Abstract

The Debye-Wolf electromagnetic diffraction integral is now routinely used to describe focusing by high numerical (NA) lenses. We obtain an eigenfunction expansion of the electric vector field in the focal region in terms of Bessel and generalized prolate spheroidal functions. Our representation has many optimal and desirable properties which offer considerable simplification to the evaluation and analysis of the Debye-Wolf integral. It is potentially also useful in implementing two-dimensional apodization techniques to synthesize electromagnetic field distributions in the focal region of a high NA lenses. Our work is applicable to many areas, such as optical microscopy, optical data storage and lithography.

© 2008 Optical Society of America

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References

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  1. B. Richards "Diffraction in systems of high relative aperture," in Astronomical Optics and Related Subjects, Z. Kopal, ed., (North Holland Publishing Company, 1955), pp. 352-359.
  2. E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London A,  253, 349-357 (1959).
    [CrossRef]
  3. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London A,  253, 358-379 (1959).
    [CrossRef]
  4. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Ed. (Cambridge University Press, 1952).
    [PubMed]
  5. I. S. Gradshtyen and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, 1980).
  6. G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575-578 (1979)
    [CrossRef]
  7. R. Kant, "An analytical solution of vector diffraction for focusing optical systems with seidel aberrations" J. Mod. Opt. 40, 2293-2310 (1993).
    [CrossRef]
  8. P. Töoröok, S. J. Hewlett and P. Varga, "On the series expansion of high aperture, vectorial diffraction integrals," J. Mod. Opt. 44, 493-503 (1997).
    [CrossRef]
  9. S. S. Sherif and P. Török, "Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula," J. Mod. Opt. 52, 857-876 (2005).
    [CrossRef]
  10. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, "Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system," J. Opt. Soc. Am. A 20, 2281-2292 (2003).
    [CrossRef]
  11. C. J. R. Sheppard and P. Török, "Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion," J. Mod. Opt. 44, 803-818 (1997).
    [CrossRef]
  12. S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high-numerical-aperture focusing," J. Mod. Opt. 51, 2007-2019 (2004).
  13. M. R. Foreman, Department of Physics, Imperial College London, Prince Consort Rd. London SW7 2BZ, United Kingdom, S. S. Sherif, P. R. T. Munro and P. Török are preparing a manuscript to be called "Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region"
  14. G. Arfken, Mathematical Methods for Physicists, 3rd Edition (Academic Press, 1985).
  15. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  16. D. Slepian, "Prolate spheroidal wave functions, Fourier analysis and uncertainty- IV: Extensions to many dimensions; generalized prolate spheroidal functions," Bell Syst. Tech. J. 43, 3009-3057 (1964).
  17. J. C. Heurtley, "Hyperspheroidal functions-optical resonators with circular mirrors," in Proceedings of Symposium on Quasi-Optics, J. Fox, ed., (Polytechnic Press, 1964), pp. 367-371.
  18. B. R. Frieden, "Evaluation, design and extrapolation methods for optical signals, based on the prolate functions," Progress in Optics 9, E. Wolf, ed., (Pergamon, 1971), pp. 311- 407.
  19. P. Török, P. D. Higdon and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148,300-315 (1998)
    [CrossRef]
  20. S. Zhang and J. Jin, Computation of Special Functions (Wiley, 1996).
  21. P. E. Falloon, Hybrid Computation of the Spheroidal Harmonics and Application to the Generalized Hydrogen Molecular Ion Problem (University of Western Australia, 2001).
    [PubMed]
  22. W. Latham and M. Tilton, "Calculation of prolate functions for optical analysis," Appl. Opt. 26, 2653-2658 (1987).
    [CrossRef] [PubMed]
  23. P. Török and P. Munro, "The use of Gauss-Laguerre vector beams in STED microscopy," Opt. Express 12, 3605-3617 (2004).http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-15-3605
    [CrossRef] [PubMed]

2005

S. S. Sherif and P. Török, "Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula," J. Mod. Opt. 52, 857-876 (2005).
[CrossRef]

2004

S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high-numerical-aperture focusing," J. Mod. Opt. 51, 2007-2019 (2004).

P. Török and P. Munro, "The use of Gauss-Laguerre vector beams in STED microscopy," Opt. Express 12, 3605-3617 (2004).http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-15-3605
[CrossRef] [PubMed]

2003

1998

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

1997

P. Töoröok, S. J. Hewlett and P. Varga, "On the series expansion of high aperture, vectorial diffraction integrals," J. Mod. Opt. 44, 493-503 (1997).
[CrossRef]

C. J. R. Sheppard and P. Török, "Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion," J. Mod. Opt. 44, 803-818 (1997).
[CrossRef]

1993

R. Kant, "An analytical solution of vector diffraction for focusing optical systems with seidel aberrations" J. Mod. Opt. 40, 2293-2310 (1993).
[CrossRef]

1987

1979

1964

D. Slepian, "Prolate spheroidal wave functions, Fourier analysis and uncertainty- IV: Extensions to many dimensions; generalized prolate spheroidal functions," Bell Syst. Tech. J. 43, 3009-3057 (1964).

1959

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London A,  253, 349-357 (1959).
[CrossRef]

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

Agrawal, G. P.

Braat, J. J. M.

Dirksen, P.

Hewlett, S. J.

P. Töoröok, S. J. Hewlett and P. Varga, "On the series expansion of high aperture, vectorial diffraction integrals," J. Mod. Opt. 44, 493-503 (1997).
[CrossRef]

Higdon, P. D.

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

Janssen, A. J. E. M.

Kant, R.

R. Kant, "An analytical solution of vector diffraction for focusing optical systems with seidel aberrations" J. Mod. Opt. 40, 2293-2310 (1993).
[CrossRef]

Latham, W.

Munro, P.

Pattanayak, D. N.

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. London A,  253, 358-379 (1959).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and P. Török, "Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion," J. Mod. Opt. 44, 803-818 (1997).
[CrossRef]

Sherif, S. S.

S. S. Sherif and P. Török, "Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula," J. Mod. Opt. 52, 857-876 (2005).
[CrossRef]

S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high-numerical-aperture focusing," J. Mod. Opt. 51, 2007-2019 (2004).

Slepian, D.

D. Slepian, "Prolate spheroidal wave functions, Fourier analysis and uncertainty- IV: Extensions to many dimensions; generalized prolate spheroidal functions," Bell Syst. Tech. J. 43, 3009-3057 (1964).

Tilton, M.

Töoröok, P.

P. Töoröok, S. J. Hewlett and P. Varga, "On the series expansion of high aperture, vectorial diffraction integrals," J. Mod. Opt. 44, 493-503 (1997).
[CrossRef]

Török, P.

S. S. Sherif and P. Török, "Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula," J. Mod. Opt. 52, 857-876 (2005).
[CrossRef]

P. Török and P. Munro, "The use of Gauss-Laguerre vector beams in STED microscopy," Opt. Express 12, 3605-3617 (2004).http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-15-3605
[CrossRef] [PubMed]

S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high-numerical-aperture focusing," J. Mod. Opt. 51, 2007-2019 (2004).

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

C. J. R. Sheppard and P. Török, "Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion," J. Mod. Opt. 44, 803-818 (1997).
[CrossRef]

van de Nes, A. S.

Varga, P.

P. Töoröok, S. J. Hewlett and P. Varga, "On the series expansion of high aperture, vectorial diffraction integrals," J. Mod. Opt. 44, 493-503 (1997).
[CrossRef]

Wilson, T.

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

Wolf, E.

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London A,  253, 349-357 (1959).
[CrossRef]

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

Appl. Opt.

Bell Syst. Tech. J.

D. Slepian, "Prolate spheroidal wave functions, Fourier analysis and uncertainty- IV: Extensions to many dimensions; generalized prolate spheroidal functions," Bell Syst. Tech. J. 43, 3009-3057 (1964).

J. Mod. Opt.

C. J. R. Sheppard and P. Török, "Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion," J. Mod. Opt. 44, 803-818 (1997).
[CrossRef]

S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high-numerical-aperture focusing," J. Mod. Opt. 51, 2007-2019 (2004).

R. Kant, "An analytical solution of vector diffraction for focusing optical systems with seidel aberrations" J. Mod. Opt. 40, 2293-2310 (1993).
[CrossRef]

P. Töoröok, S. J. Hewlett and P. Varga, "On the series expansion of high aperture, vectorial diffraction integrals," J. Mod. Opt. 44, 493-503 (1997).
[CrossRef]

S. S. Sherif and P. Török, "Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula," J. Mod. Opt. 52, 857-876 (2005).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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

Opt. Express

Proc. Roy. Soc. London A

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London A,  253, 349-357 (1959).
[CrossRef]

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

Other

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Ed. (Cambridge University Press, 1952).
[PubMed]

I. S. Gradshtyen and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, 1980).

B. Richards "Diffraction in systems of high relative aperture," in Astronomical Optics and Related Subjects, Z. Kopal, ed., (North Holland Publishing Company, 1955), pp. 352-359.

J. C. Heurtley, "Hyperspheroidal functions-optical resonators with circular mirrors," in Proceedings of Symposium on Quasi-Optics, J. Fox, ed., (Polytechnic Press, 1964), pp. 367-371.

B. R. Frieden, "Evaluation, design and extrapolation methods for optical signals, based on the prolate functions," Progress in Optics 9, E. Wolf, ed., (Pergamon, 1971), pp. 311- 407.

M. R. Foreman, Department of Physics, Imperial College London, Prince Consort Rd. London SW7 2BZ, United Kingdom, S. S. Sherif, P. R. T. Munro and P. Török are preparing a manuscript to be called "Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region"

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

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

S. Zhang and J. Jin, Computation of Special Functions (Wiley, 1996).

P. E. Falloon, Hybrid Computation of the Spheroidal Harmonics and Application to the Generalized Hydrogen Molecular Ion Problem (University of Western Australia, 2001).
[PubMed]

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of the Debye-Wolf integral

Fig. 2.
Fig. 2.

(a) Monotonically decreasing eigenvalues of the circular prolate spheroidal functions as order (N,n) increases (c=20). A value of λ |N|,n =10-4 was used to determine a suitable truncation point for the infinite series inherent in the eigenfunction expansion; (b) Finite summation limit for Bessel terms required for different defocus distances again based on a cutoff point of 10-4

Fig. 3.
Fig. 3.

Two dimensional in-focus eigenfunctions of order (N,n) for focusing by a high NA optical system.


Fig. 4.
Fig. 4.

Variation of eigenvalues λ |N|,n of the generalized prolate spheroidal functions with numerical aperture of the focusing optical system.

Fig. 5.
Fig. 5.

Pointwise error on the electric field and intensity distributions at the Gaussian focal plane when calculated using our eigenfunction expansion as shown in the insets as compared to direct integration (x polarized incident illumination, NA=0.966). Strong horizontal and vertical lines are seen due to zeros in the field distributions.

Fig. 6.
Fig. 6.

As Fig. 5 except electric field and intensity distributions are calculated at a defocused plane (z=λ).

Equations (37)

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E ( x p , y p , z p ) = ik 2 π Θ a ( s x , s y ) s z exp [ ik ( s x x p + s y y p + s z z p ) ] d s x d s y
s = ( sin θ cos ϕ , sin θ sin ϕ , cos θ )
( x p , y p , z p ) = r p ( sin θ p cos ϕ p , sin θ p sin ϕ p , cos θ p )
E x ( ρ p , ϕ p , z p ) = iA π 0 α 0 2 π cos θ sin θ { cos θ + ( 1 cos θ ) sin 2 ϕ }
× exp [ ik ρ p sin θ cos ( ϕ ϕ p ) ] exp [ ik z p cos θ ] d θ d ϕ
E y ( ρ p , ϕ p , z p ) = iA π 0 α 0 2 π cos θ sin θ ( 1 cos θ ) cos ϕ sin ϕ
× exp [ ik ρ p sin θ cos ( ϕ ϕ p ) ] exp [ ik z p cos θ ] d θ d ϕ
E z ( ρ p , ϕ p , z p ) = iA π 0 α 0 2 π cos θ sin 2 θ cos ϕ
× exp [ ik ρ p sin θ cos ( ϕ ϕ p ) ] exp [ ik z p cos θ ] d θ d ϕ
exp [ ik z p cos θ ] = m = i m J m ( kz p ) exp [ im θ ] = m = J m ( kz p ) exp [ im ( π 2 θ ) ]
E x ( ρ p , ϕ p , z p ) = iA π m = J m ( kz p ) 0 α 0 2 π cos θ sin θ { cos θ + ( 1 cos θ ) sin 2 ϕ }
× exp [ ik ρ p sin θ cos ( ϕ ϕ p ) ] exp [ im ( π 2 θ ) ] d θ d ϕ
E x ( ρ p , ϕ p , z p ) = iA π m = J m ( kz p ) 0 α 0 2 π 1 1 u 2 4 { 1 u 2 + ( 1 1 u 2 ) sin 2 ϕ }
× exp [ ik ρ p u cos ( ϕ ϕ p ) ] exp [ im ( π 2 sin 1 ( u ) ) ] u d u d ϕ
exp [ im ( π 2 sin 1 ( u ) ) ] = { T m ( u ) + i m m 1 u 2 U m 1 ( u ) m 0 1 m = 0 .
a m x ( u , ϕ ) = [ 1 u 2 + ( 1 1 u 2 ) sin 2 ϕ 1 u 2 4 ] { T m ( u ) + i m m 1 u 2 U m 1 ( u ) m 0 1 m = 0
a m x ( u , ϕ ) = N = n = 0 A m , N , n x Φ N , n ( u , c ) exp ( iN ϕ ) ,
A m , N , n x = 1 2 π λ N , n 0 2 π 0 α a m x ( u , ϕ ) Φ N , n ( u , c ) exp ( iN ϕ ) udud ϕ ,
E x ( ρ p , ϕ p , z p ) = iA π m = J m ( kz p ) N = n = 0 A m , N , n x
× 0 α 0 2 π Φ N , n ( u , c ) exp ( iN ϕ ) × exp [ ikr p u cos ( ϕ ϕ p ) ] udud ϕ .
E x ( ρ p , ϕ p , z p ) = 2 i A m = J m ( kz p ) N = i N exp ( iN ϕ p )
× n = 0 A m , N , n x 0 α Φ N , n ( u , c ) J N ( k ρ p u ) udu
E x ( ρ p , ϕ p , z p ) = 2 i A m = J m ( kz p ) N = i N exp ( iN ϕ p )
× n = 0 A m , N , n x 0 α Φ N , n ( u , c ) J N ( k ρ p u ) udu
0 r 0 Φ N , n ( c , r ) J N ( ω r ) rdr = ( 1 ) n ( r 0 Ω ) λ N , n Φ N , n ( c , r 0 ω Ω )
E x ( ρ p , ϕ p , z p ) = 2 i A ( α k ρ P max )
× m = N = n = 0 i N A m , N , n x ( 1 ) n λ N , n J m ( kz p ) e iN ϕ p Φ N , n ( c , α ρ p ρ p max )
E y ( ρ p , ϕ p , z p ) = 2 i A ( α k ρ p max )
× m = N = n = 0 i N A m , N , n y ( 1 ) n λ N , n J m ( kz p ) e iN ϕ p Φ N , n ( c , α ρ p ρ p max )
E z ( ρ p , ϕ p , z p ) = 2 i A ( α k ρ p max )
× m = N = n = 0 i N A m , N , n z ( 1 ) n λ N , n J m ( kz p ) e iN ϕ p Φ N , n ( c , α ρ p ρ p max )
a m y ( u , ϕ ) = [ ( 1 1 u 2 ) cos ϕ sin ϕ 1 u 2 4 ] { T m ( u ) + i m m 1 u 2 U m 1 ( u ) m 0 1 m = 0
a m z ( u , ϕ ) = [ u cos ϕ 1 u 2 4 ] { T m ( u ) + i m m 1 u 2 U m 1 ( u ) m 0 1 m = 0
a ( u , ϕ ) = g ( u , ϕ ) e i Ψ ( u , ϕ ) A ( u , ϕ ) e ( u , ϕ )
e = ( e x ( u , ϕ ) e y ( u , ϕ ) )
A = ( ( 1 + 1 u 2 ) ( 1 1 u 2 ) cos 2 ϕ ( 1 1 u 2 ) sin 2 ϕ ( 1 1 u 2 ) sin 2 ϕ ( 1 + 1 u 2 ) + ( 1 1 u 2 ) cos 2 ϕ 2 u cos ϕ 2 u sin φ )
ε = ( K L I k , l direct I k , l exp ansion K L I k , l direct ) · 100 ,

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