Abstract

To analyze the self-healing of a partially obstructed optical beam, we represent it by two orthogonal field components. The first component is an exact copy of the unobstructed beam, attenuated by a factor that is computed by a simple formula. The second component represents a pure distortion field, due to its orthogonality respect to the first one. This approach provides a natural measure of the beam damage, due to the obstruction, and the degree of self-healing, during propagation of the obstructed beam. As interesting results, derived in our approach, we obtain that the self-healing reaches a limit degree at the far field propagation domain, and that certain relatively small phase obstructions may produce a total damage on the beam. The theory is illustrated considering a Gaussian beam, distorted by different amplitude and phase obstructions. In the case of a soft Gaussian obstruction we obtain simple formulas for the far field limit values of the beam damage and the self-healing degree.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]

2017 (1)

2014 (2)

X. Chu and W. Wen, “Quantitative description of the self-healing ability of a beam,” Opt. Express 22(6), 6899–6904 (2014).
[Crossref] [PubMed]

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gauss beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

2012 (3)

X. Chu, “Analytical study on the self-healing property of Bessel beam,” Eur. Phys. J. D 66(10), 259 (2012).
[Crossref]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

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]

2011 (1)

2009 (1)

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]

2008 (1)

2007 (1)

2006 (1)

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8(5), 477–482 (2006).
[Crossref]

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

2004 (1)

2002 (1)

Z. Bouchal, “Resistance of nondiffracting vortex beams against amplitude and phase perturbations,” Opt. Commun. 210(3-6), 155–164 (2002).
[Crossref]

Agarwal, G. S.

Aiello, A.

Alcalá-Ochoa, N.

Anguiano-Morales, M.

Bouchal, Z.

Z. Bouchal, “Resistance of nondiffracting vortex beams against amplitude and phase perturbations,” Opt. Commun. 210(3-6), 155–164 (2002).
[Crossref]

Broky, J.

Brown, C. T. A.

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8(5), 477–482 (2006).
[Crossref]

Cai, Y.

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gauss beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

Cannan, D.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

Chávez-Cerda, S.

Chen, Z.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

Christodoulides, D. N.

Chu, X.

X. Chu and W. Wen, “Quantitative description of the self-healing ability of a beam,” Opt. Express 22(6), 6899–6904 (2014).
[Crossref] [PubMed]

X. Chu, “Analytical study on the self-healing property of Bessel beam,” Eur. Phys. J. D 66(10), 259 (2012).
[Crossref]

de la Hoz, P.

Dennis, M. R.

Dholakia, K.

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]

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8(5), 477–482 (2006).
[Crossref]

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Dogariu, A.

Fischer, P.

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8(5), 477–482 (2006).
[Crossref]

Forbes, A.

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]

Hradil, Z.

Hu, Y.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

Iturbe-Castillo, M. D.

Kozawa, Y.

Leuchs, G.

Li, T.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

Lindberg, J.

Little, H.

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8(5), 477–482 (2006).
[Crossref]

Litvin, I. A.

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]

Lopez-Mariscal, C.

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8(5), 477–482 (2006).
[Crossref]

Martínez, A.

Mazilu, M.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Mclaren, M. G.

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]

Morandotti, R.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

Mourka, A.

Paúr, M.

Rehácek, J.

Ring, J. D.

Sánchez-Soto, L. L.

Sato, S.

Sibbett, W.

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8(5), 477–482 (2006).
[Crossref]

Siviloglou, G. A.

Smith, R. L.

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8(5), 477–482 (2006).
[Crossref]

Stoklasa, B.

Tao, S. H.

Vyas, S.

Wang, F.

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gauss beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

Wen, W.

Wu, G.

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gauss beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

Yin, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

Yuan, X.

Zhang, P.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

Zhang, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

Appl. Opt. (1)

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Eur. Phys. J. D (1)

X. Chu, “Analytical study on the self-healing property of Bessel beam,” Eur. Phys. J. D 66(10), 259 (2012).
[Crossref]

J. Opt. A (1)

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8(5), 477–482 (2006).
[Crossref]

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

Opt. Commun. (2)

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]

Z. Bouchal, “Resistance of nondiffracting vortex beams against amplitude and phase perturbations,” Opt. Commun. 210(3-6), 155–164 (2002).
[Crossref]

Opt. Express (4)

Phys. Rev. A (1)

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gauss beam,” Phys. Rev. A 89(4), 043807 (2014).
[Crossref]

Phys. Rev. Lett. (1)

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref] [PubMed]

Other (1)

V. Arrizón, D. Aguirre-Olivas, G. Mellado-Villaseñor, and S. Chávez-Cerda, “Self-healing in scaled propagation invariant beams,” arXiv:1503.03125 (2015).

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Figures (6)

Fig. 1
Fig. 1 Transverse amplitudes of the fields βf(x,y,z) (blue), f0(x,y,z) (red), and e(x,y,z) (yelow) for wB/wA = 1/4. The propagation distances are cZR(wA) where c is (a) 0, (b) 1/4, (c) 1/2, (d) 1, (e) 10, and (f) 20.
Fig. 2
Fig. 2 Similar transverse amplitudes to those in Fig. 1, for wB/wA = 1/2.
Fig. 3
Fig. 3 Beam damage (a, c) and self-healing degree (b, d) for a Gaussian beam obstructed by a soft transmittance [Eq. (14)]. The waist radii wB/wA is 1/4 (blue), 1/2 (red), and 3/4 (yellow). The dashed lines mark the far field limit values for damage and self-healing. We considered the cases of scaled (a, b) and fixed (c, d) assessing domains.
Fig. 4
Fig. 4 Similar results to those in Fig. 3 (a,b), obtained for the evaluation domain which is complementary to ΩZ. Again the dashed lines mark the far field limit values.
Fig. 5
Fig. 5 Similar results to those in Fig. 1, for the binary obstruction transmittance t(x,y) = 1−circ(r/R), with R = wA/4.
Fig. 6
Fig. 6 Amplitudes (a-c) and phases (d-f) of the Gaussian beam f(x,y,z) (blue) and the damaged field f0(x,y,z) (red) generated by a binary phase obstruction t(x,y) = 1−2circ(r/R). The radius R is chosen to produce the attenuation factor β = 0. The propagation distance is 0 (a, d), zR(wA) (b, e), and 10 zR(wA) (c, f).

Equations (22)

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

f O (x,y,0)=t(x,y)f(x,y,0),
f O (x,y,0)=f(x,y,0)o(x,y)f(x,y,0).
f O (x,y,0)=βf(x,y,0)+e(x,y,0),
g 1 (x,y,z)| g 2 (x,y,z) Ω = Ω g 1 (x,y,z) g 2 * (x,y,z) dxdy,
β= f O (x,y,0)| f(x,y,0) f(x,y,0)| f(x,y,0) .
f O (x,y,z)=βf(x,y,z)+e(x,y,z),
βf(x,y,z)| βf(x,y,z) f 0 (x,y,z)| f 0 (x,y,z) + e(x,y,z)| e(x,y,z) f 0 (x,y,z)| f 0 (x,y,z) =1.
D T = e(x,y,z)| e(x,y,z) f 0 (x,y,z)| f 0 (x,y,z) .
D Ω Z (z)= e(x,y,z)| e(x,y,z) Ω Z f 0 (x,y,z)| f 0 (x,y,z) ,
SH(z)= D Ω 0 (0) D Ω Z (z) D T ,
Σ | f(x,y,0) | 2 a(x,y)exp[iϕ(x,y)]dxdy=0,
g w S (x,y,z)= Z R ( w S ) Z R ( w S )+iz exp[ ik r 2 2R(z, w S ) ]exp[ r 2 w 2 (z, w S ) ],
f(x,y,0)= g w A (x,y,0),
t(x,y)=1 g w B (x,y,0),
f 0 (x,y,0)= g w A (x,y,0) g w C (x,y,0),
β=1 2 2+ ( w A / w B ) 2 .
e(x,y,0)=(1β) g w A (x,y,0) g w C (x,y,0).
f(x,y,z)= g w A (x,y,z),
f 0 (x,y,z)= g w A (x,y,z) g w C (x,y,z),
e(x,y,z)=(1β) g w A (x,y,z) g w C (x,y,z),
D T =(1β)/2,
D lim = 1β 2 β 2 { β 2 exp( 2 α 2 1β 1+β )+(1 β 2 )[ 2exp( 2 α 2 1+β )exp(2 α 2 ) ] }.

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