Abstract

We theoretically and experimentally investigate the effect of imperfect vector symmetry on radially polarized beams focused by an aplanatic solid immersion lens at a numerical aperture of 3.3. We experimentally achieve circularly symmetric focused spot with a full-width-half-maximum of ~λ0/5.7 at λ0 = 1310nm, free-space wavelength. The tight spatial confinement and overall circular symmetry of the focused radially polarized beam are found to be sensitive to perturbations of its cylindrical polarization symmetry. The addition of a liquid crystal based variable retarder to the optical path can effectively ensure the vector symmetry and achieve circularly symmetric focused spots at such high numerical aperture conditions.

© 2014 Optical Society of America

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    [CrossRef]
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2013 (3)

2012 (1)

X. Li, T.-H. Lan, C.-H. Tien, M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).
[CrossRef] [PubMed]

2011 (1)

2009 (3)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[CrossRef]

S. Ramachandran, P. P. Kristensen, M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009).
[CrossRef] [PubMed]

S. H. Goh, C. J. R. Sheppard, “High aperture focusing through a spherical interface: Application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282(5), 1036–1041 (2009).
[CrossRef]

2008 (4)

F. H. Köklü, J. I. Quesnel, A. N. Vamivakas, S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “Widefield subsurface microscopy of integrated circuits,” Opt. Express 16(13), 9501–9506 (2008).
[CrossRef] [PubMed]

K. A. Serrels, E. Ramsay, R. J. Warburton, D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

G. M. Lerman, U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16(7), 4567–4581 (2008).
[CrossRef] [PubMed]

2007 (3)

M. A. Ahmed, A. Voss, M. M. Vogel, T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32(22), 3272–3274 (2007).
[CrossRef] [PubMed]

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

2006 (1)

2005 (2)

2004 (2)

C. J. R. Sheppard, A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43(22), 4322–4327 (2004).
[CrossRef] [PubMed]

S. F. Pereira, A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004).
[CrossRef]

2003 (1)

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (2)

L. Novotny, R. D. Grober, K. Karrai, “Reflected image of a strongly focused spot,” Opt. Lett. 26(11), 789–791 (2001).
[CrossRef] [PubMed]

S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78(26), 4071 (2001).
[CrossRef]

1997 (1)

1990 (1)

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57(24), 2615 (1990).
[CrossRef]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Agarwal, K.

Ahmed, M. A.

Atatüre, M.

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

Badolato, A.

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Beversluis, M. R.

Bifano, T.

Chen, R.

Chen, X.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Choudhury, A.

Dimarcello, F. V.

Dorn, R.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Dreiser, J.

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

Fleming, J.

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Ghalmi, S.

Goh, S. H.

S. H. Goh, C. J. R. Sheppard, “High aperture focusing through a spherical interface: Application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282(5), 1036–1041 (2009).
[CrossRef]

Goldberg, B. B.

Y. Lu, T. Bifano, S. Unlü, B. B. Goldberg, “Aberration compensation in aplanatic solid immersion lens microscopy,” Opt. Express 21(23), 28189–28197 (2013).
[CrossRef] [PubMed]

F. H. Köklü, J. I. Quesnel, A. N. Vamivakas, S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “Widefield subsurface microscopy of integrated circuits,” Opt. Express 16(13), 9501–9506 (2008).
[CrossRef] [PubMed]

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78(26), 4071 (2001).
[CrossRef]

A. Yurt, M. D. W. Grogan, S. Ramachandran, B. B. Goldberg, M. S. Ünlü, “Vortex beams in the vectorial focusing regime,” Opt. Lett. (submitted to).

Golowich, S.

Graf, T.

Grober, R. D.

Grogan, M. D. W.

A. Yurt, M. D. W. Grogan, S. Ramachandran, B. B. Goldberg, M. S. Ünlü, “Vortex beams in the vectorial focusing regime,” Opt. Lett. (submitted to).

Gu, M.

X. Li, T.-H. Lan, C.-H. Tien, M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).
[CrossRef] [PubMed]

H. Lin, B. Jia, M. Gu, “Generation of an axially super-resolved quasi-spherical focal spot using an amplitude-modulated radially polarized beam,” Opt. Lett. 36(13), 2471–2473 (2011).
[CrossRef] [PubMed]

Hayashi, S.

Ichimura, I.

Imamoglu, A.

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

Ippolito, S. B.

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Jia, B.

Karrai, K.

Kino, G. S.

Köklü, F. H.

Kozawa, Y.

Kristensen, P. P.

Lan, T.-H.

X. Li, T.-H. Lan, C.-H. Tien, M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).
[CrossRef] [PubMed]

Leger, J. R.

Lerman, G. M.

Leuchs, G.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Levy, U.

Li, X.

X. Li, T.-H. Lan, C.-H. Tien, M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).
[CrossRef] [PubMed]

Lin, H.

Lu, Y.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Mansfield, S. M.

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57(24), 2615 (1990).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Monberg, E.

Novotny, L.

Pereira, S. F.

S. F. Pereira, A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004).
[CrossRef]

Phang, J. C. H.

Quabis, S.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Quesnel, J. I.

Ramachandran, S.

Ramsay, E.

K. A. Serrels, E. Ramsay, R. J. Warburton, D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

Reid, D. T.

K. A. Serrels, E. Ramsay, R. J. Warburton, D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[CrossRef]

Sato, S.

Serrels, K. A.

K. A. Serrels, E. Ramsay, R. J. Warburton, D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Sheppard, C. J. R.

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Stranick, S. J.

Swan, A. K.

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

Tien, C.-H.

X. Li, T.-H. Lan, C.-H. Tien, M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).
[CrossRef] [PubMed]

Unlü, S.

Ünlü, M. S.

F. H. Köklü, J. I. Quesnel, A. N. Vamivakas, S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “Widefield subsurface microscopy of integrated circuits,” Opt. Express 16(13), 9501–9506 (2008).
[CrossRef] [PubMed]

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78(26), 4071 (2001).
[CrossRef]

A. Yurt, M. D. W. Grogan, S. Ramachandran, B. B. Goldberg, M. S. Ünlü, “Vortex beams in the vectorial focusing regime,” Opt. Lett. (submitted to).

Vamivakas, A. N.

F. H. Köklü, J. I. Quesnel, A. N. Vamivakas, S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “Widefield subsurface microscopy of integrated circuits,” Opt. Express 16(13), 9501–9506 (2008).
[CrossRef] [PubMed]

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

van de Nes, A. S.

S. F. Pereira, A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004).
[CrossRef]

Vogel, M. M.

Voss, A.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

Warburton, R. J.

K. A. Serrels, E. Ramsay, R. J. Warburton, D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

Wisk, P.

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[CrossRef]

Yan, M. F.

Yilmaz, S. T.

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

Yurt, A.

A. Yurt, M. D. W. Grogan, S. Ramachandran, B. B. Goldberg, M. S. Ünlü, “Vortex beams in the vectorial focusing regime,” Opt. Lett. (submitted to).

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[CrossRef]

Q. Zhan, J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41(22), 4630–4637 (2002).
[CrossRef] [PubMed]

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57(24), 2615 (1990).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78(26), 4071 (2001).
[CrossRef]

Nano Lett. (1)

A. N. Vamivakas, M. Atatüre, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoǧlu, M. S. Ünlü, “Strong extinction of a far-field laser beam by a single quantum dot,” Nano Lett. 7(9), 2892–2896 (2007).
[CrossRef] [PubMed]

Nat. Commun. (1)

X. Li, T.-H. Lan, C.-H. Tien, M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).
[CrossRef] [PubMed]

Nat. Photonics (2)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[CrossRef]

K. A. Serrels, E. Ramsay, R. J. Warburton, D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

New J. Phys. (1)

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Figures (6)

Fig. 1
Fig. 1

The intensity map of RPB and its projection through an analyzer with the orientation shown with arrows. The numbers show the phase retardance between the two orthogonal Hermite-Gauss modes that form the RPB.

Fig. 2
Fig. 2

(a) The electric field intensity map of the RPB on the focal plane. The left, center and right columns correspond to the transverse, longitudinal component and focussed spot, respectively. The top, middle and bottom rows correspond to the results in the case of φ = 0, φ = 0.3λ and φ = 0.5λ, respectively. The scale bar refers to a length of λ/2. (b) The plot shows various metrics as a function of phase retardance. The blue (solid) line, green (dashed) and red (dotted) lines refer to ratio of | E lon (0,0) | 2 / max | E tra | 2 , max | E x | 2 / max | E y | 2 and | E lon (0,0) | 2 / max | E lon | 2 , respectively.

Fig. 3
Fig. 3

Schematic of the laser scanning microscope. S: Sample, aSIL: aplanatic solid immersion lens, OL: 20X objective lens, M: silver coated mirror, L: doublet lens, SM: scanning mirrors, NPB: non-polarizing beam splitter, VR: variable retarder, VF: vortex fiber, GF: grating filters, MG: microbend grating, PC: polarization control components, LD: laser diode, MMF: multi-mode fiber, PD: photodetector.

Fig. 4
Fig. 4

(a) The InGaAs camera images of RPB after the vortex fiber. The arrows show the orientation of the analyzer. (b) The InGaAs camera images of RPB at the pupil plane of the camera. The top, middle and bottom row corresponds to the residual phase retardance due to the optics of the microscope, after the compensation is applied and after additional half-wave plate is introduced. The arrow shows the orientation of the analyzer. The scalebar indicates a length of 2mm. (c) The InGaAs camera images of the reflection image of the focused spots. The top, middle and bottom row corresponds to the residual phase retardance due to the optics of the microscope, after the compensation is applied and then the additional half-wave plate is introduced.

Fig. 5
Fig. 5

(a) Top row: A laser scanning microscope image of aluminum structures on the silicon substrate. No compensation applied. Bottom row: Plot of a mean differential of a hundred linecuts obtained on the longest lines in vertical (blue) and horizontal (green) directions in the image. The error bars show the standard deviation on the mean. The scale bar shows a length of 2 microns. (b) The same as (a) except after the phase retardance compensation. (c) The same as (b) except an additional half-wave plate is introduced after the compensation.

Fig. 6
Fig. 6

(a) Laser scanning image of resolution targets after the vector symmetry of the RPB is recovered (φ = 0). The pitch values are shown on the left of the image. The scale bar indicates a length of 5 microns (b) Higher magnification image of the indicated region in (a). (c) The same region of interest as shown in (b) except with a phase retardance of 0.5λ. (d) Modulation transfer function (MTF) of compensated (φ = 0) and uncompensated (φ = 0.5 λ) RPB calculated from the LSFs shown in Fig. 5.

Equations (4)

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RPB = HG 10 e x + HG 01 e y
RPB m = HG 10 e x +a e ikϕ HG 01 e y
I 10 = 0 θ 1max d θ 1 cos θ 1 f w ( θ 2 )sin θ 1 k 2 k 3 sin 2 θ 2 e iz k 3 1 k 2 2 / k 3 2 sin 2 θ 2 × J 0 ( k 2 ρsin θ 2 ) t p12 ( θ 2 ) t p23 ( θ 2 ) I 11 = 0 θ 1max d θ 1 cos θ 1 f w ( θ 2 )sin θ 1 sin θ 2 e iz k 3 1 k 2 2 / k 3 2 sin 2 θ 2 × J 1 ( k 2 ρsin θ 2 )( t s12 ( θ 2 ) t s23 ( θ 2 )+3 t p12 ( θ 2 ) t p23 ( θ 2 ) 1 k 2 2 / k 3 2 sin 2 θ 2 ) I 12 = 0 θ 1max d θ 1 cos θ 1 f w ( θ 2 )sin θ 1 sin θ 2 e iz k 3 1 k 2 2 / k 3 2 sin 2 θ 2 × J 1 ( k 2 ρsin θ 2 )( t s12 ( θ 2 ) t s23 ( θ 2 ) t p12 ( θ 2 ) t p23 ( θ 2 ) 1 k 2 2 / k 3 2 sin 2 θ 2 ) I 13 = 0 θ 1max d θ 1 cos θ 1 f w ( θ 2 ) k 2 k 3 sin θ 1 sin 2 θ 2 e iz k 3 1 k 2 2 / k 3 2 sin 2 θ 2 × J 2 ( k 2 ρsin θ 2 ) t p12 ( θ 2 ) t p23 ( θ 2 ) I 14 = 0 θ 1max d θ 1 cos θ 1 f w ( θ 2 )sin θ 1 sin θ 2 e iz k 3 1 k 2 2 / k 3 2 sin 2 θ 2 × J 3 ( k 2 ρsin θ 2 )( t s12 ( θ 2 ) t s23 ( θ 2 ) t p12 ( θ 2 ) t p23 ( θ 2 ) 1 k 2 2 / k 3 2 sin 2 θ 2 )
E HG 10 ( ρ,ϕ,z )=if k 1 k 2 k 3 e i k 1 f [ i I 11 cosϕ+i I 14 cos3ϕ i I 12 sinϕ+i I 14 sin3ϕ -2I 10 +2 I 13 cos2ϕ ][ e x e y e z ] E HG 01 ( ρ,ϕ,z )=if k 1 k 2 k 3 e i k 1 f [ i I 12 cosϕi I 14 cos3ϕ i I 11 sinϕi I 14 sin3ϕ -2I 10 2 I 13 cos2ϕ ][ e x e y e z ]

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