Abstract

Focusing of a vectorial (electromagnetic) optical beam through a high numerical aperture can be investigated by means of the Richards–Wolf diffraction integral. However, such an integral extends from two-dimensional to four-dimensional, greatly increasing the computation time and therefore limiting the applicability, when light with decreased spatial coherence is considered. Here, we advance an effective protocol for the fast calculation of the statistical properties of a tightly focused field produced by a random electromagnetic beam with arbitrary state of spatial coherence and polarization. The novel method relies on a vectorial pseudo-mode representation and a fast algorithm of the wave-vector space Fourier transform. The procedure is demonstrated for several types of radially (fully) polarized but spatially partially coherent Schell-model beams. The simulations show that the computation time for obtaining the focal spectral density distribution with 512 × 512 spatial points for a low coherence beam is less than 100 seconds, while with the conventional quadruple Richards–Wolf integral more than 100 hours is required. The results further indicate that spatial coherence can be viewed as an effective degree of freedom to govern both the transverse and longitudinal components of a tightly focused field with potential applications in reverse shaping of focal fields and optical trapping control.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

2019 (9)

M. W. Hyde IV, “Generating electromagnetic Schell-model sources using complex screens with spatially varying autoand cross-correlation functions,” Results Phys. 15, 102663 (2019).
[Crossref]

M. W. Hyde IV, X. Xiao, and D. G. Voelz, “Generating electromagnetic nonuniformly correlated beams,” Opt. Lett. 44(23), 5719–5722 (2019).
[Crossref]

H. Xu, R. Zhang, Z. Sheng, and J. Qu, “Focus shaping of partially coherent radially polarized vortex beam with tunable topological charge,” Opt. Express 27(17), 23959–23969 (2019).
[Crossref]

J. C. G. de Sande, R. Martínez-Herrero, G. Piquero, M. Santarsiero, and F. Gori, “Pseudo-Schell model sources,” Opt. Express 27(4), 3963–3977 (2019).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent surface plasmon polariton vortex fields,” Phys. Rev. A 100(5), 053833 (2019).
[Crossref]

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

K. Y. Bliokh, M. A. Alonso, and M. R. Dennis, “Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects,” Rep. Prog. Phys. 82(12), 122401 (2019).
[Crossref]

G. Rui, Y. Li, S. Zhou, Y. Wang, B. Gu, Y. Cui, and Q. Zhan, “Optically induced rotation of Rayleigh particles by arbitrary photonic spin,” Photonics Res. 7(1), 69–79 (2019).
[Crossref]

H. Hu, Q. Gan, and Q. Zhan, “Generation of a nondiffracting superchiral optical needle for circular dichroism imaging of spare subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
[Crossref]

2018 (7)

2017 (2)

2016 (5)

Z. Chen, T. Zeng, and J. Ding, “Reverse engineering approach to focus shaping,” Opt. Lett. 41(9), 1929–1932 (2016).
[Crossref]

A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence (invited),” J. Opt. Soc. Am. A 33(12), 2431–2442 (2016).
[Crossref]

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref]

M. Neugebauer, P. Woźniak, A. Bag, G. Leuch, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7(1), 11286 (2016).
[Crossref]

M. W. Hyde IV, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

2015 (8)

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[Crossref]

M. W. Hyde IV, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

S. Yang, S. A. Ponomarenko, and Z. Chen, “Coherent pseudo-mode decomposition of a new partially coherent source class,” Opt. Lett. 40(13), 3081–3084 (2015).
[Crossref]

S. Roy, K. Ushakova, Q. van den Berg, S. F. Pereira, and H. P. Urbach, “Radially polarized light for detection and nanolocalization of dielectric particles on a planar substrate,” Phys. Rev. Lett. 114(10), 103903 (2015).
[Crossref]

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Royd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref]

S. You, C. Kuang, K. C. Toussaint, R. Zhou, X. Xia, and X. Liu, “Iterative phase-retrieval method for generating stereo array of polarization-controlled focal spots,” Opt. Lett. 40(15), 3532–3535 (2015).
[Crossref]

2014 (6)

2013 (1)

2012 (3)

2011 (1)

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

2010 (2)

2009 (3)

2007 (2)

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
[Crossref]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1(4), 228–231 (2007).
[Crossref]

2006 (2)

2005 (3)

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 7(5), 232–237 (2005).
[Crossref]

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71(5), 056607 (2005).
[Crossref]

K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22(3), 561–568 (2005).
[Crossref]

2004 (3)

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]

2000 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 253(1274), 358–379 (1959).
[Crossref]

Aiello, A.

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref]

M. Neugebauer, P. Banzer, T. Bauer, S. Orlov, N. Lindlein, A. Aiello, and G. Leuchs, “Geometric spin Hall effect of light in tightly focused polarization-tailored light beams,” Phys. Rev. A 89(1), 013840 (2014).
[Crossref]

Alonso, M. A.

K. Y. Bliokh, M. A. Alonso, and M. R. Dennis, “Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects,” Rep. Prog. Phys. 82(12), 122401 (2019).
[Crossref]

Bag, A.

A. Bag, M. Neugebauer, P. Woźniak, G. Leuchs, and P. Banzer, “Transverse Kerker scattering for Angstrom localization of nanoparticles,” Phys. Rev. Lett. 121(19), 193902 (2018).
[Crossref]

M. Neugebauer, P. Woźniak, A. Bag, G. Leuch, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7(1), 11286 (2016).
[Crossref]

Banzer, P.

A. Bag, M. Neugebauer, P. Woźniak, G. Leuchs, and P. Banzer, “Transverse Kerker scattering for Angstrom localization of nanoparticles,” Phys. Rev. Lett. 121(19), 193902 (2018).
[Crossref]

M. Neugebauer, J. S. Eismann, T. Bauer, and P. Banzer, “Magnetic and electric transverse spin density of spatially confined light,” Phys. Rev. X 8(2), 021042 (2018).
[Crossref]

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref]

M. Neugebauer, P. Woźniak, A. Bag, G. Leuch, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7(1), 11286 (2016).
[Crossref]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Royd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref]

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref]

M. Neugebauer, P. Banzer, T. Bauer, S. Orlov, N. Lindlein, A. Aiello, and G. Leuchs, “Geometric spin Hall effect of light in tightly focused polarization-tailored light beams,” Phys. Rev. A 89(1), 013840 (2014).
[Crossref]

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8(1), 23–27 (2014).
[Crossref]

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010).
[Crossref]

Basu, S.

M. W. Hyde IV, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Bauer, T.

M. Neugebauer, J. S. Eismann, T. Bauer, and P. Banzer, “Magnetic and electric transverse spin density of spatially confined light,” Phys. Rev. X 8(2), 021042 (2018).
[Crossref]

T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer, “Optical polarization Möbius strips and points of purely transverse spin density,” Phys. Rev. Lett. 117(1), 013601 (2016).
[Crossref]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Royd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347(6225), 964–966 (2015).
[Crossref]

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref]

M. Neugebauer, P. Banzer, T. Bauer, S. Orlov, N. Lindlein, A. Aiello, and G. Leuchs, “Geometric spin Hall effect of light in tightly focused polarization-tailored light beams,” Phys. Rev. A 89(1), 013840 (2014).
[Crossref]

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8(1), 23–27 (2014).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, M. A. Alonso, and M. R. Dennis, “Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects,” Rep. Prog. Phys. 82(12), 122401 (2019).
[Crossref]

Bose-Pillai, S.

M. W. Hyde IV, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

Brown, T. G.

Cai, Y.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

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Photonics Res. (1)

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Figures (6)

Fig. 1.
Fig. 1. Geometry and notation related to tight focusing of a partially coherent vector beam by a high numerical aperture objective lens with focal length $f$. In transmission through the objective the mode vector ${\Phi }_n (\mathbf {r})$ transforms into ${\Phi }^{\textrm {(t)}}_n (\mathbf {r}, \theta )$ and the related wave vector $\mathbf {k}$ points towards the focal point and forms an angle $\theta$ with respect to the $z$ axis. $\phi$ is the azimuthal angle of position vector $\mathbf {r}$. The focal plane is located at $z=0$.
Fig. 2.
Fig. 2. Analytical (left) and numerical (middle) representations of the degree of coherence $g(\Delta \mathbf {r})$ for a GSM beam (top row), an MGSM beam of beam order $M = 10$ (second row), an LGSM beam of beam order $l = 5$ (third row), and an HGSM beam of beam order $m_x = 1, m_y = 0$ (bottom row). The spatial coherence width is $\delta _0 = 0.2$ mm in all cases. (Right) Cross-line ($\Delta y=0$) of the analytical (gray solid curves) and numerical expressions (black dashed curves).
Fig. 3.
Fig. 3. Similarity $s$ between the numerical and analytical expressions of $g(\Delta \mathbf {r})$ versus the number $N$ of the complex random screens for the LGSM beam with $l=5$ and $\delta _0 = 0.2$ mm. The shaded region and the solid curve illustrate, respectively, the standard deviation and the mean value of the similarities for ten independent simulations.
Fig. 4.
Fig. 4. Numerical focal-plane distributions of the normalized total spectral density $S(\mathbf {r})$ (left), transverse spectral density $S_x(\mathbf {r})+S_y(\mathbf {r})$ (middle), and the longitudinal spectral density $S_z(\mathbf {r})$ (right) of a tightly focused radially polarized GSM beam (top row), MGSM beam of order $M = 10$ (second row), LGSM beam of order $l=5$ (third row), and HGSM beam of order $m_x = 1, m_y = 0$ (bottom row). The parameters are: $\lambda = 632.8$ nm, $\delta _0 = 0.2$ mm, $w_0 = 1$ mm, $\textrm {NA} = 0.95$, $f = 3$ mm, and $n_{\textrm {t}} = 1$.
Fig. 5.
Fig. 5. Numerical focal-plane distributions of the normalized total spectral density $S(\mathbf {r})$ (left), transverse spectral density $S_x(\mathbf {r})+S_y(\mathbf {r})$ (middle), and the longitudinal spectral density $S_z(\mathbf {r})$ (right) of a tightly focused radially polarized LGSM beam of order $l=5$, for different values of the incident-field spatial coherence width $\delta _0$. The parameters are: $\lambda = 632.8$ nm, $w_0 = 1$ mm, $\textrm {NA} = 0.95$, $f = 3$ mm, and $n_{\textrm {t}} = 1$.
Fig. 6.
Fig. 6. Numerically obtained distributions of the normalized longitudinal spectral density, $S_z(\mathbf {r}, z)$, in the $xz$ plane ($y=0$) of the focused radially polarized (a) GSM beam, (b) MGSM beam of beam order $M=10$, (c) LGSM beam of beam order $l=5$, and (d) HGSM beam of beam order $m_x = 1$, $m_y = 0$. The parameters are: $\lambda = 632.8$ nm, $\delta _0 = 0.2$ mm, $w_0 = 1$ mm, $\textrm {NA} = 0.95$, $f = 3$ mm, and $n_{\textrm {t}} = 1$.

Equations (42)

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W ( r 1 , r 2 ) = E ( r 1 ) E T ( r 2 ) .
W ( r 1 , r 2 ) = n α n Φ n ( r 1 ) Φ n T ( r 2 ) ,
D Φ n T ( r 1 ) W ( r 1 , r 2 ) d 2 r 1 = α n Φ n T ( r 2 ) .
Φ n ( r ) = Φ n (rad) ( r ) e ^ r + Φ n (azi) ( r ) e ^ ϕ ,
e ^ r = cos ϕ e ^ x + sin ϕ e ^ y ,
e ^ ϕ = sin ϕ e ^ x + cos ϕ e ^ y ,
e ^ r (t) = cos θ e ^ r + sin θ e ^ z ,
e ^ ϕ (t) = e ^ ϕ .
Φ n (t) ( r , θ ) = t r Φ n (rad) ( r ) e ^ r (t) + t ϕ Φ n (azi) ( r ) e ^ ϕ (t) ,
Ψ n ( r , z ) = i f λ Ω Φ n (t) ( r , θ ) e i ( k z z k x x k y y ) d Ω ,
k x = k sin θ cos ϕ ,
k y = k sin θ sin ϕ ,
k z = k cos θ ,
d Ω = sin θ d θ d ϕ .
d Ω = 1 k 2 cos θ d k x d k y .
r = f sin θ .
x = f k x / k ,
y = f k y / k .
Ψ n ( r , z ) = i f λ k r < R Φ n (t) ( k x , k y ) e i k z z k z 1 e i ( k x x + k y y ) d k x d k y ,
Ψ n ( r , z ) = i f λ k F [ Φ n (t) ( k x , k y ) e i k z z k z 1 ] k x , k y ,
W ( r 1 , z 1 ; r 2 , z 2 ) = n α n Ψ n ( r 1 , z 1 ) Ψ n T ( r 2 , z 2 ) ,
W ( r 1 , r 2 ) = S ( r 1 ) S ( r 2 ) g ( Δ r ) e ^ r ( r 1 ) e ^ r T ( r 2 ) ,
S ( r ) = 2 r 2 w 0 2 e 2 r 2 / w 0 2 ,
g ( Δ r ) = 1 N n T n ( r 1 ) T n ( r 2 ) ,
T n ( r ) = F 1 [ τ n ( f x , f y ) ] f x , f y ,
τ n ( f x , f y ) = a n ( f x , f y ) + i b n ( f x , f y ) 2 p ( f x , f y ) .
p ( f x , f y ) = g ( Δ r ) e i 2 π ( Δ x f x + Δ y f y ) d 2 Δ r ,
g ( Δ r ) = exp [ Δ r 2 / ( 2 δ 0 2 ) ] ,
p ( f x , f y ) = 2 π δ 0 2 exp [ 2 π 2 δ 0 2 ( f x 2 + f y 2 ) ] .
g ( Δ r ) = 1 C 0 m = 1 M ( M m ) ( 1 ) m 1 m exp ( Δ r 2 2 m δ 0 2 ) ,
p ( f x , f y ) = 2 π δ 0 2 C 0 m = 1 M ( M m ) ( 1 ) m 1 exp [ 2 π 2 m δ 0 2 ( f x 2 + f y 2 ) ] .
g ( Δ r ) = L l 0 ( Δ r 2 2 δ 0 2 ) exp ( Δ r 2 2 δ 0 2 ) ,
p ( f x , f y ) = ( π 2 l + 1 δ 0 2 l + 2 2 l + 1 / l ! ) ( f x 2 + f y 2 ) l exp [ 2 π 2 δ 0 2 ( f x 2 + f y 2 ) ] .
g ( Δ r ) = H 2 m x [ Δ x / ( 2 δ 0 ) ] H 2 m y [ Δ y / ( 2 δ 0 ) ] H 2 m x ( 0 ) H 2 m y ( 0 ) exp ( Δ r 2 2 δ 0 2 ) ,
p ( f x , f y ) = 2 3 ( m x + m y ) + 1 π 2 ( m x + m y ) δ 0 2 ( m x + m y + 1 ) H 2 m x ( 0 ) H 2 m y ( 0 ) f x 2 m x f y 2 m y exp [ 2 π 2 δ 0 2 ( f x 2 + f y 2 ) ] .
Φ n ( r ) = S ( r ) / N T n ( r ) e ^ r ,
Φ n (t) ( r , θ ) = cos θ S ( r ) / N T n ( r ) e ^ r (t) .
Φ n (t) ( r , θ ) = cos θ S ( r ) / N T n ( r ) { cos θ cos ϕ cos θ sin ϕ sin θ } .
θ = arccos ( k z / k ) ,
ϕ = arg ( k x i k y ) ,
S ( r , z ) = S x ( r , z ) + S y ( r , z ) + S z ( r , z ) ,
s = [ g ( A ) ( Δ r ) g ( N ) ( Δ r ) d 2 Δ r ] 2 [ g ( A ) ( Δ r ) ] 2 d 2 Δ r [ g ( N ) ( Δ r ) ] 2 d 2 Δ r ,

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