Abstract

In the framework of the paraxial approximation, we derive the analytical expressions for describing the effect of the Gouy phase of Laguerre–Gauss beams on the polarization evolution of partially coherent vortex fields whose electric field vector at some transverse plane exhibits a radially polarized behavior. At each transverse plane, the polarization distribution across the beam profile is characterized by means of the percentage of irradiance associated with the radial or azimuthal components. The propagation laws for these percentages are also presented. As an illustrative example, we analyze a radially polarized partially coherent vortex beam.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
  8. R. Martínez-Herrero and P. M. Mejías, “Propagation of light fields with radial or azimuthal polarization distribution at a transverse plane,” Opt. Express 16(12), 9021–9033 (2008).
    [Crossref] [PubMed]
  9. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
  10. R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281(8), 1976–1980 (2008).
    [Crossref]
  11. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  12. R. Martínez-Herrero and P. M. Mejías, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32(11), 1504–1506 (2007).
    [Crossref] [PubMed]
  13. R. Martínez-Herrero and P. M. Mejías, “Propagation and parametric characterization of the polarization structure of paraxial radially and azimuthally polarized beams,” Opt. Laser Technol. 44(2), 482–485 (2012).
    [Crossref]
  14. A. E. Siegman, Lasers (University Science Books, 1986).
  15. R. Martínez-Herrero and P. M. Mejías, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. 1(5), 556–558 (1984).
    [Crossref]
  16. R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
    [Crossref] [PubMed]
  17. E. W. Weisstein, “Associated Laguerre Polynomial,” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html

2012 (2)

Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 29(11), 2439–2443 (2012).
[Crossref] [PubMed]

R. Martínez-Herrero and P. M. Mejías, “Propagation and parametric characterization of the polarization structure of paraxial radially and azimuthally polarized beams,” Opt. Laser Technol. 44(2), 482–485 (2012).
[Crossref]

2011 (1)

2009 (3)

2008 (3)

R. Martínez-Herrero and P. M. Mejías, “Propagation of light fields with radial or azimuthal polarization distribution at a transverse plane,” Opt. Express 16(12), 9021–9033 (2008).
[Crossref] [PubMed]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281(8), 1976–1980 (2008).
[Crossref]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

2007 (1)

2004 (1)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

1984 (1)

R. Martínez-Herrero and P. M. Mejías, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. 1(5), 556–558 (1984).
[Crossref]

Bu, J.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Gao, B. Z.

Gori, F.

Hashimoto, N.

Hibi, T.

Horanai, H.

Kozawa, Y.

Kurihara, M.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Martínez-Herrero, R.

R. Martínez-Herrero and P. M. Mejías, “Propagation and parametric characterization of the polarization structure of paraxial radially and azimuthally polarized beams,” Opt. Laser Technol. 44(2), 482–485 (2012).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]

R. Martínez-Herrero and P. M. Mejías, “Propagation of light fields with radial or azimuthal polarization distribution at a transverse plane,” Opt. Express 16(12), 9021–9033 (2008).
[Crossref] [PubMed]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281(8), 1976–1980 (2008).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32(11), 1504–1506 (2007).
[Crossref] [PubMed]

R. Martínez-Herrero and P. M. Mejías, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. 1(5), 556–558 (1984).
[Crossref]

Mejías, P. M.

R. Martínez-Herrero and P. M. Mejías, “Propagation and parametric characterization of the polarization structure of paraxial radially and azimuthally polarized beams,” Opt. Laser Technol. 44(2), 482–485 (2012).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]

R. Martínez-Herrero and P. M. Mejías, “Propagation of light fields with radial or azimuthal polarization distribution at a transverse plane,” Opt. Express 16(12), 9021–9033 (2008).
[Crossref] [PubMed]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281(8), 1976–1980 (2008).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32(11), 1504–1506 (2007).
[Crossref] [PubMed]

R. Martínez-Herrero and P. M. Mejías, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. 1(5), 556–558 (1984).
[Crossref]

Moh, K. J.

Morita, R.

Nemoto, T.

Oka, K.

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281(8), 1976–1980 (2008).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Ramírez-Sánchez, V.

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281(8), 1976–1980 (2008).
[Crossref]

Sato, A.

Sato, S.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Tokizane, Y.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Yokoyama, H.

Yuan, X. C.

Zhan, Q.

Zhu, S. W.

J. Opt. Soc. Am. (1)

R. Martínez-Herrero and P. M. Mejías, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. 1(5), 556–558 (1984).
[Crossref]

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

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Opt. Commun. (1)

R. Martínez-Herrero, P. M. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281(8), 1976–1980 (2008).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

R. Martínez-Herrero and P. M. Mejías, “Propagation and parametric characterization of the polarization structure of paraxial radially and azimuthally polarized beams,” Opt. Laser Technol. 44(2), 482–485 (2012).
[Crossref]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Other (4)

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

E. W. Weisstein, “Associated Laguerre Polynomial,” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html

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

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Figures (2)

Fig. 1
Fig. 1 Monomodal beams: (a) asymptotic behavior of the parameter ρ ˜ az as function of spiral charge m, (b) parameter ρ ˜ az in terms of propagation distance z in units of Rayleigh range z 0 . The p index is fixed as p=1 , and the topological charge m takes on the values of 1, 3, and 4.
Fig. 2
Fig. 2 (a) Parameter ρ ˜ az versus propagation distance z in units of Rayleigh range for a beam (superposition of two spatial modes p = 1, q = 0) with topological charge m = 2. The modes are assumed to be coherent ( τ=0 ), partially coherent ( τ= π 3 ), and incoherent ( τ= π 2 ). (b) Parameter ρ ˜ az versus propagation distance z in units of Rayleigh range for a beam (superposition of two spatial modes p = 2, q = 0) with topological charge m = 2. The modes are assumed to be coherent ( τ=0 ), partially coherent ( τ= π 3 ), and incoherent ( τ= π 2 ).

Equations (42)

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E ( r ,z )=( E rad ( r ,z ), E az ( r ,z ) )
W ^ ij ( r 1 , r 2 ,z )= E i ( r 1 ,z ) * E j ( r 2 ,z )
ρ ˜ j (z)= 0 0 2π ρ j ( r,θ,z )I( r,θ,z )rdrdθ 0 0 2π I( r,θ,z )rdrdθ
I( r,θ,z )=< | E (r,θ,z) | 2 >
ρ j (r,θ,z)= < | E j (r,θ,z) | 2 > < | E (r,θ,z) | 2 >
W ^ ( r 1 , r 2 ,0 )=Γ( r 1 , r 2 ,0) r 1 r 2 ( 1 0 0 0 )
μ 2 ( r 1 , r 2 ,0 )= | Γ( r 1 , r 2 ,0) | 2 Γ( r 1 , r 1 ,0)Γ( r 2 , r 2 ,0)
W rad,rad ( r 1 , r 2 ,z )=[ r 1 r 2 + iz k ( r 2 r 1 r 1 r 2 )+ z 2 k 2 2 r 1 r 2 ]Γ( r 1 , r 2 ,z)
W az,az ( r 1 , r 2 ,z )= z 2 k 2 r 1 r 2 2 θ 1 θ 2 Γ( r 1 , r 2 ,z)
W rad,az ( r 1 , r 2 ,z )= W az,rad * ( r 2 , r 1 ,z )=( iz k r 1 r 2 θ 2 + z 2 k 2 r 2 2 r 1 θ 2 )Γ( r 1 , r 2 ,z)
ρ ˜ az (z)= z 2 0 0 2π r 2 ( 2 Γ( r 1 , r 2 ,z) θ 1 θ 2 ) r 1 = r 2 =r rdrdθ k 2 0 0 2π r 2 Γ( r , r ,0)rdrdθ
Γ( r 1 , r 2 ,z)=Γ( r 1 , r 2 ,z) e im( θ 2 θ 1 )
u p m ( r,θ,z )= f p m (r,z) e imθ
f p m (r,z)= a mp ω( z ) ( 2 r ω( z ) ) | m | L p | m | ( 2 r 2 ω ( z ) 2 ) e ik r 2 2q( z ) e i φ mp ( z )
φ mp (z)= ( 2p+m+1 )α(z) ω(z)
tan( α( z ) )= z z 0 ,
Γ( r 1 , r 2 ,z)= p,q B pq m u p m ( r 1 , θ 1 ,z ) * u q m ( r 2 , θ 2 ,z )
B pq m = 0 0 Γ( r 1 , r 2 ,0) f p m ( r 1 ,0) f q m ( r 2 ,0) * r 1 r 2 d r 1 d r 2
0 | B p,q m | 2 B p,p m B q,q m
ρ ˜ az (z)= m 2 z 2 z 2 + z 2 0 p,q B pq m a mp a mq I mpq e 2iα( z )( pq ) p,q B pq m a mp a mq γ mpq
I mpq = 0 x | m |1 L p | m | (x) L q | m | (x) e x dx= b=0 min(p,q) (| m |1+b)! b!
γ mpq = 0 x | m |+1 L p | m | (x) L q | m | (x) e x dx=(| m |+2p+1) (| m |+p)! p! δ pq δ | pq |=1 (| m |+p)! (p1)!
Γ( r 1 , r 2 ,z)= B pp m u p m ( r 1 , θ 1 ,z ) * u p m ( r 2 , θ 2 ,z ),
ρ ˜ az (z)= m 2 z 2 I mpp ( z 2 + z 2 0 ) γ mpp
Γ( r 1 , r 2 ,0)= u q m ( r 1 , θ 1 ,0 ) * u q m ( r 2 , θ 2 ,0 )+ u p m ( r 1 , θ 1 ,0 ) * u p m ( r 2 , θ 2 ,0 ) +cosτ( u q m ( r 1 , θ 1 ,0 )* u p m ( r 2 , θ 2 ,0 )+ u p m ( r 1 , θ 1 ,0 )* u q m ( r 2 , θ 2 ,0 ) )
ρ ˜ az (z)= m 2 z 2 z 2 + z 2 0 ( ρ 0 +2 ρ 1 cosτcos(2(pq)α(z)) σ 0 2 σ 1 cosτ )
ρ 0 = a mpp 2 I mpp + a mqq 2 I mqq
ρ 1 =2 a mpp a mqq I mpq
σ 0 = a mpp 2 γ mpp + a mqq 2 γ mqq
σ 1 =2 a mpp a mqq (m+p)! (p1)! δ | pq |=1
cos(2α( z 1 ))= σ 1 ρ 0 σ 0 ρ 1
z k = z 0 tan( (2k1)π 4(pq) )
A(z)= 0 0 2π r 2 ( 2 Γ( r 1 , r 2 ,z) θ 1 θ 2 ) r 1 = r 2 =r rdrdθ
B= 0 0 2π r 2 Γ( r , r ,0)rdrdθ
A( z )= 2π m 2 ω 2 ( z ) p,q B pq m a mp a mq e i( φ mp ( z ) φ mq ( z ) ) 0 r 2 ( 2 r ω( z ) ) 2| m | L p | m | ( 2 r 2 ω 2 ( z ) ) L q | m | ( 2 r 2 ω 2 ( z ) )exp( 2 r 2 ω 2 ( z ) )rdr
B= 2π ω 2 ( 0 ) p,q B pq m a mp a mq 0 r 2 ( 2 r ω( 0 ) ) 2| m | L p | m | ( 2 r 2 ω 2 ( 0 ) ) L q | m | ( 2 r 2 ω 2 ( 0 ) )exp( 2 r 2 ω 2 ( 0 ) )rdr
A( z )= π m 2 ω 2 ( z ) p,q B pq m a mp a mq e i( φ mp ( z ) φ mq ( z ) ) 0 x | m |1 L p | m | ( x ) L q | m | ( x )exp( x )dx
B= π ω 2 ( 0 ) 4 p,q B pq m a mp a mq 0 x 0 | m |+1 L p | m | ( x 0 ) L q | m | ( x 0 )exp( x 0 )d x 0
I mpq = 0 x | m |1 L p | m | (x) L q | m | (x) e x dx= a=0 min(p,q) (| m |1+a)! a! ,
γ mpq = 0 x | m |+1 L p | m | (x) L q | m | (x) e x dx=(| m |+2p+1) (| m |+p)! p! δ pq δ | pq |=1 (| m |+p)! (p1)!
ρ ˜ az (z)= z 2 A( z ) k 2 B
ρ ˜ az (z)= 4 z 2 m 2 k 2 ω 2 ( 0 ) ω 2 ( z ) p,q B p,q m a mp a mq I m,p,q e 2iα( z )( pq ) p,q B p,q m a mp a mq γ m,p,q

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