Abstract

Deflection angles of light rays passing through a refractive index field can be measured by the background-oriented schlieren (BOS) technique. Assuming that the deflection angle is sufficiently small and the paraxial approximation can apply to the light rays, a vector consisting of deflection angles in two orthogonal directions is shown to be derived from a gradient of a scalar potential. The scalar potential can be written as an integration of the refractive index field over the light ray path. Thus, a method to reconstruct an axisymmetric 3D refractive index field with the scalar potential is proposed here. An arbitrary measured deflection angle vector, however, is generally written not only with a scalar potential but with a vector potential. Thus, the Poisson’s equation is derived to extract a scalar potential from a measured deflection angle vector. The axisymmetric 3D refractive index field is able to be reconstructed using the Abel transformation [1] of the scalar potential derived by applying the 2D Fourier transformation to the Poisson’s equation. The scalar potential reconstruction method is validated by reconstructing a spherically symmetric refractive index field where a deflection angle vector field is able to be calculated accurately.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. N. H. Abel, “Auflösung Einer Mechanischen Aufgabe,” J. Reine Angew. Math. 1826(1), 153–157 (1826).
    [Crossref]
  2. J. C. Owens, “Optical Refractive Index of Air: Dependence on Pressure, Temperature and Composition,” Appl. Opt. 6(1), 51–59 (1967).
    [Crossref] [PubMed]
  3. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35(9), 1566–1573 (1996).
    [Crossref] [PubMed]
  4. J. W. Hosch and J. P. Walters, “High spatial resolution schlieren photography,” Appl. Opt. 16(2), 473–482 (1977).
    [Crossref] [PubMed]
  5. J. B. Schmidt, Z. D. Schaefer, T. R. Meyer, S. Roy, S. A. Danczyk, and J. R. Gord, “Ultrafast time-gated ballistic-photon imaging and shadowgraphy in optically dense rocket sprays,” Appl. Opt. 48(4), B137–B144 (2009).
    [Crossref] [PubMed]
  6. W. L. Howes, “Rainbow schlieren and its applications,” Appl. Opt. 23(14), 2449–2460 (1984).
    [Crossref] [PubMed]
  7. W. L. Howes, “Rainbow schlieren vs Mach-Zehnder interferometer: a comparison,” Appl. Opt. 24(6), 816–822 (1985).
    [Crossref] [PubMed]
  8. A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, and D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37(3), 479–485 (1998).
    [Crossref] [PubMed]
  9. C. D. Perciante and J. A. Ferrari, “Visualization of two-dimensional phase gradients by subtraction of a reference periodic pattern,” Appl. Opt. 39(13), 2081–2083 (2000).
    [Crossref] [PubMed]
  10. J. H. Massig, “Measurement of phase objects by simple means,” Appl. Opt. 38(19), 4103–4105 (1999).
    [Crossref] [PubMed]
  11. Y. Kawata, R. Juškaitis, T. Tanaka, T. Wilson, and S. Kawata, “Differential phase-contrast microscope with a split detector for the readout system of a multilayered optical memory,” Appl. Opt. 35(14), 2466–2470 (1996).
    [Crossref] [PubMed]
  12. M. R. Ayres and R. R. McLeod, “Scanning transmission microscopy using a position-sensitive detector,” Appl. Opt. 45(33), 8410–8418 (2006).
    [Crossref] [PubMed]
  13. S. B. Mehta and C. J. R. Sheppard, “Quantitative phase-gradient imaging at high resolution with asymmetric illumination-based differential phase contrast,” Opt. Lett. 34(13), 1924–1926 (2009).
    [Crossref] [PubMed]
  14. M. Raffel, “Background-oriented schlieren (BOS) techniques,” Exp. Fluids 56(3), 60 (2015).
    [Crossref]
  15. R. Beermann, L. Quentin, A. Pösch, E. Reithmeier, and M. Kästner, “Background oriented schlieren measurement of the refractive index field of air induced by a hot, cylindrical measurement object,” Appl. Opt. 56(14), 4168–4179 (2017).
    [Crossref] [PubMed]
  16. T. Sueishi, M. Ishii, and M. Ishikawa, “Tracking background-oriented schlieren for observing shock oscillations of transonic flying objects,” Appl. Opt. 56(13), 3789–3798 (2017).
    [Crossref] [PubMed]
  17. A. B. Gojani, B. Kamishi, and S. Obayashi, “Measurement sensitivity and resolution for background oriented schlieren during image recording,” J. Visualization 16(3), 201–207 (2013).
    [Crossref]
  18. M. Ota, K. Hamada, H. Kato, and K. Maeno, “Computed-tomographic density measurement of supersonic flow field by colored-grid background oriented schlieren (CGBOS) technique,” Meas. Sci. Technol. 22(10), 104011 (2011).
    [Crossref]
  19. L. Venkatakrishnan and G. E. A. Meier, “Density measurements using the Background Oriented Schlieren technique,” Exp. Fluids 37(2), 237–247 (2004).
    [Crossref]
  20. F. A. Mier and M. J. Hargather, “Color gradient background-oriented schlieren imaging,” Exp. Fluids 57(6), 95 (2016).
    [Crossref]
  21. M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. 27(10), 1956–1959 (1988).
    [Crossref] [PubMed]
  22. H. Chehouani and M. El Fagrich, “Adaptation of the Fourier-Hankel method for deflection tomographic reconstruction of axisymmetric field,” Appl. Opt. 52(3), 439–448 (2013).
    [Crossref] [PubMed]
  23. R. Beermann, L. Quentin, A. Pösch, E. Reithmeier, and M. Kästner, “Background oriented schlieren measurement of the refractive index field of air induced by a hot, cylindrical measurement object,” Appl. Opt. 56(14), 4168–4179 (2017).
    [Crossref] [PubMed]
  24. H. Ohno and K. Toya, “Reconstruction method of axisymmetric refractive index fields with background-oriented schlieren,” Appl. Opt. 57(30), 9062–9069 (2018).
    [Crossref] [PubMed]
  25. J. Bladel, On Helmholtz’s theorem in finite regions (Midwestern Universities Research Association, 1958)
  26. E. W. Marchand, Gradient Index Optics (Academic, 1978).
  27. J. Chaves, Introduction to Nonimaging optics (CRC, 2008).
  28. R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Academic, 2005).
  29. C. V. Open, https://opencv.org/
  30. G. Farneback, “Fast and Accurate Motion Estimation using Orientation Tensors and Parametric Motion Models,” Proceedings of 15th international conference on pattern recognition I, 135–139 (2000)
    [Crossref]
  31. G. Farneback, “Very High Accuracy Velocity Estimation using Orientation Tensors, Parametric Motion, and Simultaneous Segmentation of the Motion Field,” Proceedings of 8th IEEE international conference on computer vision I, 171–177 (2001)
    [Crossref]
  32. T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” Lect. Notes Comput. Sci. 3024, 25–36 (2004).
    [Crossref]
  33. G. Bradski and A. Kaehler, Learning OpenCV: Computer Vision with the OpenCV Library (O’Reilly Media, 2008).
  34. R. N. Strickland and D. W. Sweeney, “Optical flow computation in combustion image sequences,” Appl. Opt. 27(24), 5213–5220 (1988).
    [Crossref] [PubMed]

2018 (1)

2017 (3)

2016 (1)

F. A. Mier and M. J. Hargather, “Color gradient background-oriented schlieren imaging,” Exp. Fluids 57(6), 95 (2016).
[Crossref]

2015 (1)

M. Raffel, “Background-oriented schlieren (BOS) techniques,” Exp. Fluids 56(3), 60 (2015).
[Crossref]

2013 (2)

H. Chehouani and M. El Fagrich, “Adaptation of the Fourier-Hankel method for deflection tomographic reconstruction of axisymmetric field,” Appl. Opt. 52(3), 439–448 (2013).
[Crossref] [PubMed]

A. B. Gojani, B. Kamishi, and S. Obayashi, “Measurement sensitivity and resolution for background oriented schlieren during image recording,” J. Visualization 16(3), 201–207 (2013).
[Crossref]

2011 (1)

M. Ota, K. Hamada, H. Kato, and K. Maeno, “Computed-tomographic density measurement of supersonic flow field by colored-grid background oriented schlieren (CGBOS) technique,” Meas. Sci. Technol. 22(10), 104011 (2011).
[Crossref]

2009 (2)

2006 (1)

2004 (2)

L. Venkatakrishnan and G. E. A. Meier, “Density measurements using the Background Oriented Schlieren technique,” Exp. Fluids 37(2), 237–247 (2004).
[Crossref]

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” Lect. Notes Comput. Sci. 3024, 25–36 (2004).
[Crossref]

2000 (1)

1999 (1)

1998 (1)

1996 (2)

1988 (2)

1985 (1)

1984 (1)

1977 (1)

1967 (1)

1826 (1)

N. H. Abel, “Auflösung Einer Mechanischen Aufgabe,” J. Reine Angew. Math. 1826(1), 153–157 (1826).
[Crossref]

Abel, N. H.

N. H. Abel, “Auflösung Einer Mechanischen Aufgabe,” J. Reine Angew. Math. 1826(1), 153–157 (1826).
[Crossref]

Agrawal, A. K.

Ayres, M. R.

Beermann, R.

Brox, T.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” Lect. Notes Comput. Sci. 3024, 25–36 (2004).
[Crossref]

Bruhn, A.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” Lect. Notes Comput. Sci. 3024, 25–36 (2004).
[Crossref]

Butuk, N. K.

Chehouani, H.

Ciddor, P. E.

Danczyk, S. A.

El Fagrich, M.

Farneback, G.

G. Farneback, “Fast and Accurate Motion Estimation using Orientation Tensors and Parametric Motion Models,” Proceedings of 15th international conference on pattern recognition I, 135–139 (2000)
[Crossref]

G. Farneback, “Very High Accuracy Velocity Estimation using Orientation Tensors, Parametric Motion, and Simultaneous Segmentation of the Motion Field,” Proceedings of 8th IEEE international conference on computer vision I, 171–177 (2001)
[Crossref]

Ferrari, J. A.

Gojani, A. B.

A. B. Gojani, B. Kamishi, and S. Obayashi, “Measurement sensitivity and resolution for background oriented schlieren during image recording,” J. Visualization 16(3), 201–207 (2013).
[Crossref]

Gollahalli, S. R.

Gord, J. R.

Griffin, D.

Hamada, K.

M. Ota, K. Hamada, H. Kato, and K. Maeno, “Computed-tomographic density measurement of supersonic flow field by colored-grid background oriented schlieren (CGBOS) technique,” Meas. Sci. Technol. 22(10), 104011 (2011).
[Crossref]

Hargather, M. J.

F. A. Mier and M. J. Hargather, “Color gradient background-oriented schlieren imaging,” Exp. Fluids 57(6), 95 (2016).
[Crossref]

Hosch, J. W.

Howes, W. L.

Ishii, M.

Ishikawa, M.

Juškaitis, R.

Kalal, M.

Kamishi, B.

A. B. Gojani, B. Kamishi, and S. Obayashi, “Measurement sensitivity and resolution for background oriented schlieren during image recording,” J. Visualization 16(3), 201–207 (2013).
[Crossref]

Kästner, M.

Kato, H.

M. Ota, K. Hamada, H. Kato, and K. Maeno, “Computed-tomographic density measurement of supersonic flow field by colored-grid background oriented schlieren (CGBOS) technique,” Meas. Sci. Technol. 22(10), 104011 (2011).
[Crossref]

Kawata, S.

Kawata, Y.

Maeno, K.

M. Ota, K. Hamada, H. Kato, and K. Maeno, “Computed-tomographic density measurement of supersonic flow field by colored-grid background oriented schlieren (CGBOS) technique,” Meas. Sci. Technol. 22(10), 104011 (2011).
[Crossref]

Massig, J. H.

McLeod, R. R.

Mehta, S. B.

Meier, G. E. A.

L. Venkatakrishnan and G. E. A. Meier, “Density measurements using the Background Oriented Schlieren technique,” Exp. Fluids 37(2), 237–247 (2004).
[Crossref]

Meyer, T. R.

Mier, F. A.

F. A. Mier and M. J. Hargather, “Color gradient background-oriented schlieren imaging,” Exp. Fluids 57(6), 95 (2016).
[Crossref]

Nugent, K. A.

Obayashi, S.

A. B. Gojani, B. Kamishi, and S. Obayashi, “Measurement sensitivity and resolution for background oriented schlieren during image recording,” J. Visualization 16(3), 201–207 (2013).
[Crossref]

Ohno, H.

Ota, M.

M. Ota, K. Hamada, H. Kato, and K. Maeno, “Computed-tomographic density measurement of supersonic flow field by colored-grid background oriented schlieren (CGBOS) technique,” Meas. Sci. Technol. 22(10), 104011 (2011).
[Crossref]

Owens, J. C.

Papenberg, N.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” Lect. Notes Comput. Sci. 3024, 25–36 (2004).
[Crossref]

Perciante, C. D.

Pösch, A.

Quentin, L.

Raffel, M.

M. Raffel, “Background-oriented schlieren (BOS) techniques,” Exp. Fluids 56(3), 60 (2015).
[Crossref]

Reithmeier, E.

Roy, S.

Schaefer, Z. D.

Schmidt, J. B.

Sheppard, C. J. R.

Strickland, R. N.

Sueishi, T.

Sweeney, D. W.

Tanaka, T.

Toya, K.

Venkatakrishnan, L.

L. Venkatakrishnan and G. E. A. Meier, “Density measurements using the Background Oriented Schlieren technique,” Exp. Fluids 37(2), 237–247 (2004).
[Crossref]

Walters, J. P.

Weickert, J.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” Lect. Notes Comput. Sci. 3024, 25–36 (2004).
[Crossref]

Wilson, T.

Appl. Opt. (18)

J. C. Owens, “Optical Refractive Index of Air: Dependence on Pressure, Temperature and Composition,” Appl. Opt. 6(1), 51–59 (1967).
[Crossref] [PubMed]

P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35(9), 1566–1573 (1996).
[Crossref] [PubMed]

J. W. Hosch and J. P. Walters, “High spatial resolution schlieren photography,” Appl. Opt. 16(2), 473–482 (1977).
[Crossref] [PubMed]

J. B. Schmidt, Z. D. Schaefer, T. R. Meyer, S. Roy, S. A. Danczyk, and J. R. Gord, “Ultrafast time-gated ballistic-photon imaging and shadowgraphy in optically dense rocket sprays,” Appl. Opt. 48(4), B137–B144 (2009).
[Crossref] [PubMed]

W. L. Howes, “Rainbow schlieren and its applications,” Appl. Opt. 23(14), 2449–2460 (1984).
[Crossref] [PubMed]

W. L. Howes, “Rainbow schlieren vs Mach-Zehnder interferometer: a comparison,” Appl. Opt. 24(6), 816–822 (1985).
[Crossref] [PubMed]

A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, and D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37(3), 479–485 (1998).
[Crossref] [PubMed]

C. D. Perciante and J. A. Ferrari, “Visualization of two-dimensional phase gradients by subtraction of a reference periodic pattern,” Appl. Opt. 39(13), 2081–2083 (2000).
[Crossref] [PubMed]

J. H. Massig, “Measurement of phase objects by simple means,” Appl. Opt. 38(19), 4103–4105 (1999).
[Crossref] [PubMed]

Y. Kawata, R. Juškaitis, T. Tanaka, T. Wilson, and S. Kawata, “Differential phase-contrast microscope with a split detector for the readout system of a multilayered optical memory,” Appl. Opt. 35(14), 2466–2470 (1996).
[Crossref] [PubMed]

M. R. Ayres and R. R. McLeod, “Scanning transmission microscopy using a position-sensitive detector,” Appl. Opt. 45(33), 8410–8418 (2006).
[Crossref] [PubMed]

R. Beermann, L. Quentin, A. Pösch, E. Reithmeier, and M. Kästner, “Background oriented schlieren measurement of the refractive index field of air induced by a hot, cylindrical measurement object,” Appl. Opt. 56(14), 4168–4179 (2017).
[Crossref] [PubMed]

T. Sueishi, M. Ishii, and M. Ishikawa, “Tracking background-oriented schlieren for observing shock oscillations of transonic flying objects,” Appl. Opt. 56(13), 3789–3798 (2017).
[Crossref] [PubMed]

M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. 27(10), 1956–1959 (1988).
[Crossref] [PubMed]

H. Chehouani and M. El Fagrich, “Adaptation of the Fourier-Hankel method for deflection tomographic reconstruction of axisymmetric field,” Appl. Opt. 52(3), 439–448 (2013).
[Crossref] [PubMed]

R. Beermann, L. Quentin, A. Pösch, E. Reithmeier, and M. Kästner, “Background oriented schlieren measurement of the refractive index field of air induced by a hot, cylindrical measurement object,” Appl. Opt. 56(14), 4168–4179 (2017).
[Crossref] [PubMed]

H. Ohno and K. Toya, “Reconstruction method of axisymmetric refractive index fields with background-oriented schlieren,” Appl. Opt. 57(30), 9062–9069 (2018).
[Crossref] [PubMed]

R. N. Strickland and D. W. Sweeney, “Optical flow computation in combustion image sequences,” Appl. Opt. 27(24), 5213–5220 (1988).
[Crossref] [PubMed]

Exp. Fluids (3)

M. Raffel, “Background-oriented schlieren (BOS) techniques,” Exp. Fluids 56(3), 60 (2015).
[Crossref]

L. Venkatakrishnan and G. E. A. Meier, “Density measurements using the Background Oriented Schlieren technique,” Exp. Fluids 37(2), 237–247 (2004).
[Crossref]

F. A. Mier and M. J. Hargather, “Color gradient background-oriented schlieren imaging,” Exp. Fluids 57(6), 95 (2016).
[Crossref]

J. Reine Angew. Math. (1)

N. H. Abel, “Auflösung Einer Mechanischen Aufgabe,” J. Reine Angew. Math. 1826(1), 153–157 (1826).
[Crossref]

J. Visualization (1)

A. B. Gojani, B. Kamishi, and S. Obayashi, “Measurement sensitivity and resolution for background oriented schlieren during image recording,” J. Visualization 16(3), 201–207 (2013).
[Crossref]

Lect. Notes Comput. Sci. (1)

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” Lect. Notes Comput. Sci. 3024, 25–36 (2004).
[Crossref]

Meas. Sci. Technol. (1)

M. Ota, K. Hamada, H. Kato, and K. Maeno, “Computed-tomographic density measurement of supersonic flow field by colored-grid background oriented schlieren (CGBOS) technique,” Meas. Sci. Technol. 22(10), 104011 (2011).
[Crossref]

Opt. Lett. (1)

Other (8)

G. Bradski and A. Kaehler, Learning OpenCV: Computer Vision with the OpenCV Library (O’Reilly Media, 2008).

J. Bladel, On Helmholtz’s theorem in finite regions (Midwestern Universities Research Association, 1958)

E. W. Marchand, Gradient Index Optics (Academic, 1978).

J. Chaves, Introduction to Nonimaging optics (CRC, 2008).

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Academic, 2005).

C. V. Open, https://opencv.org/

G. Farneback, “Fast and Accurate Motion Estimation using Orientation Tensors and Parametric Motion Models,” Proceedings of 15th international conference on pattern recognition I, 135–139 (2000)
[Crossref]

G. Farneback, “Very High Accuracy Velocity Estimation using Orientation Tensors, Parametric Motion, and Simultaneous Segmentation of the Motion Field,” Proceedings of 8th IEEE international conference on computer vision I, 171–177 (2001)
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Light ray path q from a far initial point of background to a far end point in (x, y, z) Cartesian coordinate system. The center of the coordinate system is O. Refractive index field is constant n0 with radius over Rc. Deflection angle vector is represented by ε.
Fig. 2
Fig. 2 (a) Perspective view of light ray path passing through spherical refractive index field. Y-axis is tilted with an angle Θ from y-axis on y-z plane. The light ray is assumed to be on the x-Y plane. (b) Cross-sectional view of light ray path on x-Y plane including the coordinate system center O. Cylindrical coordinates (ρ2, θ2) is taken for the light ray path (Q) on the plane with initial point (Q)0. The refractive index field is axisymmetric about the origin O where the field is constant n0 with the radius over Rc. The deflection angle is represented byε2.
Fig. 3
Fig. 3 Accurately calculated deflection angle fields for y- and z-components in the y-z plane are contoured with gray scale on the left-hand and the right-hand respectively. The refractive index field is represented by Eq. (40) where Δa is set to 0.01.
Fig. 4
Fig. 4 Scalar potential field derived from the 2D Fourier transformation of the accurately calculated deflection angle vector field in the y-z plane. The real part of the scalar field is contoured with gray scale on the left-hand, and the imaginary part of the scalar fields is contoured with gray scale on the right-hand.
Fig. 5
Fig. 5 Reconstructed refractive index deviation fields on the y-axis with respect to y normalized by Rc are plotted with dashed lines for Δa = 0.01, 0.1, and 1.0 respectively. The original refractive index deviation fields are also plotted with solid lines.
Fig. 6
Fig. 6 Cross-sectional view of axisymmetric 3D refractive index deviation fields in the x = 0 plane. The reconstructed refractive index deviation is contoured with a color scale in the left-hand. Original profile is also contoured with the color scale in the right-hand.
Fig. 7
Fig. 7 Reconstructed refractive index deviation fields with respect to the cylindrical radius ρ normalized by Rc for several normalized z by Rc of 0.0, 0.3, 0.4, 0.7, and 0.9 are plotted with dashed lines. The normalized quantity is denoted with a bar symbol. The original profiles are also plotted with solid lines.

Equations (44)

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S = A B n d s ,
d s = d x 1 + y 2 + z 2 ,
L ( x , y , z ) = n ( x , y , z ) 1 + y 2 + z 2 ,
S = A B L ( x , y , z ) d x .
d d x ( n y 1 + y 2 + z 2 ) = 1 + y 2 + z 2 n y ,
d d x ( n z 1 + y 2 + z 2 ) = 1 + y 2 + z 2 n z ,
q = ( x y z ) = ( 1 tan α y tan α z ) ( x x i ) + q 0 .
n = n 0 + Δ n ,
q = ( 1 tan α y tan α z ) ( x x i ) + q 0 + ( 0 Δ y ( x ) Δ z ( x ) ) .
q = ( 1 y z ) = ( 1 tan α y tan α z ) + ( 0 Δ y Δ z ) .
q ( 1 α y α z ) + ( 0 Δ y Δ z ) .
( 0 Δ y Δ z ) = ( 0 tan ( α y + ε y ) tan ( α y ) tan ( α z + ε z ) tan ( α z ) ) .
( 0 Δ y Δ z ) ( 0 ε y ε z ) .
d d x ( n ( α y + ε y ) 1 + ( α y + ε y ) 2 + ( α z + ε z ) 2 ) = 1 + ( α y + ε y ) 2 + ( α z + ε z ) 2 n y ,
d d x ( n ( α z + ε z ) 1 + ( α y + ε y ) 2 + ( α z + ε z ) 2 ) = 1 + ( α y + ε y ) 2 + ( α z + ε z ) 2 n z .
d d x ( n 0 ε y ) = n y ,
d d x ( n 0 ε z ) = n z ,
ε y = 1 n 0 R c + R c [ Δ n ( x , y , z ) y | y = y ( x ) , z = z ( x ) ] d x ,
ε z = 1 n 0 R c + R c [ Δ n ( x , y , z ) z | y = y ( x ) , z = z ( x ) ] d x ,
ε y ( y 0 , z 0 ) = 1 n 0 R c + R c [ Δ n ( x , y , z ) y | y = y 0 , z = z 0 ] d x ,
ε z ( y 0 , z 0 ) = 1 n 0 R c + R c [ Δ n ( x , y , z ) z | y = y 0 , z = z 0 ] d x ,
ϕ ( y 0 , z 0 ) = 1 n 0 R c + R c Δ n ( x , y 0 , z 0 ) d x .
ε ( 0 ε y ε z ) = ϕ .
ε = ϕ + × Ψ .
ε = ϕ + × Ψ = Δ ϕ = ( 2 x 2 + 2 y 2 + 2 z 2 ) ϕ ( y , z ) = ( 2 y 2 + 2 z 2 ) ϕ ( y , z ) .
ϕ = ϕ ˜ exp ( i k y y + i k z z ) d k y d k z ,
ε = ε ˜ exp ( i k y y + i k z z ) d k y d k z .
ϕ ˜ = i k y ε ˜ y + k z ε ˜ z k y 2 + k z 2 .
ϕ ( y , z ) = [ i k y ε ˜ y ( k y , k z ) + k z ε ˜ z ( k y , k z ) k y 2 + k z 2 exp ( i k y y + i k z z ) ] d k y d k z .
ϕ ( y , z ) = 2 n 0 y Δ n ( ρ , z ) ρ d ρ ρ 2 y 2 ,
r = x 2 + y 2 + z 2 = ρ 2 + z 2 .
Δ n ( ρ , z ) = n 0 π ρ ϕ ( y , z ) y d y y 2 ρ 2 .
ϕ ( y , z ) y = [ k y ( k y ε ˜ y ( k y , k z ) + k z ε ˜ z ( k y , k z ) ) k y 2 + k z 2 exp ( i k y y + i k z z ) ] d k y d k z .
Δ n ( ρ , z ) = 2 n 0 π d k z exp ( i k z z ) 0 d k y [ i k y ( k y ε ˜ y + k z ε ˜ z ) k y 2 + k z 2 ρ R c sin ( k y y ) d y y 2 ρ 2 ] ,
ε 2 = π 2 e r c d ρ 2 ρ 2 n 2 ρ 2 2 e 2 ,
e = H n 0 cos α ,
n ( r c , z ) r c = e .
ε 2 = 2 e e R c [ 1 ρ 2 ρ 2 2 e 2 1 ρ 2 n 2 ρ 2 2 e 2 ] d ρ 2 2 e r c e 1 ρ 2 n 2 ρ 2 2 e 2 d ρ 2 .
ε ( y , z ) = ε ( H cos Θ , H sin Θ ) = ( 0 tan 1 ( tan ( ε 2 ) cos Θ ) tan 1 ( tan ( ε 2 ) sin Θ ) ) .
n 2 = 1 + Δ a ( 1 ( r R c ) 2 ) ,
n = n 0 = 1.
r ¯ c = 1 + Δ a ( 1 + Δ a ) 2 4 Δ a e ¯ 2 Δ a ,
r ¯ c = r c R c ,
e ¯ = e R c = H R c .

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