Abstract

We present details of a novel imaging algorithm based on the extended Nijboer–Zernike (ENZ) theory of diffraction. We derive integral expressions relating the electric field distribution in the entrance pupil of an optical system to the electric field in its focal region. The evaluation of these integrals is made possible by means of a highly accurate and efficient series expansion similar to those occurring in standard ENZ theory. Based on these results an ENZ imaging scheme is constructed and evaluated in detail with attention to the convergence properties and computational complexity of the new method.

© 2009 Optical Society of America

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2008

S. van Haver, O. T. A. Janssen, A. J. E. M. Janssen, J. J. M. Braat, H. P. Urbach, and S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U, 1-8 (2008).

O. T. A. Janssen, S. van Haver, A. J. E. M. Janssen, J. J. M. Braat, H. P. Urbach, and S. F. Pereira, “Extended Nijboer-Zernike (ENZ) based mask imaging: efficient coupling of electromagnetic field solvers and the ENZ imaging algorithm,” Proc. SPIE 6924, 692410 (2008).
[CrossRef]

2007

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

S. van Haver, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory--Erratum,” J. Eur. Opt. Soc. Rapid Publ. 2, 07011e (2007).
[CrossRef]

X. Wei, A. J. Wachters, and H. P. Urbach, “Finite-element model for three-dimensional optical scattering problems,” J. Opt. Soc. Am. A 24, 866-881 (2007).
[CrossRef]

2006

S. van Haver, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory,” J. Eur. Opt. Soc. Rapid Publ. 1, 06004 (2006).
[CrossRef]

2005

2004

P. Török and P. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605-3617 (2004).
[CrossRef] [PubMed]

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175-182 (2004).
[CrossRef]

2003

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
[CrossRef]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. van de Nes, “Extended Nijboer-Zernike representation of the field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281-2292 (2003).
[CrossRef]

2002

2000

J. A. Roden and S. D. Gedney, “Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microwave Opt. Technol. Lett. 27, 334-339 (2000).
[CrossRef]

1997

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]

1996

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
[CrossRef]

D. Flagello, T. Milster, and A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53-64 (1996).
[CrossRef]

1994

D. T. Prescott and N. V. Shuley, “Extensions to the FDTD method for the analysis of infinitely periodic arrays,” IEEE Microw. Guid. Wave Lett. 4, 352-354 (1994).
[CrossRef]

1988

1987

C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354-1360 (1987).
[CrossRef]

C. Colautti, B. Ruiz, E. E. Sicre, and M. Garavaglia, “Walsh functions: Analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385-1391 (1987).
[CrossRef]

1966

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

1959

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]

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

1955

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91-103 (1955).
[CrossRef]

1953

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London, Ser. A 217, 408-432 (1953).
[CrossRef]

1947

A. Maréchal, “Study of the combined effects of diffraction and geometrical aberrations on the image of a luminous point,” Rev. Opt., Theor. Instrum. 26, 257-277 (1947).

1943

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116-128 (1943).
[CrossRef]

1939

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

1934

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica (Amsterdam) 1, 689-704 (1934).
[CrossRef]

1925

J. Picht, “Über den Schwingungsvorgang, der einem Beliebigen (Astigmatischen) Strahlenbündel Entspricht,” Ann. Phys. 77, 685-782 (1925).

1919

V. S. Ignatowsky, “Diffraction of a lens of arbitrary aperture,” Trans. Opt. Inst. 1, 1-36 (1919).

A. E. Conrady, “Star discs,” Mon. Not. R. Astron. Soc. 79, 575-593 (1919).

1918

A. E. Conrady, “The five aberrations of lens-systems,” Mon. Not. R. Astron. Soc. 79, 60-66 (1918).

1885

E. Lommel, “Die Beugungserscheinungen einer Kreisrunden Oeffnung und eines Kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233-328 (1885).

1873

E. Abbe, “Eiträge zur Theorie des Mikroskops und der Mikroskopischen Wahrnemung,” Arch. Mikrosk. Anat. Entwichlungsmech. 9, 413-468 (1873).
[CrossRef]

1834

G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283-291 (1834).

Abbe, E.

E. Abbe, “Eiträge zur Theorie des Mikroskops und der Mikroskopischen Wahrnemung,” Arch. Mikrosk. Anat. Entwichlungsmech. 9, 413-468 (1873).
[CrossRef]

Adam, K.

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
[CrossRef]

Airy, G. B.

G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283-291 (1834).

Baida, F.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
[CrossRef]

Besbes, M.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Bienstman, P.

P. Lalanne, M. Besbes, J. P. Hugonin, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. Baida, B. Guizal, and D. van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Braat, J. J. M.

S. van Haver, O. T. A. Janssen, A. J. E. M. Janssen, J. J. M. Braat, H. P. Urbach, and S. F. Pereira, “General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory,” Proc. SPIE 6924, 69240U, 1-8 (2008).

O. T. A. Janssen, S. van Haver, A. J. E. M. Janssen, J. J. M. Braat, H. P. Urbach, and S. F. Pereira, “Extended Nijboer-Zernike (ENZ) based mask imaging: efficient coupling of electromagnetic field solvers and the ENZ imaging algorithm,” Proc. SPIE 6924, 692410 (2008).
[CrossRef]

S. van Haver, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory--Erratum,” J. Eur. Opt. Soc. Rapid Publ. 2, 07011e (2007).
[CrossRef]

S. van Haver, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory,” J. Eur. Opt. Soc. Rapid Publ. 1, 06004 (2006).
[CrossRef]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, A. S. van de Nes, and S. van Haver, “Extended Nijboer-Zernike approach to aberration and birefringence retrieval in a high-numerical-aperture optical system,” J. Opt. Soc. Am. A 22, 2635-2650 (2005).
[CrossRef]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. van de Nes, “Extended Nijboer-Zernike representation of the field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281-2292 (2003).
[CrossRef]

J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858-870 (2002).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, Vol. 51, E.Wolf, ed. (Elsevier, 2008), pp. 349-468.
[CrossRef]

Chu, L. J.

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

Cobb, N.

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
[CrossRef]

Colautti, C.

C. Colautti, B. Ruiz, E. E. Sicre, and M. Garavaglia, “Walsh functions: Analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385-1391 (1987).
[CrossRef]

Conrady, A. E.

A. E. Conrady, “Star discs,” Mon. Not. R. Astron. Soc. 79, 575-593 (1919).

A. E. Conrady, “The five aberrations of lens-systems,” Mon. Not. R. Astron. Soc. 79, 60-66 (1918).

Dirksen, P.

S. van Haver, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory--Erratum,” J. Eur. Opt. Soc. Rapid Publ. 2, 07011e (2007).
[CrossRef]

S. van Haver, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory,” J. Eur. Opt. Soc. Rapid Publ. 1, 06004 (2006).
[CrossRef]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, A. S. van de Nes, and S. van Haver, “Extended Nijboer-Zernike approach to aberration and birefringence retrieval in a high-numerical-aperture optical system,” J. Opt. Soc. Am. A 22, 2635-2650 (2005).
[CrossRef]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. van de Nes, “Extended Nijboer-Zernike representation of the field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281-2292 (2003).
[CrossRef]

J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858-870 (2002).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, Vol. 51, E.Wolf, ed. (Elsevier, 2008), pp. 349-468.
[CrossRef]

Duffieux, P. M.

P. M. Duffieux, L'Intégrale de Fourier et ses Applications à l'Optique (privately published, 1946).

Flagello, D.

Freude, W.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175-182 (2004).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on the prolate functions,” in Progress in Optics Vol. IX, E.Wolf, ed. (Pergamon, 1971), pp. 311-407.
[CrossRef]

Fujii, M.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175-182 (2004).
[CrossRef]

Garavaglia, M.

C. Colautti, B. Ruiz, E. E. Sicre, and M. Garavaglia, “Walsh functions: Analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385-1391 (1987).
[CrossRef]

Gedney, S. D.

J. A. Roden and S. D. Gedney, “Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microwave Opt. Technol. Lett. 27, 334-339 (2000).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2004).

Granet, G.

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of the geometry under consideration. O is the origin, R 0 is the radius of the entrance pupil sphere P 0 , n ̂ is the normal to the pupil surface, and Q 0 is a general point in the entrance pupil where we locally define the orthogonal linear p and s polarization states.

Fig. 2
Fig. 2

Definition of the local basis for a general point Q 0 on the entrance pupil sphere with an axial cross-section (left-hand graph) and a cross-section perpendicular to the z axis (right-hand graph).

Fig. 3
Fig. 3

Schematic representation of the Köhler illumination.

Fig. 4
Fig. 4

For three different objects (top row), the RMS error present in the Zernike approximation of the electric field components in the entrance pupil (bottom row) is shown. The system settings for the three objects are, from left to right: normal incidence TM polarized plane wave illumination with an object side numerical aperture of 0.525, and for the middle and right object, normal incidence TE polarization with an object side numerical aperture of 0.2375.

Fig. 5
Fig. 5

RMS error present in the V n , j m ( r , f ) functions when approximated with the series expansion in which the infinite sum is replaced by a summation over t = 1 , 2 , , t max (see Appendix A).

Fig. 6
Fig. 6

For the same three objects as in Fig. 4, the RMS error for the intensity in the image volume is shown as a function of the RMS error in the Zernike expansion of the pupil fields. The area over which the RMS intensity error is calculated is a circle that circumscribes the square images. The final RMS value was obtained by averaging the RMS intensity errors from various through-focus images.

Fig. 7
Fig. 7

Relation between the computation time and the total number of Zernike coefficients used in the expansion of the entrance pupil field is shown.

Equations (69)

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k ̂ 0 = sin α 0 cos θ x ̂ + sin α 0 sin θ y ̂ + cos α 0 z ̂ ,
p ̂ 0 = cos α 0 cos θ x ̂ + cos α 0 sin θ y ̂ sin α 0 z ̂ ,
s ̂ 0 = sin θ x ̂ + cos θ y ̂ ,
E 0 , z = ( E 0 , x cos θ + E 0 , y sin θ ) tan α 0 .
E 0 , p ( ρ , θ ) = E 0 ( ρ , θ ) p ̂ 0 = E 0 , x cos θ + E 0 , y sin θ cos α 0 ,
E 0 , s ( ρ , θ ) = E 0 ( ρ , θ ) s ̂ 0 = E 0 , x sin θ + E 0 , y cos θ .
ϵ v E 2 d S = constant ,
n 0 E 0 2 d S 0 = n 1 E 1 2 d S 1 ,
n 0 R 0 2 d k x , 0 d k y , 0 k 0 k z , 0 E 0 2 = n 1 R 1 2 d k x , 1 d k y , 1 k 1 k z , 1 E 1 2 ,
E 1 = R 0 R 1 k z , 1 k z , 0 d k x , 0 d k y , 0 d k x , 1 d k y , 1 E 0 .
k x , 0 = M k x , 1 , k y , 0 = M k y , 1 ,
E 1 = ( M R 0 R 1 ) n 1 n 0 k z , 1 k 1 k 0 k z , 0 E 0 = ( M R 0 R 1 ) n 1 n 0 ( 1 s 0 2 ρ 2 ) 1 4 [ 1 ( n 1 2 n 0 2 ) M 2 s 0 2 ρ 2 ] 1 4 E 0 ,
E 1 = f 1 R p n 0 n 1 ( 1 s 0 2 ρ 2 ) 1 4 [ 1 ( n 1 2 n 0 2 ) M 2 s 0 2 ρ 2 ] 1 4 E 0 = f 1 R p n 0 n 1 T R ( ρ ) E 0 ,
T R ( ρ ) = ( 1 s 0 2 ρ 2 ) 1 4 [ 1 ( n 1 2 n 0 2 ) M 2 s 0 2 ρ 2 ] 1 4
T I = A ( ρ , θ ) exp [ i Φ ( ρ , θ ) ] ,
E 1 , s ( ρ , θ ) = f 1 T I T R R p n 0 n 1 E 0 , s ( ρ , θ ) = f 1 T I T R R p n 0 n 1 ( E 0 , x ( ρ , θ ) sin θ + E 0 , y ( ρ , θ ) cos θ ) ,
E 1 , p ( ρ , θ ) = f 1 T I T R R p n 0 n 1 E 0 , p ( ρ , θ ) = f 1 T I T R R p n 0 n 1 E 0 , x ( ρ , θ ) cos θ + E 0 , y ( ρ , θ ) sin θ ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 ,
E 1 , x ( ρ , θ ) = E 1 , p ( ρ , θ ) cos ( α 1 ) cos θ E 1 , s ( ρ , θ ) sin θ ,
E 1 , y ( ρ , θ ) = E 1 , p ( ρ , θ ) cos ( α 1 ) sin θ + E 1 , s ( ρ , θ ) cos θ ,
E 1 , z ( ρ , θ ) = E 1 , p ( ρ , θ ) sin ( α 1 ) ,
E 1 , x ( ρ , θ ) = f 1 T I ( ρ , θ ) T R ( ρ ) R p n 0 n 1 [ ( 1 s 0 2 ρ 2 ) 1 2 ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 × { E 0 , x ( ρ , θ ) cos 2 θ + E 0 , y ( ρ , θ ) cos θ sin θ } + { E 0 , x ( ρ , θ ) sin 2 θ E 0 , y ( ρ , θ ) cos θ sin θ } ] ,
E 1 , y ( ρ , θ ) = f 1 T I ( ρ , θ ) T R ( ρ ) R p n 0 n 1 [ ( 1 s 0 2 ρ 2 ) 1 2 ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 × { E 0 , x ( ρ , θ ) cos θ sin θ + E 0 , y ( ρ , θ ) sin 2 θ } + { E 0 , x ( ρ , θ ) cos θ sin θ + E 0 , y ( ρ , θ ) cos 2 θ } ] ,
E 1 , z ( ρ , θ ) = f 1 T I ( ρ , θ ) T R ( ρ ) s 0 ρ R p ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 n 0 n 1 { E 0 , x ( ρ , θ ) cos θ + E 0 , y ( ρ , θ ) sin θ } .
E 0 , x ( ρ , θ ) T I ( ρ , θ ) = n , m β n , x m R n m ( ρ ) exp ( i m θ ) ,
E 0 , y ( ρ , θ ) T I ( ρ , θ ) = n , m β n , y m R n m ( ρ ) exp ( i m θ ) .
E 1 , x ( ρ , θ ) = f 1 T R ( ρ ) 2 R p ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 n 0 n 1 n , m R n m ( ρ ) exp ( i m θ ) × { β n , x m [ { ( 1 s 0 2 ρ 2 ) 1 2 + ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 } { ( 1 n 1 2 M 2 n 0 2 ) s 0 2 ρ 2 cos 2 θ [ ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 + ( 1 s 0 2 ρ 2 ) 1 2 ] } ] β n , y m [ ( 1 n 1 2 M 2 n 0 2 ) s 0 2 ρ 2 sin 2 θ [ ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 + ( 1 s 0 2 ρ 2 ) 1 2 ] ] } ,
E 1 , y ( ρ , θ ) = f 1 T R ( ρ ) 2 R p ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 n 0 n 1 n , m R n m ( ρ ) exp ( i m θ ) × { β n , y m [ { ( 1 s 0 2 ρ 2 ) 1 2 + ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 } + { ( 1 n 1 2 M 2 n 0 2 ) s 0 2 ρ 2 cos 2 θ [ ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 + ( 1 s 0 2 ρ 2 ) 1 2 ] } ] β n , x m [ ( 1 n 1 2 M 2 n 0 2 ) s 0 2 ρ 2 sin 2 θ [ ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 + ( 1 s 0 2 ρ 2 ) 1 2 ] ] } ,
E 1 , z ( ρ , θ ) = f 1 T R ( ρ ) s 0 R p ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 n 0 n 1 × n , m ρ R n m ( ρ ) exp ( i m θ ) { β n , x m cos θ + β n , y m sin θ } .
E 2 ( r , ϕ , f ) = i n 1 s 0 2 λ 0 exp ( i f u 0 ) C E i ( ρ , θ + π ) ( 1 s 0 2 ρ 2 ) 1 2 × exp { i f u 0 [ 1 ( 1 s 0 2 ρ 2 ) 1 2 ] } exp { i 2 π r ρ cos ( θ ϕ ) } ρ d ρ d θ ,
E 2 ( r , ϕ , f ) = i π n 1 f 1 s 0 2 λ 0 [ 1 M f 1 R 1 ] n 0 n 1 exp ( i f u 0 ) n , m ( i ) m exp [ i m ϕ ] × [ β n , x m ( V n , 0 m + s 0 2 ( n 0 2 n 1 2 M 2 2 n 0 2 ) { V n , + 2 m exp [ + 2 i ϕ ] + V n , 2 m exp [ 2 i ϕ ] } i s 0 2 ( n 0 2 n 1 2 M 2 2 n 0 2 ) { V n , + 2 m exp [ + 2 i ϕ ] V n , 2 m exp [ 2 i ϕ ] } i s 0 { V n , + 1 m exp [ + i ϕ ] V n , 1 m exp [ i ϕ ] } ) + β n , y m ( i s 0 2 ( n 0 2 n 1 2 M 2 2 n 0 2 ) { V n , + 2 m exp [ + 2 i ϕ ] V n , 2 m exp [ 2 i ϕ ] } V n , 0 m s 0 2 ( n 0 2 n 1 2 M 2 2 n 0 2 ) { V n , + 2 m exp [ + 2 i ϕ ] + V n , 2 m exp [ 2 i ϕ ] } s 0 { V n , + 1 m exp [ + i ϕ ] + V n , 1 m exp [ i ϕ ] } ) ] ,
V n , j m ( r , f ) = 0 1 ρ j { ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 + ( 1 s 0 2 ρ 2 ) 1 2 } j + 1 ( 1 s 0 2 ρ 2 ) 1 4 ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 3 4 × exp [ i f u 0 ( 1 1 s 0 2 ρ 2 ) ] R n m ( ρ ) J m + j ( 2 π r ρ ) ρ d ρ .
d 2 P = L ( x , y ; 0 , 0 ) d S S d Ω O = L ( x , y ; 0 , 0 ) d S S d S O cos ( α ) f C 2 ,
P j = L ( x j , y j ; 0 , 0 ) S j S O cos ( α j ) f C 2 ,
P j = ϵ 0 c n 0 E O 2 S O cos ( α j ) .
E O = L ( x j , y j ; 0 , 0 ) S j ϵ 0 c n 0 f C 2 .
x j f C = η j cos ( σ j ) s obj ,
y j f C = η j sin ( σ j ) s obj ,
k ̂ j = ( η j s obj cos ( σ j ) x ̂ , η j s obj sin ( σ j ) y ̂ , 1 η j 2 s obj 2 z ̂ ) .
E tot = E sca + E zero ,
E sca ( r ) = Ω ( n × E sca ( r ) ) G ͇ H ( r , r ) ( n × H sca ( r ) ) G ͇ E ( r , r ) d r 2 .
F [ E sca ] ( k x , k y , z ) = Ω [ n × E sca ( r ) ] F [ G ͇ H ] ( r , k sc ) ( n × H sca ( r ) ) F [ G ͇ E ] ( r , k sc ) d r 2 ,
E 0 ( k x , sc , k y , sc , z ) = F [ E sca ] k z , sc k sc .
E 2 ( r , ϕ , f ) = C 1 ( f ) n , m [ β n , x m C 2 ( m , n , r , ϕ , f ) + β n , y m C 3 ( m , n , r , ϕ , f ) ] ,
N f × N r × N ϕ × 2 N Z ,
V n , j m ( r , f ) = 0 1 ρ j { ( 1 s 0 2 ρ 2 ) 1 2 + ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 } j + 1 ( 1 s 0 2 ρ 2 ) 1 4 ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 3 4 × exp [ i f u 0 ( 1 1 s 0 2 ρ 2 ) ] R n m ( ρ ) J m + j ( 2 π r ρ ) ρ d ρ .
exp [ i f u 0 ( 1 1 s 0 2 ρ 2 ) ] { ( 1 s 0 2 ρ 2 ) 1 2 + ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 } j + 1 ( 1 s 0 2 ρ 2 ) 1 4 ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 3 4 = exp { g + i f ρ 2 } t = 0 B t ρ 2 t .
F ( ρ ) = i f u 0 ( 1 1 s 0 2 ρ 2 ) + ( j + 1 ) ln [ ( 1 s 0 2 ρ 2 ) 1 2 + ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 ] 1 4 ln ( 1 s 0 2 ρ 2 ) 3 4 ln ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) ,
A ( ρ ) = 1 1 s 0 2 ρ 2 = n = 0 a n ρ 2 n ,
B ( ρ ) = ln [ ( 1 s 0 2 ρ 2 ) 1 2 + ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) 1 2 ] = n = 0 b n ρ 2 n ,
C ( ρ ) = ln ( 1 s 0 2 ρ 2 ) + 3 ln ( 1 n 1 2 M 2 s 0 2 ρ 2 n 0 2 ) = n = 0 c n ρ 2 n ,
F ( ρ ) = i f u 0 A ( ρ ) + ( j + 1 ) B ( ρ ) 1 4 C ( ρ ) = n = 0 f n ρ 2 n ,
f n = i f u 0 a n + ( j + 1 ) b n 1 4 c n , n = 0 , 1 , .
a 0 = 0 ; a n = ( 1 2 n ) ( 1 ) n s 0 2 n , n = 1 , 2 , .
f ( x ) = ln [ ( 1 x ) 1 2 + ( 1 a x ) 1 2 ] .
f ( x ) = 1 2 1 ( 1 x ) 1 2 + ( 1 a x ) 1 2 ( 1 ( 1 x ) 1 2 + a ( 1 a x ) 1 2 ) = 1 2 ( 1 x ) 1 2 ( 1 a x ) 1 2 ( 1 x ) ( 1 a x ) ( 1 a x ) 1 2 + a ( 1 x ) 1 2 ( 1 x ) 1 2 ( 1 a x ) 1 2 = ( 1 ( 1 a x ) 1 2 ( 1 x ) 1 2 ) / 2 x = 1 2 t = 1 [ ( 1 ) t r = 0 t ( 1 2 r ) ( 1 2 t r ) a r ] x t 1 ,
f ( x ) = ln 2 t = 1 [ ( 1 ) t 2 t r = 0 t ( 1 2 r ) ( 1 2 t r ) a r ] x t .
b 0 = ln 2 ;
b n = ( 1 ) n s 0 2 n 2 n r = 0 n ( 1 2 r ) ( 1 2 n r ) n 1 2 r M 2 r n 0 2 r , n = 1 , 2 , .
c 0 = 0 ; c n = ( s 0 ) 2 n ( 1 + 3 ( n 1 M / n 0 ) 2 n ) n , n = 1 , 2 , .
F ( ρ ) = g + i f ρ 2 + n = 0 A n ρ 2 n ,
β 2 k 0 = n = k 2 k + 1 k + 1 ( n k ) ( n + k + 1 n ) f n , k = 0 , 1 , ,
β 0 0 R 0 0 ( ρ ) + β 2 0 R 2 0 ( ρ ) = ( β 0 0 β 2 0 ) + 2 β 2 0 ρ 2
g = β 0 0 β 2 0 , f = ( 2 i ) β 2 0 ,
A 0 = f 0 g , A 1 = f 1 i f ; A n = f n , n = 2 , 3 , .
exp ( n = 0 A n ρ 2 n ) = t = 0 B t ρ 2 t .
B 0 = exp ( A 0 ) , B t + 1 = j = 0 t t + 1 j t + 1 A t + 1 j B j , t = 0 , 1 , .
R n m ( ρ ) = ρ m s = 0 p C s ρ 2 s , C s = ( 1 ) p s ( q + s p ) ( p s ) ,
V n , j m ( r , f ) = s = 0 p t = 0 C s B t T j + m + 2 s + 2 t m + j ( r , f ) ,
T l k ( r , f ) = 0 1 ρ l e i f ρ 2 J k ( 2 π r ρ ) ρ d ρ .

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