Abstract

In our previous article [J. Opt. Soc. Am. A 32, 1236 (2015) [CrossRef]  ] there is an issue concerning the comparison of plane wave spectrum solutions of paraxial and Helmholtz equations. We compared the angular plane wave spectrum of Helmholtz solutions with the plane wave spectrum of the paraxial solutions in terms of normalized projections of paraxial wave vectors. We show that the proper comparison of plane wave spectra must be done in terms of angles. The results presented in our previous work are corrected accordingly. The most important change is that Wünsche’s T2 operator leads to a valid method.

© 2016 Optical Society of America

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References

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  1. R. Mahillo-Isla and M. J. González-Morales, “Angular spectral framework to test full corrections of paraxial solutions,” J. Opt. Soc. Am. A 32, 1236–1242 (2015).
    [Crossref]
  2. M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [Crossref]
  3. A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation. An application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [Crossref]
  4. A. M. Targirdzhanov and A. P. Kiselev, “Complexified spherical waves and their sources. A review,” Opt. Spectrosc. 119, 257–267 (2015).
    [Crossref]
  5. R. Mahillo-Isla, M. J. González-Morales, and C. Dehesa-Martnez, “Transition between free-space Helmholtz equation solutions with plane sources and parabolic wave equation solutions,” J. Opt. Soc. Am. A 28, 1003–1006 (2011).
    [Crossref]

2015 (2)

R. Mahillo-Isla and M. J. González-Morales, “Angular spectral framework to test full corrections of paraxial solutions,” J. Opt. Soc. Am. A 32, 1236–1242 (2015).
[Crossref]

A. M. Targirdzhanov and A. P. Kiselev, “Complexified spherical waves and their sources. A review,” Opt. Spectrosc. 119, 257–267 (2015).
[Crossref]

2011 (1)

1992 (1)

1981 (1)

M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[Crossref]

Bélanger, P.-A.

M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[Crossref]

Couture, M.

M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[Crossref]

Dehesa-Martnez, C.

González-Morales, M. J.

Kiselev, A. P.

A. M. Targirdzhanov and A. P. Kiselev, “Complexified spherical waves and their sources. A review,” Opt. Spectrosc. 119, 257–267 (2015).
[Crossref]

Mahillo-Isla, R.

Targirdzhanov, A. M.

A. M. Targirdzhanov and A. P. Kiselev, “Complexified spherical waves and their sources. A review,” Opt. Spectrosc. 119, 257–267 (2015).
[Crossref]

Wünsche, A.

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Equations (17)

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2 ψ P + 2 i k z ψ P + 2    k 2 ψ P = 0 ,
ψ P ( r ) = i k 8 π 2 C q q d q C β d β F ( q , β ) exp [ i k p ( q , β ) · r ] ,
q = N ( α ) sin α ,
N ( α ) = [ 2 2 cos α ( 1 + sin 2 α ) 1 / 2 ] 1 / 2 sin 2 α .
ψ P ( r ) = i k 8 π 2 C α sf ( α ) d α C β d β F ( α , β ) exp ( i k p · r ) ,
sf ( α ) = 2 2 cos α ( 1 + sin 2 α ) 1 / 2 sin 3 α ( 1 + sin 2 α ) 1 / 2 ,
k p = k N ( α ) [ n ( β ) sin α + e z cos α ] ,
2 sin γ 2 = q ,
γ = 2 arcsin N ( α ) sin α 2 .
F H ( α , β ) = F ( 2 arcsin N ( α ) sin α 2 , β ) .
F ( α , β ) F H ( α , β ) = 1 8 α F ( α , β ) | α = 0 α 3 + O ( α 4 ) .
sin γ = q .
γ = arcsin [ N ( α ) sin α ] .
F H ( α , β ) = cos α F { arcsin [ N ( α ) sin α ] , β } .
F ( α , β ) F H ( α , β ) = 1 2 F ( α , β ) | α = 0 α 2 + 1 2 α F ( α , β ) | α = 0 α 3 + O ( α 4 ) .
F H ( α , β ) = F { arcsin [ N ( α ) sin α ] , β } .
F ( α , β ) F H ( α , β ) = 1 8 α F ( α , β ) | α = 0 α 5 + O ( α 6 ) ,

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