Abstract

We explore the diffraction limited focusing and confocal imaging properties of a high NA parabolic mirror for confocal imaging and spectroscopy of nanoparticles and single molecules. Vector field calculations of the electric fields near focus for both linear and radially polarized illumination are discussed and show that the optical field can be similar tightly focused as in the case of a high NA objective lens. Furthermore they show that a high NA parabolic mirror allows an easy orientation of the polarization of the illuminating light in all spatial directions. The simulation of confocal imaging of single molecules is discussed and yields, that the use of radially polarized excitation light gives an easy access to their orientations.

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References

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  1. W. P. Ambrose, T. Basché, and W. E. Moerner, "Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation," J. Chem. Phys. 95, 7150-7163 (1991).
    [CrossRef]
  2. L. Fleury, P. Tamarat, B. Lounis, J. Bernard, and M. Orrit. "Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K," Chem. Phys. Lett. 236, 87-95 (1995).
    [CrossRef]
  3. H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, and J. Schmidt, "An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field," Rev. Sci. Instrum. 66, 4853-4856 (1995).
    [CrossRef]
  4. Y. Durand, J. C. Woehl, B. Viellerobe, W. Göhde, and M. Orrit, "New design of a cryostat-mounted scanning near-field optical microscope for single molecule spectroscopy," Rev. Sci. Instrum. 70, 1318-1325 (1999).
    [CrossRef]
  5. J. Enderlein, T. Ruckstuhl, and S. Seeger, "Highly efficient optical detection of surface-generated fluorescence," Appl. Opt. 38, 724-732 (1999).
    [CrossRef]
  6. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000), http://www.opticsexpress.org/oearchive/source/22809.htm
    [CrossRef] [PubMed]
  7. L. Novotny, E. J. Sánchez, and X. S. Xie, "Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams," Ultramicroscopy 71, 21-29 (1998).
    [CrossRef]
  8. E. J. Sánchez, L. Novotny, and X. S. Xie, "Near-field fluorescence microscopy based on two-photon excitation with metal tips," Phys. Rev. Lett. 82, 4014-4017 (1999).
    [CrossRef]
  9. 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]
  10. V. S. Ignatovsky, "Diffraction by a parabolic mirror having arbitrary opening," Trans. Opt. Inst. Petrograd 1, paper 5 (1920), in Russian.
  11. C. J. R. Sheppard, A. Choudhury, and J. Gannaway, "Electromagnetic field near the focus of wide-angular lens and mirror systems," IEE J. Microw. Opt. Acoust. 1, 129-132 (1977).
    [CrossRef]
  12. R. Barakat, "Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration," Appl. Opt. 26, 3790-3795 (1987).
    [CrossRef] [PubMed]
  13. P. Varga and P. Török, "Focusing of electromagnetic waves by paraboloid mirrors. I. Theory," J. Opt. Soc. Am. A 17, 2081-2089 (2000).
    [CrossRef]
  14. P. Varga and P. Török, "Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results," J. Opt. Soc. Am. A 17, 2090-2095 (2000).
    [CrossRef]
  15. E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. Roy. Soc. A 253, 349-357 (1959).
    [CrossRef]
  16. J. J. Stamnes, Waves in focal regions, (Hilger, Bristol, UK, 1986), Sec. 16.1.2.
  17. R. H. Jordan and D. G. Hall, "Free-space azimuthal paraxial wave-equation: the azimuthal Bessel-Gauss beam solution," Opt. Lett. 19, 427-429 (1994).
    [CrossRef] [PubMed]
  18. D. G. Hall, "Vector-beam solutions of Maxwell's wave equation," Opt. Lett. 21, 9-11 (1996).
    [CrossRef] [PubMed]
  19. P. L. Greene and D. G. Hall, "Diffraction characteristics of the azimuthal Bessel-Gauss beam," J. Opt. Soc. Am. A 13, 962-966 (1996).
    [CrossRef]
  20. P. L. Greene and D. G. Hall, "Properties and diffraction of vector Bessel-Gauss beams," J. Opt. Soc. Am. A 15, 3020-3027 (1998).
    [CrossRef]
  21. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
    [CrossRef]
  22. J. J. Wynne, "Generation of the rotationally symmetric TE01 and TM01 modes from a wavelength-tunable laser," IEEE J. Quant. Elec. QE-10, 125-127 (1974).
    [CrossRef]
  23. R. Yamaguchi, T. Nose, and S. Sato, "Liquid crystal polarizers with axially symmetrical properties," Jpn. J. Appl. Phys. Pt. 1 28, 1730-1731 (1989).
    [CrossRef]
  24. S. C. Tidwell, D. H. Ford, and W. D. Kimura, "Generating radially polarized beams interferometrically," Appl. Opt. 29, 2234-2239 (1990).
    [CrossRef] [PubMed]
  25. E. G. Churin, J. Hoßfeld, and T. Tschudi, "Polarization configurations with singular point formed by computer generated holograms," Opt. Commun. 99, 13-17 (1993).
    [CrossRef]
  26. S. C. Tidwell, G. H. Kim, and W. D. Kimura, "Efficient radially polarized laser-beam generation with a double interferometer," Appl. Opt. 32, 5222-5229 (1993).
    [CrossRef] [PubMed]
  27. M. Stalder and M. Schadt, "Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt. Lett. 21, 1948-1950 (1996).
    [CrossRef] [PubMed]
  28. K. S. Youngworth and T. G. Brown, "Inhomogenous polarization in scanning optical microscopy," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII,J.-A.Conchello,C.J. Cogswell, T. Wilson, editors, Proc. SPIE 3919, 75-85 (2000).
  29. M. Schrader and S. W. Hell, "Wavefronts in the focus of a light microscope," J. Microsc. 184, 143-148 (1996).
  30. C. J. R. Sheppard, "Aberrations in high aperture optical systems," Optik 105, 29-33 (1997).
  31. J. Sepiol, J. Jasny, J. Keller, and U. P. Wild, "Single molecules observed by immersion mirror objective. The orientation of terrylene molecules via the direction of its transition dipole moment," Chem. Phys. Lett. 273, 444- 448 (1997).
    [CrossRef]
  32. R. M. Dickson, D. J. Norris, and W. E. Moener, "Simultaneous imaging of individual molecules aligned both parallel and perpendicular to the optic axis," Phys. Rev. Lett. 81, 5322-5325 (1998).
    [CrossRef]
  33. T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, "Polarization spectroscopy of single fluorescent molecules," J. Phys. Chem. B 103, 6839-6850 (1999).
    [CrossRef]
  34. B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
    [CrossRef] [PubMed]
  35. C. J. R. Sheppard and M. Gu, "Imaging by a high aperture optical system," J. Mod. Opt. 40, 1631-1651 (1993).
    [CrossRef]
  36. C. J. R. Sheppard and P. Török, "An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy," Bioimaging 5, 205-218 (1997).
    [CrossRef]
  37. J. Enderlein, "Theoretical study of detection of a dipole emitter through an objective with high numerical aperture," Opt. Lett. 25, 634-636 (2000).
    [CrossRef]
  38. J. D. Jackson, Classical Electrodynamics, second edition (Wiley & Sons, New York, USA, 1975), Chap. 9.2.
  39. T. Plakhotnik, E. A. Donley, and U. P. Wild, "Single-molecule spectroscopy," Annu. Rev. Phys. Chem. 48, 181- 212 (1997).
    [CrossRef] [PubMed]

Other

W. P. Ambrose, T. Basché, and W. E. Moerner, "Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation," J. Chem. Phys. 95, 7150-7163 (1991).
[CrossRef]

L. Fleury, P. Tamarat, B. Lounis, J. Bernard, and M. Orrit. "Fluorescence spectra of single pentacene molecules in p-terphenyl at 1.7 K," Chem. Phys. Lett. 236, 87-95 (1995).
[CrossRef]

H. van der Meer, J. A. J. M. Disselhorst, J. Koehler, A. C. J. Brouwer, E. J. J. Groenen, and J. Schmidt, "An insert for single-molecule magnetic-resonance spectroscopy in an external magnetic field," Rev. Sci. Instrum. 66, 4853-4856 (1995).
[CrossRef]

Y. Durand, J. C. Woehl, B. Viellerobe, W. Göhde, and M. Orrit, "New design of a cryostat-mounted scanning near-field optical microscope for single molecule spectroscopy," Rev. Sci. Instrum. 70, 1318-1325 (1999).
[CrossRef]

J. Enderlein, T. Ruckstuhl, and S. Seeger, "Highly efficient optical detection of surface-generated fluorescence," Appl. Opt. 38, 724-732 (1999).
[CrossRef]

K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000), http://www.opticsexpress.org/oearchive/source/22809.htm
[CrossRef] [PubMed]

L. Novotny, E. J. Sánchez, and X. S. Xie, "Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams," Ultramicroscopy 71, 21-29 (1998).
[CrossRef]

E. J. Sánchez, L. Novotny, and X. S. Xie, "Near-field fluorescence microscopy based on two-photon excitation with metal tips," Phys. Rev. Lett. 82, 4014-4017 (1999).
[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]

V. S. Ignatovsky, "Diffraction by a parabolic mirror having arbitrary opening," Trans. Opt. Inst. Petrograd 1, paper 5 (1920), in Russian.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, "Electromagnetic field near the focus of wide-angular lens and mirror systems," IEE J. Microw. Opt. Acoust. 1, 129-132 (1977).
[CrossRef]

R. Barakat, "Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration," Appl. Opt. 26, 3790-3795 (1987).
[CrossRef] [PubMed]

P. Varga and P. Török, "Focusing of electromagnetic waves by paraboloid mirrors. I. Theory," J. Opt. Soc. Am. A 17, 2081-2089 (2000).
[CrossRef]

P. Varga and P. Török, "Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results," J. Opt. Soc. Am. A 17, 2090-2095 (2000).
[CrossRef]

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

J. J. Stamnes, Waves in focal regions, (Hilger, Bristol, UK, 1986), Sec. 16.1.2.

R. H. Jordan and D. G. Hall, "Free-space azimuthal paraxial wave-equation: the azimuthal Bessel-Gauss beam solution," Opt. Lett. 19, 427-429 (1994).
[CrossRef] [PubMed]

D. G. Hall, "Vector-beam solutions of Maxwell's wave equation," Opt. Lett. 21, 9-11 (1996).
[CrossRef] [PubMed]

P. L. Greene and D. G. Hall, "Diffraction characteristics of the azimuthal Bessel-Gauss beam," J. Opt. Soc. Am. A 13, 962-966 (1996).
[CrossRef]

P. L. Greene and D. G. Hall, "Properties and diffraction of vector Bessel-Gauss beams," J. Opt. Soc. Am. A 15, 3020-3027 (1998).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

J. J. Wynne, "Generation of the rotationally symmetric TE01 and TM01 modes from a wavelength-tunable laser," IEEE J. Quant. Elec. QE-10, 125-127 (1974).
[CrossRef]

R. Yamaguchi, T. Nose, and S. Sato, "Liquid crystal polarizers with axially symmetrical properties," Jpn. J. Appl. Phys. Pt. 1 28, 1730-1731 (1989).
[CrossRef]

S. C. Tidwell, D. H. Ford, and W. D. Kimura, "Generating radially polarized beams interferometrically," Appl. Opt. 29, 2234-2239 (1990).
[CrossRef] [PubMed]

E. G. Churin, J. Hoßfeld, and T. Tschudi, "Polarization configurations with singular point formed by computer generated holograms," Opt. Commun. 99, 13-17 (1993).
[CrossRef]

S. C. Tidwell, G. H. Kim, and W. D. Kimura, "Efficient radially polarized laser-beam generation with a double interferometer," Appl. Opt. 32, 5222-5229 (1993).
[CrossRef] [PubMed]

M. Stalder and M. Schadt, "Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt. Lett. 21, 1948-1950 (1996).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, "Inhomogenous polarization in scanning optical microscopy," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition Processing VII,J.-A.Conchello,C.J. Cogswell, T. Wilson, editors, Proc. SPIE 3919, 75-85 (2000).

M. Schrader and S. W. Hell, "Wavefronts in the focus of a light microscope," J. Microsc. 184, 143-148 (1996).

C. J. R. Sheppard, "Aberrations in high aperture optical systems," Optik 105, 29-33 (1997).

J. Sepiol, J. Jasny, J. Keller, and U. P. Wild, "Single molecules observed by immersion mirror objective. The orientation of terrylene molecules via the direction of its transition dipole moment," Chem. Phys. Lett. 273, 444- 448 (1997).
[CrossRef]

R. M. Dickson, D. J. Norris, and W. E. Moener, "Simultaneous imaging of individual molecules aligned both parallel and perpendicular to the optic axis," Phys. Rev. Lett. 81, 5322-5325 (1998).
[CrossRef]

T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, "Polarization spectroscopy of single fluorescent molecules," J. Phys. Chem. B 103, 6839-6850 (1999).
[CrossRef]

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

C. J. R. Sheppard and M. Gu, "Imaging by a high aperture optical system," J. Mod. Opt. 40, 1631-1651 (1993).
[CrossRef]

C. J. R. Sheppard and P. Török, "An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy," Bioimaging 5, 205-218 (1997).
[CrossRef]

J. Enderlein, "Theoretical study of detection of a dipole emitter through an objective with high numerical aperture," Opt. Lett. 25, 634-636 (2000).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, second edition (Wiley & Sons, New York, USA, 1975), Chap. 9.2.

T. Plakhotnik, E. A. Donley, and U. P. Wild, "Single-molecule spectroscopy," Annu. Rev. Phys. Chem. 48, 181- 212 (1997).
[CrossRef] [PubMed]

Supplementary Material (3)

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Figures (12)

Fig. 1.
Fig. 1.

Imaging geometry for a) a parabolic mirror and b) an aplanatic objective lens (fulfilling the Abbe sine condition r=fo sinθo ). A ray incident at a distance r from the optical axis (z) reaches the sample under a certain angle θm in the mirror setup, while the same ray hits the sample under a much smaller angle θo in the lens setup. While the perpendicular component (s) of the electric field E changes sign when it is reflected on the mirror surface, it does not when it is refracted by the lens. p denotes the parallel component of E, fm and fo denote the focal spheres and s is a unit vector pointing from the focal sphere to the focus. c) Intensity distribution used for linear polarized light (solid line, Gaussian beam) and radially polarized light (dashed line, Bessel-Gauss beam), with the beam waist radius w0 =3/2× pupil radius).

Fig. 2.
Fig. 2.

Contour plots of constant |E|2 in the x, y-plane of a focused Gaussian beam polarized along the x-direction in the upper row for an objective lens (NA=1.40, n=1.518, fo =4.5 mm) and in the lower row for a parabolic mirror (NA=1.515, n=1.518, fm =4.5 mm). a) and e) show |E|2, b) and f) show |Ex|2, c) and g) show |Ey|2×8, d) and h) show |Ez|2. A logarithmic scale is used with a factor of two between adjacent contour levels. The length of the scale bar corresponds to one wavelength divided by the refractive index n.

Fig. 3.
Fig. 3.

Contourplots of constant |E|2 in the focal region of a focused Gaussian beam polarized along the x-axis in the upper row for an objective lens and in the lower row for a parabolic mirror. a) and e) show sections of |E|2 in the x,z-plane; b) and f) show sections of |Ex|2 in the x,z-plane; c) and g) show sections of |Ey|2×8 along the z-axis and the diagonal between the x- and y-axis; d) and h) show sections of |Ez|2 in the x,z-plane. A logarithmic scale is used with a factor of two between adjacent contour levels. The length of the scale bar equals one wavelength divided by the refractive index n.

Fig. 4.
Fig. 4.

(400kB) Movie showing a section through |Re(E(t))|2 in the x,z- plane close to the focus for an objective lens (a) and a parabolic mirror (b) with linear polarized illumination along the x-axis. A logarithmic scale is used with a factor of v2 between adjacent contour levels. The vectors show the orientation of the electric field. The time between successive images is one 12th of the optical period. The length of the scale bar is equal to one wavelength λ/n.

Fig. 5.
Fig. 5.

Contourplots of constant |E|2 in the x, y-plane of a focused radial polarized Bessel-Gauss beam in the upper row for an objective lens and in the lower row for a parabolic mirror. a) and d) show |E|2, b) and e) show the radial component |Er|2, c) and f) show the longitudinal component |Ez|2. A logarithmic scale is used with a factor of two between adjacent contour levels. The length of the scale bar equals one wavelength λ/n.

Fig. 6.
Fig. 6.

Contourplots of constant |E|2 in the focal region of a focused radially polarized Bessel-Gauss beam in the upper row for an objective lens and in the lower row for a parabolic mirror. a) and d) show |E|2, b) and e) show the radial component |Er|2, c) and f) show the longitudinal component |Ez|2. A logarithmic scale is used with a factor of two between adjacent contour levels. The length of the scale bar corresponds to one wavelength λ/n.

Fig. 7.
Fig. 7.

Variation of the electric field intensity, |E|2, in the x, y-plane of an objective lens (a) and a parabolic mirror (b) for linear polarized illumination along the x-axis (along the x-axis --, along the y-axis …) and radial polarization (full line). Both figures are normalized to the same optical power (1 mW) focussed by the objective lens and by the parabolic mirror. Hence the field strength can directly be obtained from √|E|2.

Fig. 8.
Fig. 8.

Movies showing the formation of coma in the focal region of a parabolic mirror illuminated with radially polarized light. Successive images show the electric field intensity |E|2 for an increasing angle (0°, 0.00318°, 0.00637°, 0.00955°, 0.01273° and 0.01592°) between the mirror axis and the k-vector of the incident wave. a) (91kB) section in the x, y-plane and b) (97kB) section in the x, z-plane. A logarithmic scale is used with a factor of two between adjacent contour levels. The length of the scale bar equals one wavelength λ, which is set to 500 nm, divided by the refractive index n.

Fig. 9.
Fig. 9.

Scheme illustrating the confocal imaging geometry of a point-dipole emitter located in the focus of a parabolic mirror with a focal length fm . sm is a unit vector pointing from the focus to the mirror surface. The light is collected by a lens with focal length fo and so is a unit vector pointing from the surface of the focal sphere of the lens to the image focus on the optical axis.

Fig. 10.
Fig. 10.

Images of |E|2, for a point dipole emitter in the focal region of a parabolic mirror. In the images a) - c) the dipole lies in the focal point at r m =(0, 0, 0) and is oriented along the x, y and z-axis, respectively. For the images d) to f) the dipole lies one wavelength away from the optical axis at r m =λ/n (cos 30°, sin 30°, 0) indicated by the arrows. The black circles in the images a) and c) have a diameter of 1.22 λ/NAo , corresponding to the diameter of the Airy disk. A logarithmic scale is used with a factor of 2 between adjacent contour levels. The scale bar length is the magnification M times the vacuum-wavelength λ.

Fig. 11.
Fig. 11.

Contour plots of the collection efficiency function (CEF) for different orientations of a dipole emitter scanned a) in the x, y-plane and b) in the x, z-plane of a parabolic mirror collector. The angles between the dipole moment p and the coordinate system, θ and φ, are shown in the lower left images. A linear intensity scale is used with a difference of 0.05 between adjacent contour levels. The maximum of the scale corresponds to a collection efficiency of 0.5. The scale bar has a length of one wavelength λ/n.

Fig. 12.
Fig. 12.

Confocal images, |E|2, of a dipole emitter for different orientations. a) and b) linear excitation polarization along the x-axis, c) and d) radial polarization. The illuminated aperture is reduced to 1.21 for radial polarization. In a) and c) the dipole is scanned in the x, y-plane, whereas in b) and d) it is scanned in the x, z-plane. The dipole orientation angles as well as the axis are shown in the lower left images. A logarithmic scale is used with a factor of 2 between adjacent contour levels. All images in a) and b) as well as in c) and d) are scaled identically with the maximum given by the highest intensity which is reached for a) and b) by the dipole oriented parallel to the polarization direction and for c) and d) by the dipole oriented along the optical axis z. The scale bar length is one wavelength λ/n.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

E ( P ) = i k f 2 π Ω E f ( θ , φ ) e i k s · r P sin θ d θ d φ
E x = i k f 2 ( I 0 , l + I 2 , l cos 2 φ P ) ,
E y = i k f 2 I 2 , l sin 2 φ P ,
E z = k f I 1 , l cos φ P .
I 0 , l = α 0 α 1 l 0 ( θ ) 2 sin θ 1 + cos θ ( 1 + cos θ ) J 0 ( k r P sin θ sin θ P ) e i k r P cos θ cos θ P d θ ,
I 1 , l = α 0 α 1 l 0 ( θ ) 2 sin 2 θ 1 + cos θ J 1 ( k r P sin θ sin θ P ) e i k r P cos θ cos θ P d θ ,
I 2 , l = α 0 α 1 l 0 ( θ ) 2 sin θ 1 + cos θ ( 1 cos θ ) J 2 ( k r P sin θ sin θ P ) e i k r P cos θ cos θ P d θ ,
E r = k f I 1 , r ,
E a = 0 ,
E z = i k f I 0 , r
I 0 , r = α 0 α 1 l 0 ( θ ) 2 sin 2 θ 1 + cos θ J 0 ( k r P sin θ sin θ P ) e i k r P cos θ cos θ P d θ ,
I 1 , r = α 0 α 1 l 0 ( θ ) sin ( 2 θ ) 1 + cos θ J 1 ( k r P sin θ sin θ P ) e i k r P cos θ cos θ P d θ ,
E de = 1 4 π ε o k 2 ( s m × p ) × s m · 1 r · exp ( i k r ω t )
E ( r o ) = i k 3 f o 16 π 3 ε o θ o , min θ o , max 0 2 π ( 1 + cos θ m ) 2 f m cos θ o [ ( p · e m , ) e o ,
( p · e ) e ] · exp ( i k ( r o s o n m r m s m ) ) sin θ o d θ o d φ
e = ( sin φ , cos φ , 0 ) ,
e m , = ( cos θ m cos φ , cos θ m sin φ , sin θ m ) ,
e o , = ( cos θ o cos φ , cos θ o sin φ , sin θ o ) ,
s m = ( sin θ m cos φ , sin θ m sin φ , cos θ m ) ,
s o = ( sin θ o cos φ , sin θ o sin φ , cos θ o ) .
θ m = 2 arctan ( M 2 sin θ o ) ,
θ o , max = arcsin ( 2 M . tan ( arcsin ( N A m n m ) 2 ) ) ,
θ o , min = arcsin ( r samp f o )
R ( r m ) = c p · E ( r m ) 2
s = ( sin θ m cos φ , sin θ m sin φ , cos θ m ) ,
E p = E 0 , p ( cos θ m cos φ , cos θ m sin φ , sin θ m ) ,
E s = E 0 , s ( sin φ , cos φ , 0 ) ,
r P = r P ( sin θ P cos φ P , sin θ P sin φ P , cos θ P )
E f = 2 l 0 ( θ ) 1 + cos θ m { cos θ m cos φ cos θ m sin φ sin θ m
exp [ i k s · r p ] = exp [ i k cos θ cos θ P ] · exp [ i k sin θ sin θ P cos ( φ φ P ) ]
0 2 π cos n φ exp ( i ρ cos ( φ ϕ ) ) d φ = 2 π i n J n ( ρ ) cos n ϕ ,
0 2 π sin n φ exp ( i ρ cos ( φ ϕ ) ) d φ = 2 π i n J n ( ρ ) sin n ϕ

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