Abstract

Two bases (one biorthogonal and one orthonormal) are proposed for the expansion of strongly focused (high numerical aperture) scalar monochromatic fields. The performance of these bases is tested and compared, both with each other and with a similar basis proposed by Alonso et al. [Opt. Express 14, 6894 (2006) ]. It is found that the orthonormal basis proposed herein exhibits the lowest truncation error of these three bases for the same truncation order for the examples considered. Additionally, this basis is advantageous because it allows for the expansion of fields without rotational symmetry.

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References

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  1. A. E. Siegman, Lasers (Univ. Science Books, 1986), pp. 642-652.
  2. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 108-109.
  3. M. A. Alonso, R. Borghi, and M. Santarsiero, “New basis for rotationally symmetric nonparaxial fields in terms of spherical waves with complex foci,” Opt. Express 14, 6894-6905 (2006).
    [CrossRef] [PubMed]
  4. Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719-730 (1967).
    [CrossRef]
  5. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  6. J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40-43 (1971).
    [CrossRef]
  7. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751-760 (1976).
    [CrossRef]
  8. S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699-700 (1977).
    [CrossRef]
  9. F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
    [CrossRef]
  10. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
    [CrossRef]
  11. G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761-764 (1968).
    [CrossRef]
  12. G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697-711 (1969).
    [CrossRef]
  13. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), pp. 331-341.
  14. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed.(Cambridge U. Press, 1963), Chap. 18.6.
  15. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1753-1756 (1988).
    [CrossRef]
  16. D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401-1405 (2004).
    [CrossRef]
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), pp. 771-792.
  18. N. J. Moore and M. A. Alonso, “Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926-5933 (2008).
    [CrossRef] [PubMed]
  19. G. Szegö, Orthogonal Polynomials, 3rd ed. (American Mathematical Society, 1967), pp. 26-27.

2008

2006

2004

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401-1405 (2004).
[CrossRef]

1998

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

1988

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1753-1756 (1988).
[CrossRef]

1979

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
[CrossRef]

1977

1976

1971

J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40-43 (1971).
[CrossRef]

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

1969

1968

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761-764 (1968).
[CrossRef]

1967

Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), pp. 771-792.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), pp. 331-341.

Alonso, M. A.

Borghi, R.

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1753-1756 (1988).
[CrossRef]

Cullen, F. A. L.

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

Ding, D.

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401-1405 (2004).
[CrossRef]

Felsen, L. B.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 108-109.

Keller, J. B.

Kravtsov, Y. A.

Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

Moore, N. J.

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Santarsiero, M.

Sheppard, C. J. R.

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Sherman, G. C.

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697-711 (1969).
[CrossRef]

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761-764 (1968).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

A. E. Siegman, Lasers (Univ. Science Books, 1986), pp. 642-652.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), pp. 331-341.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), pp. 771-792.

Streifer, W.

Szegö, G.

G. Szegö, Orthogonal Polynomials, 3rd ed. (American Mathematical Society, 1967), pp. 26-27.

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed.(Cambridge U. Press, 1963), Chap. 18.6.

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1753-1756 (1988).
[CrossRef]

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed.(Cambridge U. Press, 1963), Chap. 18.6.

Yu, P. K.

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
[CrossRef]

Zhang, Y.

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401-1405 (2004).
[CrossRef]

Electron. Lett.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

J. Acoust. Soc. Am.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1753-1756 (1988).
[CrossRef]

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401-1405 (2004).
[CrossRef]

J. Opt. Soc. Am.

Opt. Express

Phys. Rev. A

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Phys. Rev. Lett.

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761-764 (1968).
[CrossRef]

Proc. R. Soc. London, Ser. A

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
[CrossRef]

Radiophys. Quantum Electron.

Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

Other

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), pp. 331-341.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed.(Cambridge U. Press, 1963), Chap. 18.6.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), pp. 771-792.

A. E. Siegman, Lasers (Univ. Science Books, 1986), pp. 642-652.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 108-109.

G. Szegö, Orthogonal Polynomials, 3rd ed. (American Mathematical Society, 1967), pp. 26-27.

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Figures (6)

Fig. 1
Fig. 1

RMS truncation error as a function of k q for expansions of a focused Gaussian beam with Δ = 0.5 and no aberrations ( α = 0 ) for the field expanded (a) with the orthogonal basis proposed in Section 4, (b) the mapping basis from [3], and (c) the biorthogonal basis proposed in Section 3. Note that the horizontal ( k q ) axis is different in each case.

Fig. 2
Fig. 2

RMS truncation error as a function of k q for expansions of a focused Gaussian beam with Δ = 0.5 and the aberration coefficient α = 10 for the field expanded (a) with the orthogonal basis proposed in Section 4 and (b) the mapping basis from [3].

Fig. 3
Fig. 3

Real and imaginary parts of the plane-wave amplitude and reconstructed plane-wave amplitude of focused Gaussian beam ( Δ = 0.5 ) with (a) no aberrations ( α = 0 ) and (b) aberration coefficient α = 10 . This reconstruction includes seven terms and is calculated at the optimal k q as found in Fig. 2 (i.e., k q = 20 ). Note that the plane-wave amplitude (and its reconstruction) are purely real when α = 0 .

Fig. 4
Fig. 4

Amplitude of a focused Gaussian beam with Δ = 0.5 reconstructed with the orthogonal basis proposed in Section 4 up to seven terms with k q = 20 for fields with (a) no aberration ( α = 0 ) and (b) aberration coefficient α = 10 .

Fig. 5
Fig. 5

RMS truncation error as a function of k q for expansions of a nonrotationally symmetric focused Gaussian beam with Δ = 0.5 and the waist variation ratio (a) δ w = 0.05 , (b) δ w = 0.1 , and (c) δ w = 0.2 , with m max = 2 and n max from 0 to 6.

Fig. 6
Fig. 6

Field amplitude at the walls of a volumetric chunk of a nonrotationally symmetric focused Gaussian beam with Δ θ = 0.5 and waist variation ratio δ w = 0.2 reconstructed using n max = 4 and m max = 2 with k q = 7.5 .

Equations (57)

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2 U ( r ) + k 2 U ( r ) = 0 .
U ( r ) = 4 π A ( u ) exp ( i k r u ) d Ω ,
Λ l m ( r ) = 4 π i l j l ( k r ) Y l m ( θ r , φ r ) ,
Y l m ( θ , φ ) = ( σ m ) m 2 l + 1 4 π ( l m ) ! ( l + m ) ! P l ( m ) ( cos θ ) exp ( i m φ ) ,
σ m = { 1 , m 0 , 1 , else . }
Λ l m ( r ) = 4 π Y l m ( θ , φ ) exp ( i k r u ) d Ω ,
A ( u ) = l = 0 m = l l c l m Y l m ( θ , φ ) ,
c l m = 4 π Y l m * ( θ , φ ) A ( u ) d Ω .
U ( r ) = l = 0 m = l l c l m Λ l m ( r ) .
A ( u ) = A 0 Y 00 ( θ , φ ) = A 0 .
U sph ( r ) = 4 π A 0 sin k r k r ,
A ( u ; r 0 ) = A 0 exp ( i k u r 0 ) and U ( r ; r 0 ) = 4 π A 0 sin k ( r r 0 ) ( r r 0 ) k ( r r 0 ) ( r r 0 ) .
U CF ( r ; ρ 0 ) = 4 π A 0 sin k ( r ρ 0 ) ( r ρ 0 ) k ( r ρ 0 ) ( r ρ 0 ) .
A CF ( u ; i q z ̂ ) = A 0 exp ( k q u z ) = A 0 exp ( k q cos θ ) .
U CF ( r ; i q z ̂ ) = 4 π A 0 sin k x 2 + y 2 + ( z i q ) 2 k x 2 + y 2 + ( z i q ) 2 .
U ( r ) = l = 0 m = l l C l m Λ l m ( r i q z ̂ ) .
f l m ( θ , φ ; q ) = Y l m ( θ , φ ) exp ( k q cos θ ) .
4 π s l m * ( θ , φ ; q ) f l m ( θ , φ ; q ) d Ω = δ l , l δ m , m ,
s l m ( θ , φ ; q ) = Y l m ( θ , φ ) exp ( k q cos θ ) .
C l m = 4 π s l m * ( θ , φ ; q ) A ( u ) d Ω .
Y n m ( θ , φ ; q ) exp ( i m φ ) sin m θ p n ( m ) ( cos θ ; k q ) exp ( k q cos θ ) .
1 1 p n ( m ) ( u z ; k q ) p n ( m ) ( u z ; k q ) ( 1 u z 2 ) m exp ( 2 k q u z ) d u z = h n ( m ) ( k q ) δ n , n ,
Y n m ( θ , φ ; q ) = σ m m exp ( i m φ ) 2 π sin m θ exp ( k q cos θ ) p n ( m ) ( cos θ ; k q ) h n ( m ) ( k q ) ,
Y n + 1 , m ( θ , φ ; q ) = [ A n ( m ) ( k q ) + B n ( m ) ( k q ) cos θ ] Y n m ( θ , φ ; q ) C n ( m ) ( k q ) Y n 1 , m ( θ , φ ; q ) ,
A n ( m ) ( k q ) = h n ( m ) ( k q ) h n + 1 ( m ) ( k q ) a n ( m ) ( k q ) ,
B n ( m ) ( k q ) = h n ( m ) ( k q ) h n + 1 ( m ) ( k q ) b n ( m ) ( k q ) ,
C n ( m ) ( k q ) = h n 1 ( m ) ( k q ) h n + 1 ( m ) ( k q ) [ b n ( m ) ( k q ) ] 2 .
Y n m ( θ , φ ; q ) 2 n ! ( k q ) m + 1 ( n + m ) ! exp ( i m φ ) 2 π p m L n m ( k q p 2 ) exp ( k q p 2 2 ) , k q 1 ,
U n m ( r ; q ) = 4 π Y n m ( θ , φ ; q ) exp ( i k u r ) d Ω .
U n + 1 , m ( r ; q ) = [ A n ( m ) ( k q ) + B n ( m ) ( k q ) i k z ] U n m ( r ; q ) C n ( m ) ( k q ) U n 1 , m ( r ; q ) .
U n m ( r ; q ) = n = 0 n α n , n ( m ) Λ n + m , m ( r i q z ̂ ) ,
α n , n ( m ) = A n 1 ( m ) ( k q ) α n 1 , n ( m ) + B n 1 ( m ) ( k q ) B n + m 1 ( m ) α n 1 , n 1 ( m ) + B n 1 ( m ) ( k q ) C n + m + 1 ( m ) B n + m + 1 ( m ) α n 1 , n + 1 ( m ) C n 1 ( m ) ( k q ) α n 2 , n ( m ) ,
B l ( m ) = ( l + 3 2 ) ( l + 1 2 ) ( l + 1 ) 2 m 2 ,
C l ( m ) = l + 3 2 l 1 2 l 2 m 2 ( l + 1 ) 2 m 2 .
( n + 2 m ) h n ( m ) ( k q ) n ( n + m + 3 / 2 ) P n + m ( u z ) ( 1 u z 2 ) m 2 ,
χ n m = 4 π Y n m * ( θ , φ ; q ) A ( θ , φ ) d Ω .
U ( r ) = n = 0 m = χ n m U n m ( r ; q ) .
U ( r ) n = 0 n max m = m max m max [ n = n n max χ n m α n , n ( m ) ] Λ n + m , m ( r i q z ̂ ) .
A ( θ ) = { exp [ tan 2 θ 2 Δ 2 + i α ( θ 4 + 2 cos θ ) ] cos θ , 0 θ π 2 , 0 , π 2 θ π , }
A ( θ , φ ) = { exp [ tan 2 θ 2 Δ 2 ( 1 + δ w cos 2 φ ) 2 ] cos θ , 0 θ π 2 0 , π 2 θ π } ,
a b p n ( u ) p k ( u ) w ( u ) d u = h n δ n , k ,
μ n = a b u n w ( u ) d u .
p n ( u ) = det [ μ 0 μ 1 μ n μ 1 μ 2 μ n + 1 μ n 1 μ n μ 2 n 1 1 u u n ] .
p n ( u ) = k n u n + k n u n 1 + ,
k n = det [ μ 0 μ 1 μ n 1 μ 1 μ 2 μ n μ n 1 μ n μ 2 n 2 ] ,
k n = det [ μ 0 μ 1 μ n 2 μ n μ 1 μ 2 μ n 1 μ n + 1 μ n 1 μ n μ 2 n 3 μ 2 n 1 ] .
p n + 1 ( u ) = [ a n + b n u ] p n ( m ) ( u ) c n p n 1 ( m ) ( u ) ,
b n = k n + 1 k n ,
a n = b n ( k n + 1 k n + 1 k n k n ) ,
c n = k n + 1 k n 1 h n [ k n ] 2 h n 1 .
a b u k p n ( u ) w ( u ) d u = 0 ,
h n = a b p n ( u ) p n ( u ) w ( u ) d u = k n k n + 1 .
c n = k n + 1 2 k n 2 = b n 2 .
μ n ( m ) = k = 0 m m ! k ! ( m k ) ! ( 1 ) k μ n + 2 k ( 0 ) .
1 1 p n ( m ) ( u z ; k q ) p n ( m ) ( u z ; k q ) ( 1 u z 2 ) m exp ( 2 k q u ) d u z = h l ( m ) ( k q ) δ n , n .
1 1 p n ( m ) p n ( m ) ( 1 u z 2 ) m exp ( 2 k q u z ) d u = exp 2 k q 2 k q 0 4 k q p n ( m ) p n ( m ) ( x k q x 2 4 k 2 q 2 ) m exp ( x ) d x .
Y n m = exp ( i m φ ) 2 π L n m [ 2 k q ( 1 cos θ ) ] h n ( m ) ( k q ) sin m θ exp ( k q cos θ ) 2 n ! ( k q ) m + 1 ( n + m ) ! exp ( i m φ ) 2 π p m L n m ( k q p 2 ) exp ( k q p 2 2 ) ,

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