Abstract

A procedure is proposed for generating rigorous closed-form orthonormal bases for the expansion of strongly focused (high-numerical-aperture), monochromatic, electromagnetic fields. The performance of three such bases is tested in terms of a parameter that determines their directional spread, for several truncation orders. Simple example fields corresponding to beams with differing polarizations focused by a thin lens are expanded in terms of these bases.

© 2009 Optical Society of America

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References

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  1. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358-379 (1959).
    [CrossRef]
  2. G. Rodríguez-Morales and S. Chávez-Cerda, “Exact nonparaxial beams of the scalar Helmholtz equation,” Opt. Lett. 29, 430-432 (2004).
    [CrossRef] [PubMed]
  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. N. J. Moore and M. A. Alonso, “Bases for the description of monochromatic, strongly focused, scalar fields,” J. Opt. Soc. Am. A 26, 1754-1761 (2009).
    [CrossRef]
  5. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), pp. 414-416.
  6. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234-244 (1974).
    [CrossRef]
  7. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 430-431.
  8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), pp. 331-341.
  9. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543-1545 (1999).
    [CrossRef]
  10. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381-1386 (1999).
    [CrossRef]
  11. J. C. Petruccelli, N. J. Moore, and M. A. Alonso, “Two methods for modeling the propagation of the coherence and polarization properties of nonparaxial fields,” submitted to Opt. Commun. (2009).

2009 (1)

2006 (1)

2004 (1)

1999 (2)

1974 (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234-244 (1974).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

Abramowitz, M.

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

Alonso, M. A.

Borghi, R.

Chávez-Cerda, S.

Devaney, A. J.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234-244 (1974).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 430-431.

Moore, N. J.

N. J. Moore and M. A. Alonso, “Bases for the description of monochromatic, strongly focused, scalar fields,” J. Opt. Soc. Am. A 26, 1754-1761 (2009).
[CrossRef]

J. C. Petruccelli, N. J. Moore, and M. A. Alonso, “Two methods for modeling the propagation of the coherence and polarization properties of nonparaxial fields,” submitted to Opt. Commun. (2009).

Petruccelli, J. C.

J. C. Petruccelli, N. J. Moore, and M. A. Alonso, “Two methods for modeling the propagation of the coherence and polarization properties of nonparaxial fields,” submitted to Opt. Commun. (2009).

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

Rodríguez-Morales, G.

Saghafi, S.

Santarsiero, M.

Sheppard, C. J. R.

Stegun, I. A.

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

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), pp. 414-416.

Wolf, E.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234-244 (1974).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

J. Math. Phys. (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234-244 (1974).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (2)

Proc. Roy. Soc. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

Other (4)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), pp. 414-416.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 430-431.

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

J. C. Petruccelli, N. J. Moore, and M. A. Alonso, “Two methods for modeling the propagation of the coherence and polarization properties of nonparaxial fields,” submitted to Opt. Commun. (2009).

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Figures (5)

Fig. 1
Fig. 1

Polarization vectors V u and u × V u of the plane-wave amplitudes over the sphere associated with (a) the TE/TM basis and (b) the quasi-linear basis.

Fig. 2
Fig. 2

RMS truncation error as a function of k q for expansions using (a) the L-type basis, (b) the TE/TM basis, and (c) the quasi-linear basis, for n max from 0 to 6, of the plane-wave amplitude of a radially polarized donut beam (left column), an azimuthally polarized donut beam (middle column), and a linearly polarized Gaussian beam (right column), focused by a thin lens with Δ=0.5.

Fig. 3
Fig. 3

Magnitude over the x - z plane of a radially polarized field focused by a thin lens with Δ = 0.5 reconstructed using the L-type basis. This reconstruction needs only the m = 0 terms and is truncated, in this case, at n max = 4 ; k q = 15 (which is approximately the optimal value for this truncation order) is used for this reconstruction.

Fig. 4
Fig. 4

Magnitude over the x - z plane of an azimuthally polarized field focused by a thin lens with Δ = 0.5 reconstructed using the L-type basis. This reconstruction needs only the m = 0 terms and in this case is truncated at n max = 4 ; k q = 15 (which is approximately the optimal value for this truncation order) is used for this reconstruction.

Fig. 5
Fig. 5

Magnitude over (a) the x - z plane (b) the y - z plane and (c) the x - y plane of a linearly-polarized Gaussian beam focused by a thin lens with Δ = 0.5 reconstructed using the quasi-linear basis. This reconstruction is truncated at n max = 4 and m max = 2 ; k q = 15 (which is approximately the optimal value for this truncation order) is used for this reconstruction. Note that the focal spot is wider in the x-direction than in the y-direction.

Equations (66)

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2 E ( r ) + k 2 E ( r ) = 0 ,
E ( r ) = 0 .
E ( r ) = 4 π A ( u ) exp ( i k u r ) d Ω ,
Λ l m ( I ) ( r ) = 4 π Z l m ( θ , φ ) exp ( i k r u ) d Ω ,
Λ l m ( II ) ( r ) = 4 π Y l m ( θ , φ ) exp ( i k r u ) d Ω ,
Y l m ( θ , φ ) = 1 l ( l + 1 ) L u Y l m ( θ , φ ) ,
Z l m ( θ , φ ) = u × Y l m ( θ , φ ) ,
L u = i u × Ω = i θ ̂ sin θ φ i φ ̂ θ ,
Y l m ( θ , φ ) = σ m exp ( i m φ ) 2 π h ¯ l ( m ) P l ( | m | ) ( cos θ ) ,
h ¯ l ( m ) = 2 2 l + 1 ( l + | m | ) ! ( l | m | ) ! ,
σ m = { 1 , m 0 , ( 1 ) m , else . }
4 π Y l m * ( θ , φ ) Y l m ( θ , φ ) d Ω = δ l , l δ m , m ,
4 π Z l m * ( θ , φ ) Z l m ( θ , φ ) d Ω = δ l , l δ m , m ,
4 π Z l m * ( θ , φ ) Y l m ( θ , φ ) d Ω = 0 ,
Λ l m ( I ) ( r ) = i × L r Λ l m ( r ) k l ( l + 1 ) = 4 π i l { l ( l + 1 ) k r j l ( k r ) Y l m ( θ r , φ r ) r ̂ i 2 [ j l ( k r ) k r + j l 1 ( k r ) j l + 1 ( k r ) ] Z l m ( θ r , φ r ) } ,
Λ l m ( II ) ( r ) = L r Λ l m ( r ) l ( l + 1 ) = 4 π i l j l ( k r ) Y l m ( θ r , φ r ) ,
Λ l m ( r ) = 4 π Y l m ( θ , φ ) exp ( i k r u ) d Ω = 4 π i l j l ( k r ) Y l m ( θ r , φ r ) .
E ( r ) = l = 1 m = l l [ η l m ( I ) Λ l m ( I ) ( r ) + η l m ( II ) Λ l m ( II ) ( r ) ] ,
η l m ( I ) = 4 π Z l m * ( θ , φ ) A ( u ) d Ω ,
η l m ( II ) = 4 π Y l m * ( θ , φ ) A ( u ) d Ω .
Y n m ( θ , φ ; k q ) = V u Y n m ( θ , φ ; k q ) ,
Z n m ( θ , φ ; k q ) = u × V u Y n m ( θ , φ ; k q ) ,
Y n m ( θ , φ ; k q ) = exp ( i m φ ) 2 π h n ( m ) ( k q ) sin | m | θ Q n ( m ) ( cos θ ; k q ) exp ( k q cos θ ) .
4 π Y n m * ( θ , φ ; k q ) Z n m ( θ , φ ; k q ) d Ω = 0 .
4 π Y n m * ( θ , φ ; k q ) Y n m ( θ , φ ; k q ) d Ω = 4 π Z n m * ( θ , φ ; k q ) Z n m ( θ , φ ; k q ) d Ω .
4 π Y n m * ( θ , φ ; k q ) Y n m ( θ , φ ; k q ) d Ω = δ n , n δ m , m
1 1 Q n ( m ) ( u z ; k q ) w ̂ ( m ) ( u z ; k q ) Q n ( m ) ( u z ; k q ) d u z = h n ( m ) ( k q ) δ n , n δ m , m ,
w ̂ ( m ) ( cos θ ; k q ) = sin | m | θ exp ( k q cos θ i m φ ) V u V u sin | m | θ exp ( k q cos θ + i m φ ) .
Q n ( m ) ( u z ; k q ) = n = 0 n α n , n ( m ) h n ( m ) ( k q ) h ¯ n + m ( m ) P n + | m | ( | m | ) ( u z ) ( 1 u z 2 ) | m | 2 .
L n m ( II ) ( r ; k q ) = 4 π Y n m ( θ , φ ; k q ) exp ( i k u r ) d Ω ,
= n = 0 n α n , n ( m ) 4 π exp ( i k u r ) V u Y n + m , m ( θ , φ ) exp ( k q cos θ ) d Ω = n = 0 n α n , n ( m ) 4 π exp [ i k u ( r i q z ̂ ) ] V u Y n + m , m ( θ , φ ) d Ω = n = 0 n α n , n ( m ) V r Λ n + m , m ( r i q z ̂ ) ,
V u = V u + exp ( k q cos θ ) [ V u exp ( k q cos θ ) exp ( k q cos θ ) V u ] ,
L n m ( I ) ( r ; k q ) = 4 π Z n m ( θ , φ ; k q ) exp ( i k u r ) d Ω = 4 π u × Y n m ( θ , φ ; k q ) exp ( i k u r ) d Ω = i k n = 0 n α n , n ( m ) × V r Λ n + m , m ( r i q z ̂ ) .
χ n m ( I ) = 4 π Z n m * ( θ , φ ; k q ) A ( u ) d Ω ,
χ n m ( II ) = 4 π Y n m * ( θ , φ ; k q ) A ( u ) d Ω .
E ( r ) = n = 0 m = [ χ n m ( I ) L n m ( I ) ( r ; k q ) + χ n m ( II ) L n m ( II ) ( r ; k q ) ] .
E ( r ) = n = 0 n max m = m max m max { i k [ n = n n max χ n m ( I ) α n , n ( m ) ] × V r Λ n + m , m ( r i q z ̂ ) + [ n = n n max χ n m ( II ) α n , n ( m ) ] V r Λ n + m , m ( r i q z ̂ ) } .
ε rms 2 = A 2 n = 0 n max m = m max m max [ | χ n m ( I ) | 2 + | χ n m ( II ) | 2 ] A 2 ,
A 2 = 4 π | A ( u ) | 2 d Ω .
w ̂ ( m ) ( u z ; k q ) = w 1 ( m ) ( u z ; k q ) 2 u z 2 + w 2 ( m ) ( u z ; k q ) u z + w 3 ( m ) ( u z ; k q ) ,
w 1 ( m ) ( u z ; k q ) = ( 1 u z 2 ) | m | + 1 exp ( 2 k q u z ) ,
w 2 ( m ) ( u z ; k q ) = 2 [ ( | m | + 1 ) u z k q ( 1 u z 2 ) ] ( 1 u z 2 ) | m | exp ( 2 k q u z ) ,
w 3 ( m ) ( u z ; k q ) = [ ( | m | + 2 k q u z ) ( | m | + 1 ) ( k q ) 2 ( 1 u z 2 ) ] ( 1 u z 2 ) | m | exp ( 2 k q u z ) .
L n m ( I ) ( r ; k q ) = n = 0 n α n , n ( m ) { ( n + | m | ) ( n + | m | + 1 ) Λ n + | m | , m ( I ) ( r i q z ̂ ) + q i k × [ z ̂ × Λ n + | m | , m ( r i q z ̂ ) ] } ,
L n m ( II ) ( r ; q ) = n = 0 n α n , n ( m ) [ ( n + | m | ) ( n + | m | + 1 ) Λ n + | m | , m ( II ) ( r i q z ̂ ) + q z ̂ × Λ n + | m | , m ( r i q z ̂ ) ] .
w ̂ ( m ) ( u z ; k q ) = ( 1 u z 2 ) | m | + 1 exp ( 2 k q u z ) .
V r = 1 k 2 × ( × z ̂ ) = z ̂ 1 k 2 z ,
L n m ( I ) ( r ; q ) = n = 0 n α n , n ( m ) 1 i k × [ z ̂ Λ n + | m | , m ( r i q z ̂ ) ] ,
L n m ( II ) ( r ; q ) = n = 0 n α n , n ( m ) 1 k 2 × { × [ z ̂ Λ n + | m | , m ( r i q z ̂ ) ] } .
w ̂ ( m ) ( u z ; k q ) = ( 1 + u z ) 2 ( 1 u z 2 ) | m | exp ( 2 k q u z ) .
V r = × ( × x ̂ ) k 2 + y ̂ × i k .
L n m ( I ) ( r ; q ) = n = 0 n α n , n ( m ) ( 1 k 2 × { × [ y ̂ Λ n + | m | , m ( r i q z ̂ ) ] } + 1 i k × [ x ̂ Λ n + | m | , m ( r i q z ̂ ) ] ) ,
L n m ( II ) ( r ; q ) = n = 0 n α n , n ( m ) ( 1 k 2 × { × [ x ̂ Λ n + | m | , m ( r i q z ̂ ) ] } 1 i k × [ y ̂ Λ n + | m | , m ( r i q z ̂ ) ] ) .
A ( θ , φ ) = 1 cos θ [ θ ̂ T ( θ ) E ρ ( i ) ( f tan θ , φ , 0 ) + φ ̂ T ( θ ) E φ ( i ) ( f tan θ , φ , 0 ) ] ,
T ( θ ) = 4 sin θ sin ( 3 θ 2 ) cos ( θ 2 ) , T ( θ ) = 4 sin θ cos ( θ 2 ) sin ( 3 θ 2 ) .
E ( i ) ( ρ , φ , 0 ) = E 0 ρ ̂ ρ a exp ( ρ 2 2 a 2 ) ,
E ( i ) ( ρ , φ , 0 ) = E 0 φ ̂ ρ a exp ( ρ 2 2 a 2 ) ,
E ( i ) ( ρ , φ , 0 ) = E 0 x ̂ exp ( ρ 2 2 a 2 ) ,
x ̂ = cos φ ρ ̂ sin φ φ ̂ .
a b p n ( u ) w ̂ ( u ) p n ( u ) d u = h n δ n , n ,
μ n , n = a b u n w ̂ ( u ) u n d u .
p n ( u ) = det [ μ 0 , 0 μ 1 , 0 μ n , 0 μ 0 , 1 μ 1 , 1 μ n , 1 μ 0 , n 1 μ 1 , n 1 μ n , n 1 1 u u n ] .
p n ( u ) = k n ( n ) u n + k n 1 ( n ) u n 1 + ,
k n ( n ) = det [ μ 0 , 0 μ 1 , 0 μ n 1 , 0 μ 0 , 1 μ 1 , 1 μ n 1 , 1 μ 0 , n 1 μ 1 , n 1 μ n 1 , n 1 ] ,
k n 1 ( n ) = det [ μ 0 , 0 μ 1 , 0 μ n 2 , 0 μ n , 0 μ 0 , 1 μ 1 , 1 μ n 2 , 1 μ n , 1 μ 0 , n 1 μ 1 , n 1 μ n 2 , n 1 μ n , n 1 ] .
h n = k n ( n ) k n 1 ( n 1 ) .

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