.org/10.1364/OE.22.006899


View Full Text Article

Acrobat PDF (1444 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Quantitative description of the self-healing ability of a beam is very important for studying or comparing the self-healing ability of different beams. As describing the similarity by using the angle of two infinite-dimensional complex vectors in Hilbert space, the angle of two intensity profiles is proposed to quantitatively describe the self-healing ability of different beams. As a special case, quantitative description of the self-healing ability of a Bessel-Gaussian beam is studied. Results show that the angle of two intensity profiles can be used to describe the self-healing ability of arbitrary beams as the reconstruction distance for quantitatively describing the self-healing ability of Bessel beam. It offers a new method for studying or comparing the self-healing ability of different beams.

© 2014 Optical Society of America

1. Introduction

Nondiffracting waves which have many interesting properties and potential applications have attracted more attentions [1

1. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves: Theory and. Applications (John Wiley, 2008).

6

6. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]

]. Self-healing ability of beams is one of the most interesting properties which make them very useful in many areas. In recent years, much works concerning the self-healing properties of nondiffracting waves has been carried out [7

7. S. Vyas, Y. Kozawa, and S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A 28(5), 837–843 (2011). [CrossRef] [PubMed]

12

12. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]

]. For example, S. Vyas et al. have studied the self-healing of tightly focused scalar and vector Bessel–Gauss beams at the focal plane [7

7. S. Vyas, Y. Kozawa, and S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A 28(5), 837–843 (2011). [CrossRef] [PubMed]

]; J. D. Ring et al. shows that Pearcey beams have the self-healing property [8

8. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012). [CrossRef] [PubMed]

]; R. Cao et al. have found that optical pillar array also has the self-healing ability [9

9. R. Cao, Y. Hua, C. Min, S. Zhu, and X. C. Yuan, “Self-healing optical pillar array,” Opt. Lett. 37(17), 3540–3542 (2012). [CrossRef] [PubMed]

]; M. Anguiano-Morales et al. have focused his study on the self-healing property of a caustic optical beam [10

10. M. Anguiano-Morales, A. Martínez, M. D. Iturbe-Castillo, S. Chávez-Cerda, and N. Alcalá-Ochoa, “Self-healing property of a caustic optical beam,” Appl. Opt. 46(34), 8284–8290 (2007). [CrossRef] [PubMed]

]; Pravin Vaity et al. have analyzed the self-healing property of optical ring lattice [11

11. P. Vaity and R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett. 36(15), 2994–2996 (2011). [CrossRef] [PubMed]

]; J. Broky et al. have investigated the self-healing properties of optical Airy beams [12

12. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]

].

Besides the properties of self-healing ability, the mechanism of many nondiffracting waves have been investigated [13

13. I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]

16

16. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010). [CrossRef] [PubMed]

]. For example, the phenomenon of reconstruction for a Bessel beam was explained by considering the dynamics of the conical waves [13

13. I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]

, 14

14. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46(7), 078001 (2007). [CrossRef]

]; the explanations of the self-healing of an Airy beam are given by using the method of uniform geometrical optics and catastrophe optics [15

15. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef] [PubMed]

, 16

16. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010). [CrossRef] [PubMed]

]. From the explanations we can see that the mechanism of the self-healing ability for different beams is different. In addition, which self-healing ability of a beam is stronger in practice is also very important. However, to the best of our knowledge, the comparison of the self-healing ability between different beams has not been made until now.

Self-healing ability of a beam describes that a beam shape partially blocked by an opaque obstacle can be reconstructed during propagation. It means that the beam shape partially blocked by an opaque obstacle will become more and more similar to that of the beam without obstruction. Similarity can be used to estimate the difference between beams, for example, Martinez-Herrero et al. in Ref. [17

17. R. Martínez-Herrero, I. Juvells, and A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38(12), 2065–2067 (2013). [CrossRef] [PubMed]

] have studied the difference between the closest field and the target function by using the similarity. Therefore, we can use similarity to describe the self-healing process. In present paper quantitative description of the self-healing ability is proposed. As an example, quantitative description of the self-healing ability for Bessel-Gaussian beam is investigated.

2. Quantitative description of the self-healing ability

To describe the self-healing procession, the optical fields of a beam with and without obstacle are denoted by E(x,y,z) and E(x,y,z)in the following. In Hilbert space E(x,y,z) and E(x,y,z) can be regarded as two infinite-dimensional complex vectors. The angle of two infinite-dimensional vectors can be defined as [18

18. J. James, Mathematics Dictionary Mathematics Dictionary, 5th ed. (Springer, 1992).

]
cos(EE)=(E,E)EE
(1)
where (E,E)presents the inner product of two infinite-dimensional complex vectors and is expressed as
(E,E)=E(x,y,z)E¯(x,y,z)dxdy.
(2)
Here E and E are the norm for the vectors which are the length of the vector in Hilbert space, and are defined as
f(x,y,z)=f(x,y,z)f¯(x,y,z)dxdy.
(3)
In present paper the over-bar refers to the conjugate. By the Schwarz inequality
(E,E)EE
(4)
from Eq. (1) we can see that cos(EE)is always in the range −1 to + 1. It is obvious that cos(EE)=±1 if E(x,y,z)=±E(x,y,z). If we assume that G(u,v,z) and G(u,v,z) are the two-dimensional Fourier transform of E(x,y,z)and E(x,y,z), by using the Plancherel theorem
E(x,y,z)E¯(x,y,z)dxdy=G(u,v,z)G¯(u,v,z)dudv
(5)
and Parseval's theorem
E(x,y,z)E¯(x,y,z)dxdy=G(u,v,z)G¯(u,v,z)dudv.
(6)
Equation (1) can be rewritten as
cos(EE)=cos(GG)=(G,G)GG
(7)
where (u,v) are the coordinates in the spatial frequency domain. Because G(u,v,z) and G(u,v,z)in free space can be represented by
(G(u,v,z)G(u,v,z))=(G(u,v,0)G(u,v,0))M(u,v)
(8)
where
M(u,v)=exp(izk2u2v2)
(9)
is the transfer function of the angular spectrum in free space. From Eqs. (7)(9), one can find that cos(GG)is independent on the propagation distance. Namely, cos(GG)in Eq. (7) is unchanged during propagation and cannot be used to define the self-healing ability. However, study shows that |E||E|depend on the propagation distance. By using the inequality
|EE||E||E|,
(10)
we use the equation
S=cos(|E||E|)=(|E|,|E|)EE
(11)
as the quantitative description of the self-healing ability. S is named as similarity of two functions in general.

3. Special case: self-healing ability of a Bessel-Gaussian beam

Self-healing of a Bessel-Gaussian beam has been studied based on geometrical optics [13

13. I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]

, 14

14. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46(7), 078001 (2007). [CrossRef]

]. By considering the dynamics of the conical waves, there is a minimum distance (reconstruction distance) behind an obstacle of radius R before reconstruction occurs [13

13. I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]

, 14

14. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46(7), 078001 (2007). [CrossRef]

]
dmin=R/tanα
(12)
where α is the axicon angle (see Fig. 1).
Fig. 1 Generation of a Bessel-Gaussian beam.
For comparison with the existing results in a special case, as an example, quantitative description of the self-healing ability of a Bessel-Gaussian beam is studied in the following. As shown in Fig. 1, Gaussian beam passing through an axicon is used to generate the Bessel-Gaussian beam [13

13. I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]

].

The angular spectrum of an axicon is
T(κ)=1κδ(κksinα)
(13)
where δ () is the DiracDelta function and κ=u2+v2 is the radial coordinate (u and v being the component of the angular spectrum along x and y-axis), k = 2π/λ is the wavenumber.

From the convolution of the angular spectrum of an axicon and a Gaussian beam, the angular spectrum at the initial plane can be obtained as
G(κ,z=0)=12w02I0(k2κw02sinα)exp{w024[κ2+(ksinα)2]}
(14)
where I0 is the zeroth-order modified Bessel function of the first kind, w0 is the waist width of the Gaussian beam. By using the transfer function of the angular spectrum in free space
M(κ,z)=exp(izk2κ2),
(15)
we can obtain the angular at z-plane as
G(κ,z)=12w02I0(k2κw02sinα)exp{w024[κ2+(ksinα)2]}exp(izk2κ2).
(16)
From Eq. (16) and using the inverse Fourier transform we can get the optical field of Gaussian- Bessel beam in free space as
E(r,z)=w02kw02k+2izJ0(w02k2sinαw02k+2izr)exp[k(r2+2z2)w02k+2iz+w02k2zicosαw02k+2iz]
(17)
where J0 is the zeroth-order Bessel function of the first kind, and r=x2+y2 [(x, y) being the transverse coordinates]. When we set z = 0 or w0from Eq. (17) one can find
E(r,z=0)=J0(krsinα)exp(r2w02)
(18)
and
E(r,z)=J0(krsinα)exp(ikzcosα).
(19)
Equations (18) and (19) agree with the existing results.

For simplicity to quantitatively study the self-healing ability, the expression of the transmission for an opaque obstacle which block the initial optical field is given by a Gaussian function as
g(r,z=0)=1exp(r2R2)
(20)
where R is the radius of the obstacle. With the same method as in Eq. (17), the optical field of a Bessel-Gaussian beam at z-plane with an obstacle can be given as
E(r,z)=E(r,z)w012kw012k+2izJ0(w012k2sinαw012k+2izr)exp[k(r2+2z2)w012k+2iz+w012k2zicosαw012k+2iz]
(21)
wherew01=w0R/w02+R2. From Eqs. (17) and (21) the self-healing ability of Bessel-Gaussian beam can be quantitatively studied.

To see the self-healing process of Bessel-Gaussian beam in free space we setα=π/12, λ=512nmin the following calculation. Intensity distributions of a Bessel-Gaussian beam with and without an opaque obstacle at initial plane are plotted in Fig. 2.
Fig. 2 Intensity distribution of a Bessel-Gaussian beam at initial plane where w0 = 4μm (a) with an opaque obstacle (R = w0/5) (b) without opaque obstacle.

It can be seen that the central spot of a Bessel-Gaussian beam in Fig. 2(a) is obstructed. To see the self-healing process, the evolution of the intensity distribution of a Bessel-Gaussian beam with and without opaque obstacle is shown in Fig. 3 where the beam travel along z-axis and y = 0.
Fig. 3 Intensity evolution of a Bessel-Gaussian beam propagating along z-axis where y = 0 and w0 = 4μm (a) with an opaque obstacle (R = w0/5) (b) without opaque obstacle.

From Fig. 3 we can see that the central spot obstructed by an opaque obstacle gradually reconstructs during propagation. Because of the circular symmetry of the Bessel-Gaussian beam with and without an opaque obstacle in present paper, only one-dimensional similarity S is investigated. From Eqs. (11), (17) and (21) the similarity can be calculated. Figure 4 shows the variation of the similarity S during propagation with different parameters.
Fig. 4 One-dimensional similarity of the Bessel-Gaussian beam during propagation (a) with different w0 where R = 0.8μm and dmin = 3μm (b) with different R where w0 = 10μm, d1min = 1.5μm, d2min = 3μm and d3min = 4.5μm.

Figure 4(a) shows the variation of the similarity S with different w0 where R = 0.8μm. It can be seen that the similarity is large with large w0 when the propagation distance is short. With the increase of propagation distance the difference of the similarity corresponding different w0 become small. For comparison with the reconstruction distance in Eq. (12), dmin is also denoted in the Figs. We can see that the similarity is about 0.95 when the propagation distance equal to dmin. Namely, the reconstruction distance can be used as a metric to estimate the self-healing ability. Small dmin means fast speed of the self-healing process. Even though dmin denotes the distance where the reconstruction occurs in geometrical optics, the self-healing process almost complete when we consider diffraction of the beam. Figure 4(b) shows the variation of the similarity S with different R where w0 = 10μm. We can see that the speed of the self-healing is different with different R. When R is small, the distance is small to reconstruct its shape. We also can see from Fig. 4(b) that can also be used to describe the distance where the reconstruction has completed.

It should be pointed out that the reconstruction distance in Ref. [13

13. I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]

] which is obtained from the geometrical optics is only used to describe the self-healing of Bessel beam generated by an axicon. From the study we can see that Eq. (11) can be used to describe the evolution of the similarity of arbitrary beams during propagation. As comparing with the reconstruction distance we can see that Eq. (11) can be regarded as a metric to estimate the self-healing ability of Bessel-Gaussian beam. Because the self-healing ability means that a beam which is partially blocked by an opaque obstacle will become more and more similar to that without obstacle during propagation, Eq. (11) can be used to describe the self-healing ability of arbitrary beam.

4. Conclusion

In Hilbert space, optical field can be regarded as an infinite-dimensional complex vector. Using the angle of two infinite-dimensional complex vectors, the similarity of two intensity distribution is used to describe the self-healing ability. As an example, the self-healing ability of a Bessel-Gaussian beam is studied. Study shows that similarity can be used to quantitatively describe the strength of the self-healing ability of any beams liking reconstruction distance for Bessel beam. However, reconstruction distance is only used to describe the self-healing of Bessel beam generated by an axicon, similarity can be used to quantitatively describe the self-healing ability of arbitrary beams. With the help of the similarity for two intensity profiles, the self-healing ability of different beams can be compared.

Acknowledgment

The project was supported by National Natural Science Foundation of China (No. 11374264).

References and links

1.

H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves: Theory and. Applications (John Wiley, 2008).

2.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]

3.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007). [CrossRef]

4.

M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett. 98(6), 061111 (2011). [CrossRef]

5.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]

6.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]

7.

S. Vyas, Y. Kozawa, and S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A 28(5), 837–843 (2011). [CrossRef] [PubMed]

8.

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012). [CrossRef] [PubMed]

9.

R. Cao, Y. Hua, C. Min, S. Zhu, and X. C. Yuan, “Self-healing optical pillar array,” Opt. Lett. 37(17), 3540–3542 (2012). [CrossRef] [PubMed]

10.

M. Anguiano-Morales, A. Martínez, M. D. Iturbe-Castillo, S. Chávez-Cerda, and N. Alcalá-Ochoa, “Self-healing property of a caustic optical beam,” Appl. Opt. 46(34), 8284–8290 (2007). [CrossRef] [PubMed]

11.

P. Vaity and R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett. 36(15), 2994–2996 (2011). [CrossRef] [PubMed]

12.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]

13.

I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]

14.

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46(7), 078001 (2007). [CrossRef]

15.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef] [PubMed]

16.

Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010). [CrossRef] [PubMed]

17.

R. Martínez-Herrero, I. Juvells, and A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38(12), 2065–2067 (2013). [CrossRef] [PubMed]

18.

J. James, Mathematics Dictionary Mathematics Dictionary, 5th ed. (Springer, 1992).

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Physical Optics

History
Original Manuscript: February 26, 2014
Revised Manuscript: March 11, 2014
Manuscript Accepted: March 11, 2014
Published: March 17, 2014

Citation
Xiuxiang Chu and Wei Wen, "Quantitative description of the self-healing ability of a beam," Opt. Express 22, 6899-6904 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6899


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves: Theory and. Applications (John Wiley, 2008).
  2. V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]
  3. X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007). [CrossRef]
  4. M. Boguslawski, P. Rose, C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett. 98(6), 061111 (2011). [CrossRef]
  5. A. Chong, W. H. Renninger, D. N. Christodoulides, F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]
  6. M. V. Berry, N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
  7. S. Vyas, Y. Kozawa, S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A 28(5), 837–843 (2011). [CrossRef] [PubMed]
  8. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012). [CrossRef] [PubMed]
  9. R. Cao, Y. Hua, C. Min, S. Zhu, X. C. Yuan, “Self-healing optical pillar array,” Opt. Lett. 37(17), 3540–3542 (2012). [CrossRef] [PubMed]
  10. M. Anguiano-Morales, A. Martínez, M. D. Iturbe-Castillo, S. Chávez-Cerda, N. Alcalá-Ochoa, “Self-healing property of a caustic optical beam,” Appl. Opt. 46(34), 8284–8290 (2007). [CrossRef] [PubMed]
  11. P. Vaity, R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett. 36(15), 2994–2996 (2011). [CrossRef] [PubMed]
  12. J. Broky, G. A. Siviloglou, A. Dogariu, D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]
  13. I. A. Litvin, M. G. Mclaren, A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]
  14. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46(7), 078001 (2007). [CrossRef]
  15. E. Greenfield, M. Segev, W. Walasik, O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef] [PubMed]
  16. Y. Kaganovsky, E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010). [CrossRef] [PubMed]
  17. R. Martínez-Herrero, I. Juvells, A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38(12), 2065–2067 (2013). [CrossRef] [PubMed]
  18. J. James, Mathematics Dictionary Mathematics Dictionary, 5th ed. (Springer, 1992).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited