Abstract

In this paper, we present the theoretical studies of a refractive index map to implement a Gauss to a J0-Bessel-Gauss convertor. We theoretically demonstrate the viability of a device that could be fabricated on a Si/Si1−yOy/Si1−x−yGexCy platform or by photo-refractive media. The proposed device is 200 μm in length and 25 μm in width, and its refractive index varies in controllable steps across the light propagation and transversal directions. The computed conversion efficiency and loss are 90%, and −0.457  dB, respectively. The theoretical results, obtained from the beam conversion efficiency, self-regeneration, and propagation through an opaque obstruction, demonstrate that a two-dimensional (2D) graded index map of the refractive index can be used to transform a Gauss beam into a J0-Bessel-Gauss beam. To the best of our knowledge, this is the first demonstration of such beam transformation by means of a 2D index-mapping that is fully integrable in silicon photonics based planar lightwave circuits (PLCs). The concept device is significant for the eventual development of a new array of technologies, such as micro optical tweezers, optical traps, beam reshaping and nonlinear beam diode lasers.

© 2012 Optical Society of America

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    [Crossref]
  7. I. A. Litvin, M. G. McLaren, and A. Forbes, “Propagation of obstructed Bessel and Bessel-Gauss beams,” Proc. SPIE 7062, 706218 (2008).
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  16. F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
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  25. A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
    [Crossref]
  26. H. Hadar and P. Monk, “The linear sampling method for solving the electromagnetic inverse medium problem,” Inverse Probl. 18, 891–906 (2002).
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  27. H. Hadar, “The interior transmission problem for anisotropic Maxwell’s equations and its applications to the inverse problem,” Math. Methods Appl. Sci. 27, 2111–2129 (2004).
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  30. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2008).
    [Crossref]
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    [Crossref]
  32. H. J. Lee, C. H. Henry, K. J. Orlowsky, R. F. Kazarinov, and T. Y. Kometani, “Refractive-index dispersion of phospho-silicate glass, thermal oxide, and silicon nitride films on silicon,” Appl. Opt. 27, 4104–4109 (1988).
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    [Crossref]
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    [Crossref]
  35. S. B. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674(2009).
    [Crossref]

2010 (2)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[Crossref]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[Crossref]

2009 (3)

S. B. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674(2009).
[Crossref]

Y. Lai, J. Ng, H. Yang, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chang, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

V. Arrizón, D. Sánchez-de-la-Llave, U. Ruiz, and G. Méndez, “Efficient generation of an arbitrary nondiffracting Bessel beam employing its phase modulation,” Opt. Lett. 34, 1456–1458 (2009).
[Crossref]

2008 (2)

I. A. Litvin, M. G. McLaren, and A. Forbes, “Propagation of obstructed Bessel and Bessel-Gauss beams,” Proc. SPIE 7062, 706218 (2008).
[Crossref]

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2008).
[Crossref]

2007 (3)

2006 (3)

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[Crossref]

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by radially polrized beam,” Opt. Lett. 31, 1726–1728 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

2005 (1)

B. Gang and L. Peijun, “Inverse medium scattering problems for electromagnetic waves,” SIAM J. Appl. Math. 65, 2049–2066 (2005).
[Crossref]

2004 (2)

M. Lei and B. Yao, “Characteristics of beam profiles of Gaussian beam passing through an axicon,” Opt. Commun. 239, 367–372 (2004).
[Crossref]

H. Hadar, “The interior transmission problem for anisotropic Maxwell’s equations and its applications to the inverse problem,” Math. Methods Appl. Sci. 27, 2111–2129 (2004).
[Crossref]

2003 (1)

A. Zakery, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1–12 (2003).
[Crossref]

2002 (2)

H. Hadar and P. Monk, “The linear sampling method for solving the electromagnetic inverse medium problem,” Inverse Probl. 18, 891–906 (2002).
[Crossref]

J. Canning, “Diffraction-free mode generation and propagation in optical waveguides,” Opt. Commun. 207, 35–39 (2002).
[Crossref]

2001 (2)

R. M. Herman and T. A. Wiggins, “Propagation and focusing of Bessel-Gauss, generalized Bessel-Gauss and modified Bessel-Gauss beams,” J. Opt. Soc. Am. A 18, 170–176 (2001).
[Crossref]

W. Fenga, W. K. Choia, L. K. Beraa, M. Jib, and C. Y. Yangb, “Optical characterization of as-prepared and rapid thermal oxidized partially strain compensated Si1xyGexCy films,” Mater. Sci. Semicond. Process. 4, 655–659 (2001).
[Crossref]

1999 (2)

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[Crossref]

S. Chavez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

1998 (2)

W. Cong, N. Chen, and B. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. 15, 2362–2364 (1998).
[Crossref]

M. Piana, “On uniqueness for anisotropic inhomogeneous inverse scattering problems,” Inverse Probl. 14, 1565–1579 (1998).
[Crossref]

1996 (1)

K. Hayata, “Are Bessel beams supportable in graded-index media?” Opt. Rev. 3, 299–300 (1996).
[Crossref]

1992 (1)

1988 (1)

1987 (3)

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

1983 (1)

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
[Crossref]

1982 (1)

A. J. Devaney, “A filtered backprojection algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
[Crossref]

1969 (1)

R. M. Lewis, “Physical optics inverse diffraction,” IEEE Trans. Antennas Propag. 17, 308–314 (1969).
[Crossref]

Arnold, C. B.

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[Crossref]

Arrizón, V.

Arroyo Carrasco, M. L.

M. M. Méndez Otero, G. C. Martínez Jimnez, M. L. Arroyo Carrasco, M. D. Iturbe Castillo, and E. A. Mart Panameño, “Generation of Bessel-Gauss beams by means of computed generated holograms for Bessel Beams,” in Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Beraa, L. K.

W. Fenga, W. K. Choia, L. K. Beraa, M. Jib, and C. Y. Yangb, “Optical characterization of as-prepared and rapid thermal oxidized partially strain compensated Si1xyGexCy films,” Mater. Sci. Semicond. Process. 4, 655–659 (2001).
[Crossref]

Berryman, J. G.

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[Crossref]

Bertete-Aguirre, H.

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[Crossref]

Brown, D. L.

Cakoni, F.

F. Cakoni, “A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media,” Inverse Probl. Imaging 1, 443–456 (2007).

Canning, J.

J. Canning, “Diffraction-free mode generation and propagation in optical waveguides,” Opt. Commun. 207, 35–39 (2002).
[Crossref]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[Crossref]

Chang, C. T.

Y. Lai, J. Ng, H. Yang, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chang, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Chavez-Cerda, S.

S. Chavez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[Crossref]

Chen, N.

W. Cong, N. Chen, and B. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. 15, 2362–2364 (1998).
[Crossref]

Chen, Y.

Choia, W. K.

W. Fenga, W. K. Choia, L. K. Beraa, M. Jib, and C. Y. Yangb, “Optical characterization of as-prepared and rapid thermal oxidized partially strain compensated Si1xyGexCy films,” Mater. Sci. Semicond. Process. 4, 655–659 (2001).
[Crossref]

Cong, W.

W. Cong, N. Chen, and B. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. 15, 2362–2364 (1998).
[Crossref]

Devaney, A. J.

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
[Crossref]

A. J. Devaney, “A filtered backprojection algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
[Crossref]

Dorn, O.

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[Crossref]

Durnin, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

Eberly, J. H.

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[Crossref]

Fenga, W.

W. Fenga, W. K. Choia, L. K. Beraa, M. Jib, and C. Y. Yangb, “Optical characterization of as-prepared and rapid thermal oxidized partially strain compensated Si1xyGexCy films,” Mater. Sci. Semicond. Process. 4, 655–659 (2001).
[Crossref]

Forbes, A.

I. A. Litvin, M. G. McLaren, and A. Forbes, “Propagation of obstructed Bessel and Bessel-Gauss beams,” Proc. SPIE 7062, 706218 (2008).
[Crossref]

Gang, B.

B. Gang and L. Peijun, “Inverse medium scattering problems for electromagnetic waves,” SIAM J. Appl. Math. 65, 2049–2066 (2005).
[Crossref]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Gu, B.

W. Cong, N. Chen, and B. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. 15, 2362–2364 (1998).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Guo, D.

Hadar, H.

H. Hadar, “The interior transmission problem for anisotropic Maxwell’s equations and its applications to the inverse problem,” Math. Methods Appl. Sci. 27, 2111–2129 (2004).
[Crossref]

H. Hadar and P. Monk, “The linear sampling method for solving the electromagnetic inverse medium problem,” Inverse Probl. 18, 891–906 (2002).
[Crossref]

Han, D.

Y. Lai, J. Ng, H. Yang, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chang, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Hayata, K.

K. Hayata, “Are Bessel beams supportable in graded-index media?” Opt. Rev. 3, 299–300 (1996).
[Crossref]

Henry, C. H.

Herman, R. M.

Huang, H.

Ilchenko, V. S.

Iturbe Castillo, M. D.

M. M. Méndez Otero, G. C. Martínez Jimnez, M. L. Arroyo Carrasco, M. D. Iturbe Castillo, and E. A. Mart Panameño, “Generation of Bessel-Gauss beams by means of computed generated holograms for Bessel Beams,” in Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Jib, M.

W. Fenga, W. K. Choia, L. K. Beraa, M. Jib, and C. Y. Yangb, “Optical characterization of as-prepared and rapid thermal oxidized partially strain compensated Si1xyGexCy films,” Mater. Sci. Semicond. Process. 4, 655–659 (2001).
[Crossref]

Kang, S. B.

S. B. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674(2009).
[Crossref]

Kazarinov, R. F.

Kometani, T. Y.

Lai, Y.

Y. Lai, J. Ng, H. Yang, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chang, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Lee, H. J.

Lei, M.

M. Lei and B. Yao, “Characteristics of beam profiles of Gaussian beam passing through an axicon,” Opt. Commun. 239, 367–372 (2004).
[Crossref]

Leonhardt, U.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2008).
[Crossref]

Lewis, R. M.

R. M. Lewis, “Physical optics inverse diffraction,” IEEE Trans. Antennas Propag. 17, 308–314 (1969).
[Crossref]

Lin, Y.

Litvin, I. A.

I. A. Litvin, M. G. McLaren, and A. Forbes, “Propagation of obstructed Bessel and Bessel-Gauss beams,” Proc. SPIE 7062, 706218 (2008).
[Crossref]

Maleki, L.

Mart Panameño, E. A.

M. M. Méndez Otero, G. C. Martínez Jimnez, M. L. Arroyo Carrasco, M. D. Iturbe Castillo, and E. A. Mart Panameño, “Generation of Bessel-Gauss beams by means of computed generated holograms for Bessel Beams,” in Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Martínez Jimnez, G. C.

M. M. Méndez Otero, G. C. Martínez Jimnez, M. L. Arroyo Carrasco, M. D. Iturbe Castillo, and E. A. Mart Panameño, “Generation of Bessel-Gauss beams by means of computed generated holograms for Bessel Beams,” in Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Matsko, A. B.

McLaren, M. G.

I. A. Litvin, M. G. McLaren, and A. Forbes, “Propagation of obstructed Bessel and Bessel-Gauss beams,” Proc. SPIE 7062, 706218 (2008).
[Crossref]

McLeod, E.

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[Crossref]

Méndez, G.

Méndez Otero, M. M.

M. M. Méndez Otero, G. C. Martínez Jimnez, M. L. Arroyo Carrasco, M. D. Iturbe Castillo, and E. A. Mart Panameño, “Generation of Bessel-Gauss beams by means of computed generated holograms for Bessel Beams,” in Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Mohageg, M.

Monk, P.

H. Hadar and P. Monk, “The linear sampling method for solving the electromagnetic inverse medium problem,” Inverse Probl. 18, 891–906 (2002).
[Crossref]

Ng, J.

Y. Lai, J. Ng, H. Yang, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chang, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Orlowsky, K. J.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Papanicolaou, G. C.

O. Dorn, H. Bertete-Aguirre, J. G. Berryman, and G. C. Papanicolaou, “A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields,” Inverse Probl. 15, 1523–1558 (1999).
[Crossref]

Peijun, L.

B. Gang and L. Peijun, “Inverse medium scattering problems for electromagnetic waves,” SIAM J. Appl. Math. 65, 2049–2066 (2005).
[Crossref]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2008).
[Crossref]

Piana, M.

M. Piana, “On uniqueness for anisotropic inhomogeneous inverse scattering problems,” Inverse Probl. 14, 1565–1579 (1998).
[Crossref]

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[Crossref]

Ruiz, U.

Sánchez-de-la-Llave, D.

Savchenkov, A. A.

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Seka, W.

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[Crossref]

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Tsai, T.

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[Crossref]

Wiggins, T. A.

Wu, F.

Xiao, J.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic representation of a Gauss to J0-Bessel-Gauss transformation via a heterogeneous medium; the transformation device in region A has a refractive index map that needs to be calculated in order to achieve the desired beam transformation. The device is embedded in a homogeneous medium B, where Bessel and Gauss beams are known solutions to the Helmholtz equation.

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Fig. 2.
Fig. 2.

Illustration of (a) superposition of transition stages in the transformation of Gauss to J0-Bessel-Gauss beams, and (b) the transformation functions f and g.

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Fig. 3.
Fig. 3.

Sketch of the proposed refractive index map for the converter as defined by the transformation functions f and g, with a device length of 200 μm and a width 25 μm. Albeit here shown in 11 steps, the refractive index varies smoothly between 1 and 2.57 in controllable steps. The device is rotationally symmetric around the propagation axis z.

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Fig. 4.
Fig. 4.

Fine resolution refractive index map for the converter, with minimum feature size d=1  nm, as defined by the transformation functions f and g. Device length is 200 μm and width is 25 μm. The refractive index varies smoothly between 1 and 2.57 in 50 controllable steps. A practical device could be fabricated on a Si/Si1yOy/Si1xyGexCy slab and the refractive index steps achieved by controlled oxidation or with photo-refractive materials. The device is rotationally symmetric around the propagation axis z.

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Fig. 5.
Fig. 5.

Transverse electric field intensity of the converted beam: (a) simulated near field computed at 5 μm from the output of the device (green), and fit, 95% confidence, to J0-Bessel-Gauss (red). Its far field profile (b), and (c) zoom to principal peak.

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Fig. 6.
Fig. 6.

Transverse electric field intensity profile for the converted beam for two different grid sizes: (a) shows the optimal grid size of 1 nm (green) and modified grid size of 10 nm (blue), both computed at the near fields, 5 μm from the output of the device. The far fields for both output beams (b), for optimal grid size (green) and modified grid (blue), and (c) magnification of the near field around the principal peak.

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Fig. 7.
Fig. 7.

Transverse electric field intensity profile for (a) the converted beam (green), its fit (red), and the scattered field profile (blue), computed at 50 μm from the scattering region. The far field for the undisturbed and perturbed beams are shown in (b), and (c) shows magnification of the fields around the principal peak.

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Equations (8)

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∇2E⃗−με(r⃗)∂2E⃗∂t2=−∇(1εE⃗·∇ε),
εmin≤ε(r⃗)≤εmax,∀  r⃗∈A.
∇2ϑ⃗(r⃗)−ω2με(r⃗)ϑ⃗(r⃗)=∇(ϑ⃗(r⃗)·∇ ln ε(r⃗)).
ϑ⃗(r⃗)={ϕ⃗(r⃗)∀r⃗∈Si,ψ⃗(r⃗)∀r⃗∈A,ζ⃗(r⃗)∀r⃗∈So,
ψ⃗(r⃗)=f(r⃗)ϕ⃗(r⃗)+g(r⃗)ζ⃗(r⃗)+γ⃗(r⃗),
∇2(f(r⃗)ϕ⃗(r⃗)+g(r⃗)ζ⃗(r⃗))︷ϖ⃗− ω2μ·(f(r⃗)ϕ⃗(r⃗)+g(r⃗)ζ⃗(r⃗))︷ω2μψ⃗·ε(r⃗)=∇[(f(r⃗)ϕ⃗(r⃗)+g(r⃗)ζ⃗(r⃗))·∇ ln ε(r⃗)].
∇[(f(r⃗)ϕ⃗(r⃗)+g(r⃗)ζ⃗(r⃗))·∇ ln ε(r⃗)]=(ψ⃗·∇)∇ ln ε(r⃗)+∇ψ(∇ ln ε(r⃗)·ψ⃗).
ω2μψ⃗ε(r⃗)+(ψ⃗·∇)∇ ln ε(r⃗)+∇ψ(∇ ln ε(r⃗)·ψ⃗)=ϖ⃗.

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