Abstract

In order to rigorously calculate the image of a scattered field, it must be propagated into the far-field before vectorial focusing theory is applied. This approach may become difficult when, for example, the scattering object is embedded in a stratified medium, requiring calculation of the appropriate Green’s tensor. We present a method for calculating the image of an arbitrary vectorial field by decomposing the field into a superposition of magnetic-dipole waves. We show that this technique can significantly simplify the calculation of the image of arbitrary vectorial fields even when the field is known within a stratified medium. The technique is more computationally efficient than existing methods however we also show that the method retains accuracy.

© 2007 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. R. Soc. London, Ser. A 253, 358-379 (1959).
    [CrossRef]
  2. P. Török and C.J.R. Sheppard, High numerical aperture focusing and imaging (Adam Hilger, Bristol, In Press).
  3. C. J. R. Sheppard and T. Wilson, "The image of a single point in microscopes of large numerical aperture," Proc. R. Soc. London, Ser. A 379, 145-58 (1982).
    [CrossRef]
  4. P. Török, P.D.  Higdon, and T. Wilson, "On the general properties of polarising conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
    [CrossRef]
  5. P. Török, P.D. Higdon, and T.Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
    [CrossRef]
  6. A.J. Poggio and E.K. Miller, "Integral equation solutions of three-dimensional scattering problems," in Computer techniques for electromagnetics, R. Mittra, ed. (Pergamon Press, Oxford, 1973) pp. 159-264.
  7. P. T¨or¨ok, P.R.T. Munro, and Em.E. Kriezis, "A rigorous near to farfield transformation for vectorial diffraction calculations and its numerical implementation," J. Opt. Soc. Am. A 23, 713-722 (2006).
    [CrossRef]
  8. K. Shimura and T.D. Milster, "Vector diffraction analysis by discrete-dipole approximation," J. Opt. Soc. Am. A 18, 2895-2900 (2001).
    [CrossRef]
  9. B. Karczewski and E. Wolf, "Comparison of three theories of electromagnetic diffraction at an aperture Part I: coherence matrices, Part II: The far field," J. Opt. Soc. Am. 56, 1207-19 (1966).
    [CrossRef]
  10. P. Török, "An imaging theory for advanced, high numerical aperture optical microscopes," DSc thesis, Hungarian academy of sciences (2004).
  11. R. Hiptmair, "Coupling of finite elements and boundary elements in electromagnetic scattering," SIAMJ.Numer. Anal. 41, 919-943 (2003).
    [CrossRef]
  12. P.D. Higdon, P. T¨or¨ok, and T. Wilson, "Imaging properties of high aperture multiphoton fluorescence scanning microscopes," J. Microsc. 193, 127-141 (1998).
    [CrossRef]
  13. D. J. Innes and A. L. Bloom, "Design of optical systems for use with laser beams," Spectra-Physics Laser Technical Bulletin 5, 1-10 (1966).
  14. P. T¨or¨ok, "Propagation of electromagnetic dipole waves through dielectric interfaces," Opt. Lett. 25, 1463-1465 (2000).
    [CrossRef]
  15. O. Haeberl’e, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. T¨or¨ok, "Point spread function of optical microscopes imaging through stratified media," Opt. Express 11, 2964-2969 (2003).
    [CrossRef] [PubMed]
  16. P. T¨or¨ok, "Focusing of electromagnetic waves through dielectric interfaces - Theory and correction of aberration," J. Opt. Mem. Neur. Net. 8(1), 10-24 (1999).
  17. P. T¨or¨ok, "Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number," J. Opt. Soc. Am. A 15(12), 3009-3015 (1998).
    [CrossRef]
  18. A.S. van de Nes, P.R.T. Munro, S.F. Pereira, J.J.M. Braat, and P. T¨or¨ok, "Cylindrical vector beam focusing through a dielectric interface: comment," Opt. Express 12(5), 967-969 (2004).
    [CrossRef]
  19. J.D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, New York, 1999).
  20. J.A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

2006 (1)

2004 (1)

2003 (2)

2001 (1)

2000 (1)

1999 (1)

P. T¨or¨ok, "Focusing of electromagnetic waves through dielectric interfaces - Theory and correction of aberration," J. Opt. Mem. Neur. Net. 8(1), 10-24 (1999).

1998 (4)

P. T¨or¨ok, "Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number," J. Opt. Soc. Am. A 15(12), 3009-3015 (1998).
[CrossRef]

P.D. Higdon, P. T¨or¨ok, and T. Wilson, "Imaging properties of high aperture multiphoton fluorescence scanning microscopes," J. Microsc. 193, 127-141 (1998).
[CrossRef]

P. Török, P.D.  Higdon, and T. Wilson, "On the general properties of polarising conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

P. Török, P.D. Higdon, and T.Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

1982 (1)

C. J. R. Sheppard and T. Wilson, "The image of a single point in microscopes of large numerical aperture," Proc. R. Soc. London, Ser. A 379, 145-58 (1982).
[CrossRef]

1966 (2)

D. J. Innes and A. L. Bloom, "Design of optical systems for use with laser beams," Spectra-Physics Laser Technical Bulletin 5, 1-10 (1966).

B. Karczewski and E. Wolf, "Comparison of three theories of electromagnetic diffraction at an aperture Part I: coherence matrices, Part II: The far field," J. Opt. Soc. Am. 56, 1207-19 (1966).
[CrossRef]

1959 (1)

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

Higdon, , P.D.

P. Török, P.D.  Higdon, and T. Wilson, "On the general properties of polarising conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

Bloom, A. L.

D. J. Innes and A. L. Bloom, "Design of optical systems for use with laser beams," Spectra-Physics Laser Technical Bulletin 5, 1-10 (1966).

Braat, J.J.M.

Higdon, P.D.

P.D. Higdon, P. T¨or¨ok, and T. Wilson, "Imaging properties of high aperture multiphoton fluorescence scanning microscopes," J. Microsc. 193, 127-141 (1998).
[CrossRef]

Hiptmair, R.

R. Hiptmair, "Coupling of finite elements and boundary elements in electromagnetic scattering," SIAMJ.Numer. Anal. 41, 919-943 (2003).
[CrossRef]

Innes, D. J.

D. J. Innes and A. L. Bloom, "Design of optical systems for use with laser beams," Spectra-Physics Laser Technical Bulletin 5, 1-10 (1966).

Karczewski, B.

Milster, T.D.

Munro, P.R.T.

Pereira, S.F.

Richards, B.

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

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, "The image of a single point in microscopes of large numerical aperture," Proc. R. Soc. London, Ser. A 379, 145-58 (1982).
[CrossRef]

Sheppard, C.J.R.

P. Török and C.J.R. Sheppard, High numerical aperture focusing and imaging (Adam Hilger, Bristol, In Press).

Shimura, K.

Török, P.

P. Török, P.D.  Higdon, and T. Wilson, "On the general properties of polarising conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

P. Török and C.J.R. Sheppard, High numerical aperture focusing and imaging (Adam Hilger, Bristol, In Press).

van de Nes, A.S.

Wilson, T.

P. Török, P.D.  Higdon, and T. Wilson, "On the general properties of polarising conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

C. J. R. Sheppard and T. Wilson, "The image of a single point in microscopes of large numerical aperture," Proc. R. Soc. London, Ser. A 379, 145-58 (1982).
[CrossRef]

Wolf, E.

B. Karczewski and E. Wolf, "Comparison of three theories of electromagnetic diffraction at an aperture Part I: coherence matrices, Part II: The far field," J. Opt. Soc. Am. 56, 1207-19 (1966).
[CrossRef]

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

Adam Hilger, Bristol (1)

P. Török and C.J.R. Sheppard, High numerical aperture focusing and imaging (Adam Hilger, Bristol, In Press).

J. Microsc. (1)

P.D. Higdon, P. T¨or¨ok, and T. Wilson, "Imaging properties of high aperture multiphoton fluorescence scanning microscopes," J. Microsc. 193, 127-141 (1998).
[CrossRef]

J. Mod. Opt. (1)

P. Török, P.D. Higdon, and T.Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

J. Opt. Mem. Neur. Net. (1)

P. T¨or¨ok, "Focusing of electromagnetic waves through dielectric interfaces - Theory and correction of aberration," J. Opt. Mem. Neur. Net. 8(1), 10-24 (1999).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Numer. Anal. (1)

R. Hiptmair, "Coupling of finite elements and boundary elements in electromagnetic scattering," SIAMJ.Numer. Anal. 41, 919-943 (2003).
[CrossRef]

Opt. Commun. (1)

P. Török, P.D.  Higdon, and T. Wilson, "On the general properties of polarising conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. R. Soc. London, Ser. A (2)

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

C. J. R. Sheppard and T. Wilson, "The image of a single point in microscopes of large numerical aperture," Proc. R. Soc. London, Ser. A 379, 145-58 (1982).
[CrossRef]

Spectra-Physics Laser Technical Bulletin (1)

D. J. Innes and A. L. Bloom, "Design of optical systems for use with laser beams," Spectra-Physics Laser Technical Bulletin 5, 1-10 (1966).

Other (4)

A.J. Poggio and E.K. Miller, "Integral equation solutions of three-dimensional scattering problems," in Computer techniques for electromagnetics, R. Mittra, ed. (Pergamon Press, Oxford, 1973) pp. 159-264.

P. Török, "An imaging theory for advanced, high numerical aperture optical microscopes," DSc thesis, Hungarian academy of sciences (2004).

J.D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, New York, 1999).

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

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Figures (11)

Fig. 1.
Fig. 1.

Configuration for statement of the m-theory of diffraction according to Török [10].

Fig. 2.
Fig. 2.

Coordinate system used to calculate image of an EMD in transmission (top) and reflection mode (bottom).

Fig. 3.
Fig. 3.

Notation for calculating the image of a laterally displaced dipole.

Fig. 4.
Fig. 4.

Labelling scheme for calculating the image of an EMD embedded in a stratified medium in reflection (left) and transmission (right).

Fig. 5.
Fig. 5.

Configuration for calculating the image of an harmonically oscillating electric dipole directly (a) and using EMD decomposition (b).

Fig. 6.
Fig. 6.

Plot showing how the proportion of power flowing through the EMD plane varies with the width of the plane.

Fig. 7.
Fig. 7.

Electric field components at the detector for tests one (top), two (middle) and three (bottom). Components were normalised by the peak magnitude of the y-component.

Fig. 8.
Fig. 8.

Plots showing (a) a comparison of aggregate error as a function of the EMD density for test one. Each curve represents a different EMD plane width and (b) component wise aggregate errors for test one as a function of the total power propagating through the EMD plane.

Fig. 9.
Fig. 9.

Plot of Ieq,s functions for test one, normalised collectively (left) and normalised individually (right).

Fig. 10.
Fig. 10.

Comparison of total aggregate error for tests one, two and three as a function of the total power propagating through the EMD plane.

Fig. 11.
Fig. 11.

Comparison of magnitude of Is,eq 0, Is,eq 1 andIs,eq 2 for test one and test three.

Equations (57)

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E ( P ) = 1 2 π p × S 0 n × E ( Q ) exp ( ι k 0 r ) r d S
p × [ n × E ( Q ) exp ( ι k 0 r ) r ] = exp ( ι k 0 r ) r ( p × ( n × E ( Q ) ) ) ( n × E ( Q ) ) × p [ exp ( ι k 0 r ) r ]
= ( n × E ( Q ) ) × [ exp ( ι k 0 r ) r ( ι k 0 r 1 r 2 ) r ]
E ( P ) = 1 2 π S 0 ( r × ( n × E ( Q ) ) ) exp ( ι k 0 r ) r 2 ι k 0 d S
E ( r d ) = ι k f 2 2 π Ω ε 0 ( s 2 x , s 2 y ) s 2 z exp ( ι k s 2 · r d ) d s 2 x d s 2 y
ε 0 ( s 2 ) = ε 0 ( s 1 ) = cos θ 2 cos θ 1 1 ( ϕ 1 ) L ( π θ 2 ) L ( π θ 1 ) ( ϕ 1 ) ε eq
= [ cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ]
L = [ cos θ 0 sin θ 0 1 0 sin θ 0 cos θ ]
ε 0 , x = cos θ 2 cos θ 1 [ p x 2 [ ( cos θ 2 + cos θ 1 ) + ( cos θ 2 cos θ 1 ) cos 2 ϕ 1 ] + p y 2 ( cos θ 2 cos θ 1 ) sin 2 ϕ 1 ]
ε 0 , y = cos θ 2 cos θ 1 [ p y 2 [ ( cos θ 1 + cos θ 2 ) + ( cos θ 1 cos θ 2 ) cos 2 ϕ 1 ] + p x 2 ( cos θ 2 cos θ 1 ) sin 2 ϕ 1 ]
ε 0 , z = cos θ 2 cos θ 1 sin θ 2 ( p x cos ϕ 1 + p y sin ϕ 1 )
E ( r d ) = ι f 2 λ π α 2 π 0 2 π cos θ 2 cos θ 1 ε 0 ( θ 2 , ϕ 2 + π ) exp ( ι k r d sin θ 2 cos ( ϕ 2 ϕ d ) ) ×
× exp ( ι k z d cos θ 2 ) exp ( ι k z dp cos θ 1 ) sin θ 2 d ϕ 2 d θ 2
= ι f 2 λ π α 2 π cos θ 2 cos θ 1 exp ( ι ( k z d cos θ 2 k z d cos θ 2 ) ) sin θ 2 ×
× 0 2 π ε 0 ( θ 2 , ϕ 2 + π ) exp ( ι k r d sin θ 2 cos ( ϕ 2 ϕ d ) ) d ϕ 2 d θ 2
0 2 π exp ( ι q ϕ ) exp ( ι ρ cos ( ϕ γ ) ) d ϕ = 2 π ι q J q ( ρ ) exp ( ι q γ )
E x = p x ( I 0 , r eq + I 2 , r eq cos 2 φ d ) + p y I 2 , r eq sin 2 φ d
E y = p y ( I 0 , r eq I 2 , r eq cos 2 φ d ) + p x I 2 , r eq sin 2 φ d
E z = 2 ι ( p x I 1 , r eq cos φ d + p y I 1 , r eq sin φ d )
I 0 , r eq = π α 2 π cos θ 2 cos θ 1 sin θ 2 ( cos θ 1 + cos θ 2 ) J 0 ( k r d sin θ 2 ) exp ( ι k ( z d cos θ 2 z dp cos θ 1 ) ) ) d θ 2
I 1 , r eq = π α 2 π cos θ 2 cos θ 1 sin 2 θ 2 J 1 ( k r d sin θ 2 ) exp ( ι k ( z d cos θ 2 z dp cos θ 1 ) ) ) d θ 2
I 2 , r eq = π α 2 π cos θ 2 cos θ 1 sin θ 2 ( cos θ 1 + cos θ 2 ) J 2 ( k r d sin θ 2 ) exp ( ι k ( z d cos θ 2 z dp cos θ 1 ) ) ) d θ 2
I 0 , t eq = 0 α 2 cos θ 2 cos θ 1 sin θ 2 ( cos θ 1 + cos θ 2 ) J 0 ( k r d sin θ 2 ) exp ( ι k ( z d cos θ 2 z dp cos θ 1 ) ) ) d θ 2
I 1 , t eq = 0 α 0 cos θ 2 cos θ 1 sin 2 θ 2 J 1 ( k r d sin θ 2 ) exp ( ι k ( z d cos θ 2 z dp cos θ 1 ) ) ) d θ 2
I 2 , t eq = 0 α 2 cos θ 2 cos θ 1 sin θ 2 ( cos θ 1 cos θ 2 ) J 2 ( k r d sin θ 2 ) exp ( ι k ( z d cos θ 2 z dp cos θ 1 ) ) ) d θ 2
exp ( ι k Δ r ) exp ( ι k s 2 · r d )
= exp ( ι k sin θ 2 ( cos ϕ 2 ( x d + β x dp ) + sin ϕ 2 ( y d + β y dp ) ) ) exp ( ι k ( z d cos θ 2 z dp cos θ 1 ) )
E t r ( r d ) = ι k 0 2 π S 0 E o a , t r ( n × E i ( Q ) , r d + β q ) d S
ε d = cos θ d cos θ 1 1 ( ϕ 1 ) L ( π θ d ) I E ( N 1 ) L ( π θ N ) ( ϕ 1 ) ε dp
I E ( N 1 ) = [ T p ( N 1 ) 0 0 0 T s ( N 1 ) 0 0 0 T p ( N 1 ) ]
T m ( N 1 ) = t m ( N 1 ) j = 1 N 2 t m ( j ) exp ( ι β j + 1 ) D m ( N 1 )
D m ( 2 ) = 1 + r m ( 1 ) r m ( 2 ) exp ( 2 ι β 2 )
D m ( 3 ) = 1 + r m ( 1 ) { r m ( 2 ) exp ( 2 ι β 2 ) + r m ( 3 ) exp [ 2 ι ( β 2 + β 3 ) ] + r m ( 2 ) r m ( 3 ) exp ( 2 ι β 3 ) }
D m ( 4 ) = 1 + r m ( 1 ) { r m ( 2 ) exp ( 2 ι β 2 ) + r m ( 3 ) exp [ 2 ι ( β 2 + β 3 ) ] + r m ( 4 ) exp [ 2 ι ( β 2 + β 3 + β 4 ) ] }
+ r m ( 2 ) { r m ( 3 ) exp ( 2 ι β 3 ) + r m ( 4 ) exp [ 2 ι ( β 3 + β 4 ) ] }
+ r m ( 3 ) r m ( 4 ) exp ( 2 ι β 4 ) + r m ( 2 ) r m ( 4 ) r m ( 3 ) r m ( 4 ) exp ( 2 ι ( β 2 + β 4 ) )
t s ( l ) = 2 n l + 1 cos θ l + 1 n l + 1 cos θ l + 1 + n l cos θ l t p ( l ) = 2 n l + 1 cos θ l + 1 n l cos θ l + 1 + n l + 1 cos θ l r s ( l ) = n l + 1 cos θ l + 1 n l cos θ l n l + 1 cos θ l + 1 + n l cos θ l r p ( l ) = n l cos θ l + 1 n l + 1 cos θ l n l cos θ l + 1 + n l + 1 cos θ l
E d ( r d ) = ι k d 2 π Ω d ε d ( s d ) s dz exp ( ι k d r d · s d ) exp ( ι k N r dp · s N ) exp ( ι k 0 Ψ i ) d s dx d s dy
I 0 , r eq , s = π α d π cos θ d cos θ 1 sin θ d ( T s cos θ N + T p cos θ d ) J 0 ( k d r dt sin θ d ) ×
× exp ( ι k 0 Ψ i ) exp ( ι ( k d z d cos θ d k N z dp cos θ N ) ) ) d θ d
I 1 , r eq , s = π α d π cos θ d cos θ 1 T p sin 2 θ d J 1 ( k d r dt sin θ d ) ×
× exp ( ι k 0 Ψ i ) exp ( ι ( k d z d cos θ d k N z dp cos θ N ) ) ) d θ d
I 2 , r eq , s = π α d π cos θ d cos θ 1 sin θ d ( T s cos θ N T p cos θ d ) J 2 ( k d r dt sin θ d ) ×
× exp ( ι k 0 Ψ i ) exp ( ι ( k d z d cos θ d k N z dp cos θ N ) ) ) d θ d
I 0 , t eq , s = 0 α d cos θ d cos θ 1 sin θ d ( T s cos θ N + T p cos θ d ) J 0 ( k d r dt sin θ d ) ×
× exp ( ι k 0 Ψ i ) exp ( ι ( k d z d cos θ d k N z dp cos θ N ) ) ) d θ d
I 1 , t eq , s = 0 α d cos θ d cos θ 1 T p sin 2 θ d J 1 ( k d r dt sin θ d ) ×
× exp ( ι k 0 Ψ i ) exp ( ι ( k d z d cos θ d k N z dp cos θ N ) ) ) d θ d
I 2 , t eq , s = 0 α d cos θ d cos θ 1 sin θ d ( T s cos θ N T p cos θ d ) J 2 ( k d r dt sin θ d ) ×
× exp ( ι k 0 Ψ i ) exp ( ι ( k d z d cos θ d k N z dp cos θ N ) ) ) d θ d
I 0 , r eq , s = I 0 , t eq , s * I 1 , r eq , s = I 1 , t eq , s * I 2 , r eq , s = I 2 , t eq , s *
P = c 2 Z k 4 12 π p 2
P plane = S 0 1 2 { E × H * } · ( k ̂ ) d S
H = c k 2 4 π ( n ̂ × p ) e ι k r r ( 1 1 ι k r )
E = 1 4 π ϵ ( k 2 ( n ̂ × p ) × n ̂ e ι k r r + ( 3 n ̂ ( n ̂ · p ) p ) ( 1 r 3 ι k r 2 ) e ι k r )
ϵ E = i = 1 N E eq ( r i ) E dir ( r i ) 2 i = 1 N E dir ( r i ) 2
ϵ E s = i = 1 N ( E eq ( r i ) E dir ( r i ) ) · s 2 i = 1 N E dir ( r i ) · s 2

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