Abstract

A rigorous method for transforming an electromagnetic near-field distribution to the far field is presented. We start by deriving a set of self-consistent integral equations that can be used to represent the electromagnetic field rigorously everywhere in homogeneous space apart from the closed interior of a volume encompassing all charges and sinks. The representation is derived by imposing a condition analogous to Sommerfeld’s radiation condition. We then examine the accuracy of our numerical implementation of the formula, also on a parallel computer cluster, by comparing the results with a case when the analytical solution is also available. Finally, an application example is shown for a nonanalytical case.

© 2006 Optical Society of America

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References

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  1. J. A. Stratton and L. J. Chu, 'Diffraction theory of electromagnetic waves,' Phys. Rev. 56, 99-107 (1939).
    [CrossRef]
  2. W. Hsu and R. Barakat, 'Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,' J. Opt. Soc. Am. A 11, 623-629 (1994).
    [CrossRef]
  3. P. Török, 'Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,' J. Opt. Soc. Am. A 15, 3009-3015 (1998).
    [CrossRef]
  4. P. Varga and P. Török, 'Electromagnetic focusing by a paraboloid mirror. I. Theory,' J. Opt. Soc. Am. A 17, 2081-2089 (2000).
    [CrossRef]
  5. P. Varga and P. Török, 'Electromagnetic focusing by a paraboloid mirror. II. Numerical results,' J. Opt. Soc. Am. A 17, 2090-2095 (2000).
    [CrossRef]
  6. P. Varga, 'Focusing of electromagnetic radiation by hyperboloidal and ellipsoidal lenses,' J. Opt. Soc. Am. A 19, 1658-1667 (2002).
    [CrossRef]
  7. A. Taflove, Computational Electrodynamics: The Finite Difference Time Domain Method, 1st ed. (Artech House, 1995).
  8. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).
  9. B. Baker and E. Copson, The Mathematical Theory of Huygens' Principle, 2nd ed. (Oxford U. Press, 1953).
  10. J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems (Springer-Verlag, 2001).
  11. J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  12. P. Török and C. Sheppard, High Numerical Aperture Focusing and Imaging (Hilger, to be published).
  13. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).
  14. J. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

2002

2000

1998

1994

1939

J. A. Stratton and L. J. Chu, 'Diffraction theory of electromagnetic waves,' Phys. Rev. 56, 99-107 (1939).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Baker, B.

B. Baker and E. Copson, The Mathematical Theory of Huygens' Principle, 2nd ed. (Oxford U. Press, 1953).

Barakat, R.

Chu, L. J.

J. A. Stratton and L. J. Chu, 'Diffraction theory of electromagnetic waves,' Phys. Rev. 56, 99-107 (1939).
[CrossRef]

Copson, E.

B. Baker and E. Copson, The Mathematical Theory of Huygens' Principle, 2nd ed. (Oxford U. Press, 1953).

Hsu, W.

Jackson, J.

J. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

Nédélec, J.-C.

J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems (Springer-Verlag, 2001).

Sheppard, C.

P. Török and C. Sheppard, High Numerical Aperture Focusing and Imaging (Hilger, to be published).

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Stratton, J. A.

J. A. Stratton and L. J. Chu, 'Diffraction theory of electromagnetic waves,' Phys. Rev. 56, 99-107 (1939).
[CrossRef]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite Difference Time Domain Method, 1st ed. (Artech House, 1995).

Török, P.

Varga, P.

J. Opt. Soc. Am. A

Phys. Rev.

J. A. Stratton and L. J. Chu, 'Diffraction theory of electromagnetic waves,' Phys. Rev. 56, 99-107 (1939).
[CrossRef]

Other

A. Taflove, Computational Electrodynamics: The Finite Difference Time Domain Method, 1st ed. (Artech House, 1995).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

B. Baker and E. Copson, The Mathematical Theory of Huygens' Principle, 2nd ed. (Oxford U. Press, 1953).

J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems (Springer-Verlag, 2001).

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

P. Török and C. Sheppard, High Numerical Aperture Focusing and Imaging (Hilger, to be published).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

J. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

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Figures (10)

Fig. 1
Fig. 1

Geometry of the problem showing the closed surface of the integration, the surface normal, the integral surface element, and the observation point.

Fig. 2
Fig. 2

Closed surface of integration for the first version of the Stratton–Chu formula.

Fig. 3
Fig. 3

Closed surface of integration for the second version of the Stratton–Chu formula.

Fig. 4
Fig. 4

(Color online) Graph showing aggregate error versus mesh density for a single dipole scatterer.

Fig. 5
Fig. 5

(Color online) Graph showing aggregate error versus propagation distance for a single dipole scatterer when a vertex spacing of λ 20 is used.

Fig. 6
Fig. 6

(Color online) Diagram showing the position of the three spheres in the plane z = 0 .

Fig. 7
Fig. 7

Intensity of the electric field scattered by three spheres on the FDTD surface for x- (top) and y- (lower) polarized incident waves. Images have been individually normalized.

Fig. 8
Fig. 8

Intensity of scattered field in the plane z = 2 m for (a) x-polarized incident waves and (b) y-polarized incident waves. Images have been individually normalized.

Fig. 9
Fig. 9

Timing data for parallel implementation of the Stratton–Chu code. The upper plot shows total execution time as a function of the number of processors, and the lower plot shows the average total amount of processor time per observation point as a function of the number of processors employed.

Fig. 10
Fig. 10

(Color online) Theoretical prediction for the total execution time as a function of the number of processors employed.

Equations (39)

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V B d V = S B m ̂ d S ,
V [ M × ( × N ) ] d V = S [ M × ( × N ) ] m ̂ d S .
V [ ( × M ) ( × N ) M × ( × N ) ] d V = S [ M × ( × N ) ] m ̂ d S .
V [ N × ( × M ) M × ( × N ) ] d V = S [ M × ( × N ) N × ( × M ) ] m ̂ d S ,
G a [ × ( × E ) ] E { × [ × ( G a ) ] } = k 2 G a E E [ k 2 G a + ( a G ) ] = E [ ( a G ) ] = [ ( a G ) E ] ,
V [ ( a G ) E ] d V = a S ( E m ̂ ) G d S ,
[ E × ( G × a ) ] m ̂ = a [ G × ( E × m ̂ ) ] ,
[ G a × ( × E ) ] m ̂ = a [ ( × E ) × m ̂ ] G = i ω μ a ( H × m ̂ ) G ,
S [ i ω μ ( m ̂ × H ) G + ( m ̂ × E ) × G + ( m ̂ E ) G ] d S = 0 .
G = exp ( i k r ) r ,
r = r s r p = ( x x p ) 2 + ( y y p ) 2 + ( z z p ) 2 ,
S i [ ] d S i + S ii [ ] d S ii + S iii [ ] d S iii + S iv [ ] d S iv + S v [ ] d S v = 0 .
S i [ ] d S i S iii [ ] d S iii + S v [ ] d S v = 0 ,
S i [ i ω μ ( m ̂ × H ) G + G ( i k 1 r ) ( m ̂ × E ) × m ̂ + G ( i k 1 r ) ( m ̂ E ) m ̂ ] d S i .
lim δ 0 Ω [ i ω μ ( m ̂ × H ) exp ( i k r ) r + E exp ( i k r ) r ( i k 1 r ) ] r = δ δ 2 d Ω = 4 π E .
S v [ i ω μ ( m ̂ × H ) exp ( i k r ) r + E exp ( i k r ) r ( i k 1 r ) ] d S v = S v { [ m ̂ × ( × E ) + i k E ] 1 r E 1 r 2 } exp ( i k r ) d S v ,
Ω { [ m ̂ × ( × E ) + i k E ] r E } exp ( i k r ) d Ω = Ω { [ r × ( × E ) + i k r E ] E } exp ( i k r ) d Ω ,
r E < K ,
r × ( × E ) + i k r E 0
E ( x p , y p , z p ) = 1 4 π S iii [ i ω μ ( m ̂ × H ) G + ( m ̂ × E ) × G + ( m ̂ E ) G ] d S iii .
H ( x p , y p , z p ) = 1 4 π S iii [ i ω ϵ ( m ̂ × E ) G ( m ̂ × H ) × G ( m ̂ H ) G ] d S iii .
E ( x p , y p , z p ) = 1 4 π S iii [ i ω μ ( m ̂ × H ) G + ( m ̂ × E ) × G + ( m ̂ E ) G ] d S iii ,
H ( x p , y p , z p ) = 1 4 π S iii [ i ω ϵ ( m ̂ × E ) G ( m ̂ × H ) × G ( m ̂ H ) G ] d S iii .
U ( r p ) = i = 1 N facets [ 1 3 j = 1 3 I ( r p , m ̂ i , r s , v i j , E v i j , H v i j ) Δ i ] ,
H = c k 2 4 π ( n ̂ × p ) exp ( i k r ) r ( 1 1 i k r ) ,
E = 1 4 π ϵ 0 { k 2 ( n ̂ × p ) × n ̂ exp ( i k r ) r + [ 3 n ̂ ( n ̂ p ) p ] ( 1 r 3 i k r 2 ) exp ( i k r ) } ,
ϵ U = i = 1 N U S C ( r i ) U A n ( r i ) 2 i = 1 N U A n ( r i ) 2 ,
δ r x x p r δ x ,
T ( N ) = T 1 N + N t p ,
4 π p E = i ω μ S p [ ( m ̂ × H ) G ] d S S p [ ( m ̂ × E ) × G ] d S S p [ ( m ̂ E ) G ] d S .
i ω μ S p [ ( m ̂ × H ) G ] d S = i ω μ S [ p ( m ̂ × H ) ] G d S i ω μ S ( m ̂ × H ) p G d S = i ω μ S ( m ̂ × H ) G d S
S p [ ( m ̂ E ) G ] d S = S ( m ̂ E ) 2 G d S = ω 2 μ ϵ S ( m ̂ E ) G d S
ω 2 μ ϵ S ( m ̂ E ) G d S = i ω μ S [ G ( × H ) ] m ̂ d S ,
i ω μ S [ G ( × H ) ] m ̂ d S = i ω μ S [ × ( G H ) ] m ̂ d S + i ω μ S ( G × H ) m ̂ d S = i ω μ S [ × ( G H ) ] m ̂ d S i ω μ S ( m ̂ × H ) G d S .
S ( × B ) m ̂ d S = S ( × B ) d S = C B d s ,
S p [ ( m ̂ E ) G ] d S = i ω μ S ( m ̂ × H ) G d S .
p E = 0 ,
4 π p × H = i ω ϵ S p × [ ( m ̂ × E ) G ] d S S p × [ ( m ̂ × H ) × G ] d S
p × H = i ω ϵ 4 π S [ i ω μ ( m ̂ × E ) G + ( m ̂ × E ) × G + ( m ̂ E ) G ] d S = i ω ϵ E .

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