Abstract

Focused electromagnetic beams are frequently modeled by either an angular spectrum of plane waves or a partial-wave sum of spherical multipole waves. The connection between these two beam models is explored here. The partial-wave expansion of an angular spectrum containing evanescent components is found to possess only odd partial waves. On the other hand, the partial-wave expansion of an alternate angular spectrum constructed so as to be free of evanescent components contains all partial waves but describes a propagating beam with a small amount of standing-wave component mixed in. A procedure is described for minimizing the standing-wave component so as to more accurately model a purely forward propagating experimental beam.

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 48-51.
  2. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), pp. 561-564.
  3. C. Yeh, S. Colak, and P. Barber, "Scattering of sharply focused beams by arbitrarily shaped dielectric particles: an exact solution," Appl. Opt. 21, 4426-4433 (1982).
    [Crossref] [PubMed]
  4. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 121.
  5. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
    [Crossref]
  6. G. Gouesbet, B. Maheu, and G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1443 (1988).
    [Crossref]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 122-125.
  8. D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Pearson Prentice Hall, 2005), p. 418.
  9. K. Gottfried and T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed. (Springer, 2003), pp. 84-85, Eqs. (354) and (355).
  10. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965), p. 761, Eq. (6.738.1).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 43-45.
  12. O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962), App. B, Eqs. (B.1) and (B.2).
  13. A. Messiah, Quantum Mechanics (Wiley, 1966), Vol. 1, p. 497, Eqs. (B.99)-(B.101).
  14. Ref. , p. 684, Eq. (6.561.14).
  15. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1965), p. 545, Eq. (10.33b).
  16. A. Messiah, Quantum Mechanics (Wiley, 1966), Vol. 2, pp. 1057-1059, Eqs. (C.16), (C.22), (C.23a), and (C.23b).
  17. J. A. Lock, "Excitation of morphology-dependent resonances and van de Hulst's localization principle," Opt. Lett. 24, 427-429 (1999).
    [Crossref]
  18. A. Messiah, Quantum Mechanics (Wiley, 1966), Vol. 1, p. 497, Eq. (B105).
  19. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1965), pp. 629-630, Eqs. (11.174) and (11.175).
  20. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965), p. 716, Eq. (6.631.1).
  21. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1965), p. 753, Eq. (13.134).
  22. B. Maheu, G. Grehan, and G. Gousebet, "Ray localization in Gaussian beams," Opt. Commun. 70, 259-262 (1989).
    [Crossref]
  23. G. Gouesbet, J. A. Lock, and G. Grehan, "Partial-wave representations of laser beams for use in light-scattering calculations," Appl. Opt. 34, 2133-2143 (1995).
    [Crossref] [PubMed]
  24. J. A. Lock and J. T. Hodges, "Far-field scattering of an axisymmetric laser beam of arbitrary profile by an on-axis spherical particle," Appl. Opt. 35, 4283-4290 (1996).
    [Crossref] [PubMed]
  25. J. A. Lock and J. T. Hodges, "Far-field scattering of a non-Gaussian off-axis axisymmetric laser beam by a spherical particle," Appl. Opt. 35, 6605-6616 (1996).
    [Crossref] [PubMed]
  26. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 208-209.

1999 (1)

1996 (2)

1995 (1)

1989 (1)

B. Maheu, G. Grehan, and G. Gousebet, "Ray localization in Gaussian beams," Opt. Commun. 70, 259-262 (1989).
[Crossref]

1988 (2)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[Crossref]

G. Gouesbet, B. Maheu, and G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1443 (1988).
[Crossref]

1982 (1)

1962 (1)

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962), App. B, Eqs. (B.1) and (B.2).

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[Crossref]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1965), p. 545, Eq. (10.33b).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1965), pp. 629-630, Eqs. (11.174) and (11.175).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1965), p. 753, Eq. (13.134).

Barber, P.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), pp. 561-564.

Colak, S.

Cruzan, O. R.

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962), App. B, Eqs. (B.1) and (B.2).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 48-51.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 43-45.

Gottfried, K.

K. Gottfried and T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed. (Springer, 2003), pp. 84-85, Eqs. (354) and (355).

Gouesbet, G.

Gousebet, G.

B. Maheu, G. Grehan, and G. Gousebet, "Ray localization in Gaussian beams," Opt. Commun. 70, 259-262 (1989).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965), p. 716, Eq. (6.631.1).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965), p. 761, Eq. (6.738.1).

Grehan, G.

Griffiths, D. J.

D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Pearson Prentice Hall, 2005), p. 418.

Hodges, J. T.

Lock, J. A.

Maheu, B.

Messiah, A.

A. Messiah, Quantum Mechanics (Wiley, 1966), Vol. 1, p. 497, Eqs. (B.99)-(B.101).

A. Messiah, Quantum Mechanics (Wiley, 1966), Vol. 2, pp. 1057-1059, Eqs. (C.16), (C.22), (C.23a), and (C.23b).

A. Messiah, Quantum Mechanics (Wiley, 1966), Vol. 1, p. 497, Eq. (B105).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965), p. 761, Eq. (6.738.1).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965), p. 716, Eq. (6.631.1).

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 122-125.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 121.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 208-209.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), pp. 561-564.

Yan, T.-M.

K. Gottfried and T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed. (Springer, 2003), pp. 84-85, Eqs. (354) and (355).

Yeh, C.

Appl. Opt. (4)

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[Crossref]

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

Opt. Commun. (1)

B. Maheu, G. Grehan, and G. Gousebet, "Ray localization in Gaussian beams," Opt. Commun. 70, 259-262 (1989).
[Crossref]

Opt. Lett. (1)

Q. Appl. Math. (1)

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962), App. B, Eqs. (B.1) and (B.2).

Other (17)

A. Messiah, Quantum Mechanics (Wiley, 1966), Vol. 1, p. 497, Eqs. (B.99)-(B.101).

Ref. , p. 684, Eq. (6.561.14).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1965), p. 545, Eq. (10.33b).

A. Messiah, Quantum Mechanics (Wiley, 1966), Vol. 2, pp. 1057-1059, Eqs. (C.16), (C.22), (C.23a), and (C.23b).

A. Messiah, Quantum Mechanics (Wiley, 1966), Vol. 1, p. 497, Eq. (B105).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1965), pp. 629-630, Eqs. (11.174) and (11.175).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965), p. 716, Eq. (6.631.1).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1965), p. 753, Eq. (13.134).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 48-51.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), pp. 561-564.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 121.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 122-125.

D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Pearson Prentice Hall, 2005), p. 418.

K. Gottfried and T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed. (Springer, 2003), pp. 84-85, Eqs. (354) and (355).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965), p. 761, Eq. (6.738.1).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 43-45.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 208-209.

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Equations (64)

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( 2 + k 2 ) ψ beam ( r ) = 0 .
r = x u x + y u y + z u z
ψ beam ( x , y , z ) = ( 1 4 π 2 ) d k x d k y A ( k x , k y ) exp [ i ( k x x + k y y ) ] exp [ i ( k 2 k x 2 k y 2 ) 1 2 z ] .
A ( k x , k y ) = d x d y ψ beam ( x , y , 0 ) exp [ i ( k x x + k y y ) ] .
A ( ρ k ) = 2 π 0 ρ d ρ ψ beam ( ρ , 0 ) J 0 ( ρ ρ k ) .
r a = x a u x + y a u y
r = r r a = x u x + y u y + z u z
ψ beam ( x , y , z ) = d x a d y a p ( x x a , y y a , z ) ψ beam ( x a , y a , 0 ) ,
p ( x , y , z ) = ( 1 4 π 2 ) d k x d k y exp [ i ( k x x + k y y ) ] exp [ i ( k 2 k x 2 k y 2 ) 1 2 z ] .
p ( x , y , 0 ) = δ ( x ) δ ( y ) .
p ( ρ , ϕ , z ) = ( 1 2 π ) 0 ρ k d ρ k J 0 ( ρ ρ k ) exp [ i ( k 2 ρ k 2 ) 1 2 z ]
= ( 1 2 π ) 0 k w d w J 0 [ ρ ( k 2 w 2 ) 1 2 ] exp ( i z w ) + ( 1 2 π ) 0 w d w J 0 [ ρ ( k 2 + w 2 ) 1 2 ] exp ( z w ) .
p ( r , θ , ϕ ) = ( i k 2 2 π ) h 1 ( 1 ) ( k r ) P 1 ( cos θ ) ,
r = ( ρ 2 + z 2 ) 1 2 ,
cos θ = z r .
p ( r , θ , ϕ ) = ( i k 2 2 π ) h 1 ( 2 ) ( k r ) P 1 ( cos θ ) .
h 1 ( 1 ) ( k r ) P 1 ( cos θ ) = v n , m [ ( n m ) ! ( n + m 1 ) ! ] { [ ( n m + 1 ) ( n + m ) ] j n ( k r < ) h n + 1 ( 1 ) ( k r > ) P n m ( cos θ < ) P n + 1 m ( cos θ > ) exp ( i m ϕ < ) exp ( i m ϕ > ) j n ( k r < ) h n 1 ( 1 ) ( k r > ) P n m ( cos θ < ) P n 1 m ( cos θ > ) exp ( i m ϕ < ) exp ( i m ϕ > ) } ,
( r > , θ > , ϕ > ) = { ( r , θ , ϕ ) , if r > a ( r a , θ a , ϕ a ) , if r < a } ,
( r < , θ < , ϕ < ) = { ( r a , θ a , ϕ a ) , if r > a ( r , θ , ϕ ) , if r < a } ,
v = { 1 , if r > a 1 , if r < a } .
ψ beam ( r , θ , ϕ ) = n = odd i n ( 2 n + 1 ) [ ( n ) ! ! ( n 1 ) ! ! ] [ h n ( 1 ) ( k r ) P n ( cos θ ) 0 r k d ρ a j n ( k ρ a ) ψ beam ( ρ a , 0 ) + j n ( k r ) P n ( cos θ ) r k d ρ a h n ( 1 ) ( k ρ a ) ψ beam ( ρ a , 0 ) ] .
0 k d ρ a j n ( k ρ a ) = { ( n 1 ) ! ! ( n ) ! ! , if n = odd ( π 2 ) ( n 1 ) ! ! ( n ) ! ! , if n = even } ,
ψ beam ( r , θ , ϕ ) = n odd i n ( 2 n + 1 ) [ α n h n ( 1 ) ( k r ) + i β n ( k r ) j n ( k r ) i γ n ( k r ) n n ( k r ) ] P n ( cos θ ) ,
α n = 0 k d ρ a j n ( k ρ a ) ψ beam ( ρ a , 0 ) 0 k d ρ a j n ( k ρ a ) ,
β n ( k r ) = r k d ρ a n n ( k ρ a ) ψ beam ( ρ a , 0 ) 0 k d ρ a j n ( k ρ a ) ,
γ n ( k r ) = r k d ρ a j n ( k ρ a ) ψ beam ( ρ a , 0 ) 0 k d ρ a j n ( k ρ a ) .
s 1 k w 0 1 .
1 w 0 ρ k k ,
p ( x , y , z ) = ( i k 2 π z ) exp ( i k z ) exp [ i k ( x 2 + y 2 ) 2 z ] ,
k x 2 + k y 2 + k z 2 k 2 + ( k x 2 + k y 2 ) 2 4 k 2 .
ξ n ( k r ) = α n γ n ( k r ) = 0 r k d ρ a j n ( k ρ a ) ψ beam ( ρ a , 0 ) 0 k d ρ a j n ( k ρ a ) ,
S ( θ ) = { 1 , for 0 θ < π 2 1 , for π 2 < θ π } ,
ϵ n = { 1 , for n = odd 0 , for n = even } ,
ψ beam ( r , θ , ϕ ) = n = 0 ϵ n i n ( 2 n + 1 ) { α n j n ( k r ) P n ( cos θ ) + i [ β n ( k r ) j n ( k r ) + ξ n ( k r ) n n ( k r ) ] S ( θ ) P n ( cos θ ) } .
ψ scatt ( r , θ , ϕ ) = n = 0 i n ( 2 n + 1 ) b n h n ( 1 ) ( k r ) P n ( cos θ ) ,
ψ interior ( r , θ , ϕ ) = n = 0 i n ( 2 n + 1 ) d n j n ( N k r ) P n ( cos θ ) ,
b n = ϵ n α n M n ( M n + i D n ) + p = 0 ϵ p ( p + 1 2 ) i p + 1 n I n p [ β p ( k a ) M p + ξ p ( k a ) D p ] ( M n + i D n ) ,
d n = ϵ n α n [ i ( k a ) 2 ] ( M n + i D n ) + p = 0 ϵ p ( p + 1 2 ) i p + 1 n I n p { [ i β p ( k a ) + ξ p ( k a ) ] ( k a ) 2 } ( M n + i D n ) ,
M n = N j n ( k a ) j n ( N k a ) j n ( k a ) j n ( N k a ) ,
D n = N n n ( k a ) j n ( N k a ) n n ( k a ) j n ( N k a ) ,
I n p = 0 π sin θ d θ S ( θ ) P n ( cos θ ) P p ( cos θ ) .
ψ beam ( r , θ , ϕ ) = ( 1 4 π ) 0 π sin θ k d θ k 0 2 π d ϕ k A ( θ k , ϕ k ) exp ( i k r ) ,
A ( θ k , ϕ k ) = n , m α n m Y n m ( θ k , ϕ k ) ,
α n m = 0 π sin θ k d θ k 0 2 π d ϕ k A ( θ k , ϕ k ) Y n m ( θ k , ϕ k ) .
ψ beam ( r , θ , ϕ ) = n , m i n α n m j n ( k r ) Y n m ( θ , ϕ ) .
α n m = [ 4 π ( 2 n + 1 ) ] 1 2 α n δ m , 0 ,
ψ beam ( r , θ , ϕ ) = n = 0 i n ( 2 n + 1 ) α n j n ( k r ) P n ( cos θ ) .
A ( θ k ) = n = 0 ( 2 n + 1 ) α n P n ( cos θ k ) ,
α n = ( 1 2 ) 0 π sin θ k d θ k A ( θ k ) P n ( cos θ k ) .
b n = α n M n ( M n + i D n ) ,
d n = α n [ 1 ( k a ) 2 ] ( M n + i D n ) .
ψ beam ( ρ , 0 ) = n even ( 2 n + 1 ) [ ( n 1 ) ! ! ( n ) ! ! ] α n j n ( k r )
α n = 0 k d ρ j n ( k ρ ) ψ beam ( ρ , 0 ) 0 k d ρ j n ( k ρ ) ,
ψ beam ( r , θ , ϕ ) = n = 0 i n ( n + 1 2 ) α n [ h n ( 1 ) ( k r ) + h n ( 2 ) ( k r ) ] P n ( cos θ ) .
ψ beam ( z ) ( i k z ) n = 0 ( n + 1 2 ) α n [ exp ( i k z ) ( 1 ) n exp ( i k z ) ] .
n even ( n + 1 2 ) α n = n odd ( n + 1 2 ) α n ,
α n = 1 2 ( α n + 1 + α n 1 ) + [ 1 2 ( α n + 1 α n 1 ) ( n + 1 2 ) ] ,
α n = 0 k d ρ j n ( k ρ ) ψ beam ( ρ , 0 ) 0 k d ρ j n ( k ρ )
ψ beam ( ρ , 0 ) = exp ( ρ 2 w 0 2 )
α n = t n [ 2 ( n + 2 ) 2 ( n ) ! ! ] M [ ( n + 1 ) 2 , n + 3 2 , 1 4 s 2 ] [ π 1 2 ( 2 s ) n + 1 ( 2 n + 1 ) ! ! ] ,
t n = { 1 , if n = even ( 2 π ) 1 2 , if n = odd } .
k ρ n + 1 2 .
α n ψ beam [ ( n + 1 2 ) k , 0 ] ,
α n exp [ s 2 ( n + 1 2 ) 2 ] .

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