Abstract

Laser beam divergence was considered using the components of the Poynting vector. A physical interpretation of the real and imaginary parts of the “longitudinal” field component (along the $z$ axis, the direction of beam propagation) is given in an exponential representation. The longitudinal field component was shown to be the reason for the formation of laser beam divergence. This fact was used to reveal fundamental differences between Laguerre–Gaussian beams and incoherent rays in the focal region. The possibility of using selective action on the longitudinal field component to decrease the divergence of Laguerre–Gaussian beams is discussed.

© 2020 Optical Society of America

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References

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    [Crossref]
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2020 (1)

2019 (1)

U. Levy, Y. Silberberg, and N. Davidson, “Mathematics of vectorial Gaussian beams,” Adv. Opt. Photonics 11, 828–892 (2019).
[Crossref]

2018 (2)

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Longitudinal component of the Poynting vector of tightly focused cylindrical vector beam,” J. Phys. Conf. Ser. 1135, 012064 (2018).
[Crossref]

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

2017 (1)

2012 (1)

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “On the longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[Crossref]

2005 (1)

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608 (2005).
[Crossref]

2004 (1)

2001 (1)

A. V. Nesterov and V. G. Niziev, “Propagation features of beams with axially symmetric polarization,” J. Opt. B 3, 215–219 (2001).
[Crossref]

2000 (1)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

1986 (1)

1966 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Cai, X.

Carnicer, A.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “On the longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[Crossref]

Davidson, N.

U. Levy, Y. Silberberg, and N. Davidson, “Mathematics of vectorial Gaussian beams,” Adv. Opt. Photonics 11, 828–892 (2019).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Forbes, A.

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

Juvells, I.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “On the longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[Crossref]

Kogelnik, H.

Kotlyar, V. V.

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Longitudinal component of the Poynting vector of tightly focused cylindrical vector beam,” J. Phys. Conf. Ser. 1135, 012064 (2018).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics (Butterworth-Heinemann, 1994), Vol. 2.

Levy, U.

U. Levy, Y. Silberberg, and N. Davidson, “Mathematics of vectorial Gaussian beams,” Adv. Opt. Photonics 11, 828–892 (2019).
[Crossref]

U. Levy and Y. Silberberg, “Weakly diverging to tightly focused Gaussian beams: continuation—symmetric beams,” J. Opt. Soc. Am. A 34, 331–335 (2017).
[Crossref]

Li, T.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics (Butterworth-Heinemann, 1994), Vol. 2.

Maluenda, D.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “On the longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields, Springer Series in Optical Sciences (Springer, 2009).

Martínez-Herrero, R.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “On the longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[Crossref]

Mejías, P. M.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “On the longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[Crossref]

R. Martinez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields, Springer Series in Optical Sciences (Springer, 2009).

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Mukunda, N.

Nalimov, A. G.

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Longitudinal component of the Poynting vector of tightly focused cylindrical vector beam,” J. Phys. Conf. Ser. 1135, 012064 (2018).
[Crossref]

Ndagano, B.

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608 (2005).
[Crossref]

A. V. Nesterov and V. G. Niziev, “Propagation features of beams with axially symmetric polarization,” J. Opt. B 3, 215–219 (2001).
[Crossref]

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608 (2005).
[Crossref]

A. V. Nesterov and V. G. Niziev, “Propagation features of beams with axially symmetric polarization,” J. Opt. B 3, 215–219 (2001).
[Crossref]

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

Piquero, G.

R. Martinez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields, Springer Series in Optical Sciences (Springer, 2009).

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Rosales-Guzmán, C.

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

Santarsiero, M.

Silberberg, Y.

U. Levy, Y. Silberberg, and N. Davidson, “Mathematics of vectorial Gaussian beams,” Adv. Opt. Photonics 11, 828–892 (2019).
[Crossref]

U. Levy and Y. Silberberg, “Weakly diverging to tightly focused Gaussian beams: continuation—symmetric beams,” J. Opt. Soc. Am. A 34, 331–335 (2017).
[Crossref]

Simon, R.

Stafeev, S. S.

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Longitudinal component of the Poynting vector of tightly focused cylindrical vector beam,” J. Phys. Conf. Ser. 1135, 012064 (2018).
[Crossref]

Sudarshant, E. C. G.

Woan, G.

G. Woan, The Cambridge Handbook of Physics Formulas (Cambridge University, 2010).

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Zheng, Y.

Zhu, Y.

Adv. Opt. Photonics (1)

U. Levy, Y. Silberberg, and N. Davidson, “Mathematics of vectorial Gaussian beams,” Adv. Opt. Photonics 11, 828–892 (2019).
[Crossref]

Appl. Opt. (1)

Eur. J. Phys. (1)

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “On the longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[Crossref]

J. Opt. (1)

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

J. Opt. B (1)

A. V. Nesterov and V. G. Niziev, “Propagation features of beams with axially symmetric polarization,” J. Opt. B 3, 215–219 (2001).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Phys. Conf. Ser. (1)

S. S. Stafeev, A. G. Nalimov, and V. V. Kotlyar, “Longitudinal component of the Poynting vector of tightly focused cylindrical vector beam,” J. Phys. Conf. Ser. 1135, 012064 (2018).
[Crossref]

J. Phys. D (1)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

Phys. Rev. E (1)

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608 (2005).
[Crossref]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Proc. R. Soc. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Other (7)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

R. Martinez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields, Springer Series in Optical Sciences (Springer, 2009).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

“Lasers and laser-related equipment–Test methods for laser beam widths, divergence angles and beam propagation ratios,” ISO 11146:2005 (2005).

G. Woan, The Cambridge Handbook of Physics Formulas (Cambridge University, 2010).

Massachusetts Institute of Technology, http://web.mit.edu/viz/EM/visualizations/notes/modules/guide13.pdf .

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics (Butterworth-Heinemann, 1994), Vol. 2.

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Figures (5)

Fig. 1.
Fig. 1. Problem of beam divergence formation. The beam incident on the lens has a radius R and a divergence $\alpha$; ${f}$ is the focal length of the lens.
Fig. 2.
Fig. 2. Decomposition of the Poynting vector into $r$ and $z$ components. The LG ring mode passes through a waist. In the waist area, the standing-wave formation zone is shown schematically.
Fig. 3.
Fig. 3. Graphical explanation of the calculation of the interference of two narrow coherent beams. The Poynting vector and the vector of the electric and magnetic fields are shown here. The local coordinate system is ${p},\;{ s},\;{ q}$; the cylinder coordinates are ${r},\;\varphi ,\;{z};\;{tg}\alpha = {R}/{ f}$.
Fig. 4.
Fig. 4. Distribution of the real part (curves) and imaginary part (filled areas) [Eq. (10)] of longitudinal field [Eq. (A2)] over the radius at different distances from the waist (parameter ${z}/{{z}_0}$).
Fig. 5.
Fig. 5. Components of the electric and magnetic fields of a Gaussian beam according to Eq. (11).

Equations (24)

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S r = E φ × H z + E z × H φ = ( E φ H z E z H φ ) e r .
E e i ( k z ω t ) + e i ( k z ω t ) = 2 cos ( k z ) e i ω t , H e i ( k z ω t ) e i ( k z ω t ) = 2 i sin ( k z ) e i ω t .
E s = E 0 e i ( kq ω t ) u s = E 0 u s e i k r r e i ( k z z ω t ) .
E 1 , 2 = ± E 0 u φ e ± i k r r e i ω t ; E 1 + 2 = E 0 u φ ( e i k r r e i k r r ) e i ω t = 2 i E 0 u φ sin ( k r r ) e i ω t ;
H p = ( H 0 u r cos α + H 0 u z sin α ) e i k r r e i ( k z z ω t ) ; H 1 , 2 = cos α ( ± H 0 u r + H 0 u z t g α ) e ± i k r r e i ω t ;
H 1 + 2 = 2 i H 0 cos α [ u r sin ( k r r ) u z i t g α cos ( k r r ) ] e i ω t .
S 1 + 2 = E 1 + 2 × H 1 + 2 = 4 E 0 H 0 cos α { [ u φ sin ( k r r ) ] × [ u r sin ( k r r ) u z i λ π w 0 cos ( k r r ) ] } e 2 i ω t ,
S z 1 + 2 = 4 E 0 H 0 u z cos α sin 2 ( k r r ) e 2 i ω t ; S r 1 + 2 = 4 E 0 H 0 u r i λ π w 0 cos α sin ( k r r ) cos ( k r r ) e 2 i ω t .
S z 1 + 2 ¯ = 2 E 0 H 0 cos α sin 2 ( k r r ) ; S r 1 + 2 ¯ = 0.
H z Re ZR 2 exp ( R 2 ) ; H z Im ( 1 R 2 ) exp ( R 2 ) .
E ~ x = g ; E ~ y = 0 ; E ~ z = 2 i ϑ r ¯ g cos φ H ~ x = 0 ; H ~ y = g ; H ~ z = 2 i ϑ r ¯ g sin φ , g = ( 2 π 1 w 0 ) exp ( x 2 + y 2 w 0 2 ) ; ϑ = 1 k 1 w 0 ; r ¯ = r w 0 = x 2 + y 2 w 0 ,
E ~ φ = g sin φ ; E ~ r = g cos φ ; E ~ z = 2 i ϑ r ¯ g cos φ , H ~ φ = g cos φ ; H ~ r = g sin φ ; H ~ z = 2 i ϑ r ¯ g sin φ .
Re E r Re H φ = g 2 ( cos φ ) 2 ( cos ω t ) 2 , Re E φ Re H r = g 2 ( sin φ ) 2 ( cos ω t ) 2 .
Re E φ Re H z = ϑ r ¯ g 2 ( sin φ ) 2 sin 2 ω t , Re E z Re H φ = ϑ r ¯ g 2 ( cos φ ) 2 sin 2 ω t .
E φ = E φ ( R ) e i θ e i ( k z ω t ) ; E φ ( R ) = ( 2 π 1 w ) R exp ( R 2 ) , θ = 2 arctg Z 2 Z z 0 2 w 0 2 Z R 2 ; R = r / w ; w 2 = w 0 2 ( 1 + Z 2 ) ; Z = z / z 0 ; z 0 = π w 0 2 λ .
H E φ = ( 2 π 1 w ) R exp ( R 2 ) ,
H z = λ π w ( 2 π 1 w ) [ ZR 2 + i ( 1 R 2 ) ] exp ( R 2 ) .
H E φ = ( 2 π 1 w 0 ) r w 0 exp ( r 2 w 0 2 ) ,
H z = i λ π w 0 ( 2 π 1 w 0 ) ( 1 r 2 w 0 2 ) exp ( r 2 w 0 2 ) .
E × H = { E r H φ E φ H r } e z + { E φ H z E z H φ } e r + { E z H r E r H z } e φ .
E = a cos ( ω t ) + b sin ( ω t ) ; H = c cos ( ω t ) + d sin ( ω t ) ,
Re E = 1 2 ( E + E ) , e i ω t = cos ω t i sin ω t .
Re E Re H = 1 4 ( E ~ e i ω t + E ~ e i ω t ) ( H ~ e i ω t + H ~ e i ω t ) .
Re E Re H ¯ = 1 4 ( E ~ H ~ + E ~ H ~ ) = 1 2 ( a c + b d ) .

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