Abstract

We recently introduced the edge-imaging condition, a necessary condition for all generalized lenses (glenses) [J. Opt. Soc. Am. A 33, 962 (2016) [CrossRef]  ] in a ray-optical transformation-optics (RTO) device that share a common edge [Opt. Express 26, 17872 (2018) [CrossRef]  ]. The edge-imaging condition states that, in combination, such glenses must image every point to itself. Here we begin the process of building up a library of combinations of glenses that satisfy the edge-imaging condition, starting with all relevant combinations of up to three glenses. As it grows, this library should become increasingly useful when constructing lens-based RTO devices.

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References

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  1. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref]
  3. R. F. Stevens and T. G. Harvey, “Lens arrays for a three-dimensional imaging system,” J. Opt. A 4, S17–S21 (2002).
    [Crossref]
  4. A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
    [Crossref]
  5. G. J. Chaplain, G. Macauley, J. Bělín, T. Tyc, E. N. Cowie, and J. Courtial, “Ray optics of generalized lenses,” J. Opt. Soc. Am. A 33, 962–969 (2016).
    [Crossref]
  6. J. Courtial, S. Oxburgh, and T. Tyc, “Direct, stigmatic, imaging with curved surfaces,” J. Opt. Soc. Am. A 32, 478–481 (2015).
    [Crossref]
  7. S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
    [Crossref]
  8. S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
    [Crossref]
  9. T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
    [Crossref]
  10. A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
    [Crossref]
  11. J. Courtial, “Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets,” Opt. Commun. 282, 2480–2483 (2009).
    [Crossref]
  12. T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
    [Crossref]
  13. J. Courtial, T. Tyc, J. Bělín, S. Oxburgh, G. Ferenczi, E. N. Cowie, and C. D. White, “Ray-optical transformation optics with ideal thin lenses makes omnidirectional lenses,” Opt. Express 26, 17872–17888 (2018).
    [Crossref]
  14. J. Bělín, T. Tyc, M. Grunwald, S. Oxburgh, E. N. Cowie, C. D. White, and J. Courtial, “Ideal-lens cloaks and new cloaking strategies,” Opt. Express (to be published).
  15. J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14, 9627–9635 (2006).
    [Crossref]
  16. A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
    [Crossref]
  17. J. Courtial, T. Tyc, S. Oxburgh, J. Bělín, E. N. Cowie, and C. D. White, “Mathematica notebooks with detailed loop-imaging-theorem calculations,” figshare, 2016, https://dx.doi.org/10.6084/m9.figshare.4269701.v1 .
  18. J. S. Choi and J. C. Howell, “Paraxial ray optics cloaking,” Opt. Express 22, 29465–29478 (2014).
    [Crossref]
  19. J. Courtial, “Calculations about interchangeability of the order of lenses and glenses with a common nodal point,” figshare, 2019, https://doi.org/10.6084/m9.figshare.10299299.v1 .

2018 (1)

2016 (3)

2015 (1)

2014 (2)

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

J. S. Choi and J. C. Howell, “Paraxial ray optics cloaking,” Opt. Express 22, 29465–29478 (2014).
[Crossref]

2011 (1)

T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[Crossref]

2009 (3)

A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
[Crossref]

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[Crossref]

J. Courtial, “Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets,” Opt. Commun. 282, 2480–2483 (2009).
[Crossref]

2006 (4)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

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

J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14, 9627–9635 (2006).
[Crossref]

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[Crossref]

2002 (1)

R. F. Stevens and T. G. Harvey, “Lens arrays for a three-dimensional imaging system,” J. Opt. A 4, S17–S21 (2002).
[Crossref]

Antoniou, G.

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

Belín, J.

Chaplain, G. J.

Choi, J. S.

Courtial, J.

J. Courtial, T. Tyc, J. Bělín, S. Oxburgh, G. Ferenczi, E. N. Cowie, and C. D. White, “Ray-optical transformation optics with ideal thin lenses makes omnidirectional lenses,” Opt. Express 26, 17872–17888 (2018).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
[Crossref]

G. J. Chaplain, G. Macauley, J. Bělín, T. Tyc, E. N. Cowie, and J. Courtial, “Ray optics of generalized lenses,” J. Opt. Soc. Am. A 33, 962–969 (2016).
[Crossref]

J. Courtial, S. Oxburgh, and T. Tyc, “Direct, stigmatic, imaging with curved surfaces,” J. Opt. Soc. Am. A 32, 478–481 (2015).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[Crossref]

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[Crossref]

J. Courtial, “Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets,” Opt. Commun. 282, 2480–2483 (2009).
[Crossref]

A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
[Crossref]

J. Bělín, T. Tyc, M. Grunwald, S. Oxburgh, E. N. Cowie, C. D. White, and J. Courtial, “Ideal-lens cloaks and new cloaking strategies,” Opt. Express (to be published).

Cowie, E. N.

Ferenczi, G.

Grunwald, M.

J. Bělín, T. Tyc, M. Grunwald, S. Oxburgh, E. N. Cowie, C. D. White, and J. Courtial, “Ideal-lens cloaks and new cloaking strategies,” Opt. Express (to be published).

Hamilton, A. C.

A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
[Crossref]

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[Crossref]

Harvey, T. G.

R. F. Stevens and T. G. Harvey, “Lens arrays for a three-dimensional imaging system,” J. Opt. A 4, S17–S21 (2002).
[Crossref]

Hendi, A.

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[Crossref]

Henn, J.

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[Crossref]

Howell, J. C.

Juzeliunas, G.

T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[Crossref]

Leonhardt, U.

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

Macauley, G.

Maceina, T.

T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[Crossref]

Miñano, J. C.

Orife, E.

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

Oxburgh, S.

J. Courtial, T. Tyc, J. Bělín, S. Oxburgh, G. Ferenczi, E. N. Cowie, and C. D. White, “Ray-optical transformation optics with ideal thin lenses makes omnidirectional lenses,” Opt. Express 26, 17872–17888 (2018).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
[Crossref]

J. Courtial, S. Oxburgh, and T. Tyc, “Direct, stigmatic, imaging with curved surfaces,” J. Opt. Soc. Am. A 32, 478–481 (2015).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

J. Bělín, T. Tyc, M. Grunwald, S. Oxburgh, E. N. Cowie, C. D. White, and J. Courtial, “Ideal-lens cloaks and new cloaking strategies,” Opt. Express (to be published).

Pendry, J. B.

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

Schurig, D.

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

Sharpe, T.

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

Smith, D. R.

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

Stevens, R. F.

R. F. Stevens and T. G. Harvey, “Lens arrays for a three-dimensional imaging system,” J. Opt. A 4, S17–S21 (2002).
[Crossref]

Tyc, T.

White, C. D.

J. Courtial, T. Tyc, J. Bělín, S. Oxburgh, G. Ferenczi, E. N. Cowie, and C. D. White, “Ray-optical transformation optics with ideal thin lenses makes omnidirectional lenses,” Opt. Express 26, 17872–17888 (2018).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

J. Bělín, T. Tyc, M. Grunwald, S. Oxburgh, E. N. Cowie, C. D. White, and J. Courtial, “Ideal-lens cloaks and new cloaking strategies,” Opt. Express (to be published).

J. Opt. (1)

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

J. Opt. A (2)

R. F. Stevens and T. G. Harvey, “Lens arrays for a three-dimensional imaging system,” J. Opt. A 4, S17–S21 (2002).
[Crossref]

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[Crossref]

J. Opt. Soc. Am. A (3)

New J. Phys. (1)

A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
[Crossref]

Opt. Commun. (2)

J. Courtial, “Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets,” Opt. Commun. 282, 2480–2483 (2009).
[Crossref]

T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[Crossref]

Opt. Express (3)

Phys. Rev. Lett. (1)

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[Crossref]

Proc. SPIE (1)

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

Science (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

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

Other (3)

J. Courtial, T. Tyc, S. Oxburgh, J. Bělín, E. N. Cowie, and C. D. White, “Mathematica notebooks with detailed loop-imaging-theorem calculations,” figshare, 2016, https://dx.doi.org/10.6084/m9.figshare.4269701.v1 .

J. Courtial, “Calculations about interchangeability of the order of lenses and glenses with a common nodal point,” figshare, 2019, https://doi.org/10.6084/m9.figshare.10299299.v1 .

J. Bělín, T. Tyc, M. Grunwald, S. Oxburgh, E. N. Cowie, C. D. White, and J. Courtial, “Ideal-lens cloaks and new cloaking strategies,” Opt. Express (to be published).

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Figures (10)

Fig. 1.
Fig. 1. Schematic of part of a transformation-optics device formed by glenses (cyan lines). The glenses divide the inside of the device into cells, three of them numbered 1 to 3; cell 0 is the outside of the device. The glens separating cells $ i $ and $ j $ ($ i \lt j $) is labelled $ {\text{G}_{ij}} $. A few light rays that start from a point light source at position $ {\text{P}_1} $ inside the space of cell 1 and travel to its image position in outside space, $ {\text{P}_0} $, are shown; rays that travel through glenses $ {\text{G}_{12}} $ and $ {\text{G}_{02}} $ are shown as solid red lines, and those that travel through glenses $ {\text{G}_{13}} $, $ {\text{G}_{23}} $, and $ {\text{G}_{02}} $ are shown as dashed red lines. $ {\text{P}_2} $ and $ {\text{P}_3} $ are intermediate images in the space of cells 2 and 3, respectively. A number of closed loops are shown as thick red arrows, the one around the common vertex between cells 1, 2, and 3 as a solid arrow, others as dotted arrows. The loops can contain any number of vertices, including none. For simplicity, the figure is drawn in 2D; in 3D, the polygonal cells from the 2D case become polyhedral cells, and the vertices become edges.
Fig. 2.
Fig. 2. Diagrams used in the derivation of conditions on the nodal points in combinations of (a) two, (b) three, (c) four, and (d) six glenses that satisfy the identity-mapping condition. The cyan lines marked as $ {\text{G}_i} $ indicate the plane of the $ i $-th glens, and $ {\text{N}_i} $ is the position of the nodal point of the $ i $-th glens. The red arrows show the trajectories of light rays that pass through the nodal point of the first glens and, in (c), also that of the second glens. In (d), the combination of glenses $ {\text{G}_1} $ to $ {\text{G}_4} $ satisfies the identity-mapping condition, and so does the combination of glenses $ {\text{G}_5} $ and $ {\text{G}_6} $. The nodal points of $ {\text{G}_1} $ to $ {\text{G}_4} $ and $ {\text{N}_1} $ to $ {\text{N}_4} $ lie on the dotted line. Glenses $ {\text{G}_5} $, $ {\text{G}_6} $, and $ {\text{G}_7} $ share a common nodal point, $ {\text{N}_{5,6,7}} $.
Fig. 3.
Fig. 3. Geometry of $ n $ glenses sharing a common edge, drawn here for $ n = 3 $. (a) 3D view; (b) orthographic projection in a plane perpendicular to the edge. E is a point on the edge; $ \hat{\bf d} $ is a unit vector in the direction of the edge; $ {\alpha _{ij}} $ is the angle from glens $ i $ to glens $ j $.
Fig. 4.
Fig. 4. Combinations of two lenses, $ {\text{L}_1} $ and $ {\text{L}_2} $, or two glenses, $ {\text{G}_1} $ and $ {\text{G}_2} $, that satisfy the edge-imaging condition. The two possible cases are (a) $ {\alpha _{12}} = 0 $ and (b) $ {\alpha _{12}} = \pi $. The common edge for which the edge-imaging condition is satisfied is highlighted by a thick dashed line. A closed loop around each of these edges is shown as a red arrow.
Fig. 5.
Fig. 5. Ray trajectory (red line) through three glenses, $ {\text{G}_1} $ to $ {\text{G}_3} $, which, in combination, perform the identity mapping. After transmission through all three glenses, the ray is restored to its original straight line. Due to the common nodal-point position N of the three glenses, the ray remains in the plane that includes the incident ray and N.
Fig. 6.
Fig. 6. Geometry of four-lens intersections for which the conditions on the focal lengths for satisfying the identity-mapping condition are known. (a) Four-lens paraxial cloak (“Rochester cloak”) [18]; (b,c) different four-lens intersection lines in the ideal-thin-lens structure that forms an omnidirectional transformation-optics device [13]. The lenses are marked $ {\text{L}_1} $ to $ {\text{L}_4} $; $ {\text{P}_1} $ to $ {\text{P}_4} $ are their principal points, which are also their nodal points.
Fig. 7.
Fig. 7. Regular star of lenses. Any number of lenses can form a regular star; the figure is drawn for six lenses. All lenses have the same focal length, $ f $, and principal point, P, and intersect along a common edge through P. All angles between neighboring lenses are equal; in a star of $ n $ lenses, neighboring lenses are at an angle $ 2\pi /n $.
Fig. 8.
Fig. 8. Placement of the principal points (which, in ideal lenses, coincide with the nodal points) according to the nodal-point conditions (Section 4) in structures of ideal lenses (cyan lines). $ {\text{P}_{i,j,...}} $ is the principal point (and therefore the nodal point) of lenses $ {\text{L}_i}, {\text{L}_j},\ldots $, placed so that the nodal-point conditions are satisfied for the intersections involving any of those lenses. (a) Example of a structure in which the conditions require one or more principal points, i.e., that of lens $ {\text{L}_1} $, to lie in two positions at the same time. In such structures, the nodal-point conditions cannot be satisfied. (b) Structure in which the nodal-point conditions can be satisfied.
Fig. 9.
Fig. 9. Mapping of an arbitrary position P in the presence of two planes, $ {\text{I}_1} $ and $ {\text{I}_2} $ (dashed black lines), that have the property that any point that lies on $ {\text{I}_1} $ or $ {\text{I}_2} $ is mapped to itself. Two light rays (solid red lines) are placed through P; $ {\text{A}_1} $ and $ {\text{B}_1} $ are the intersections of these rays with $ {\text{I}_1} $, and $ {\text{A}_2} $ and $ {\text{B}_2} $ are the intersections of these rays with $ {\text{I}_2} $.
Fig. 10.
Fig. 10. Diagrams relating to properties of co-planar glenses (cyan lines). The trajectories of a few principal rays are shown (red arrows). (a) Diagram used in the construction of the cardinal points of co-planar glenses, $ {\text{G}_1} $ and $ {\text{G}_2} $. $ \text{F}_1^ - $ and $ \text{F}_1^ + $, $ \text{F}_2^ - $ and $ \text{F}_2^ + $, and $ {\text{F}^ - } $ and $ {\text{F}^ + } $ are the object- and image-sided focal points of glenses $ {\text{G}_1} $, $ {\text{G}_2} $, and of the glens equivalent to the combination, respectively. N is the nodal point of the equivalent glens. For clarity, the glenses are drawn slightly apart. (b) The principal ray through the nodal points $ {\text{N}_1} $ and $ {\text{N}_2} $ of the individual glenses.

Equations (39)

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

n = f + + f .
P 0 = c 0 k c k j c j i P i .
c 02 c 21 = C 1 = c 02 c 23 c 31 .
I = c 12 c 23 c 31 ,
Q Q = 0 .
N 1 = N 2 = N ,
sin α 12 = 0 ,
f 2 = f 1 + ,
f 2 = f 1 + ,
N 1 = N 2 = N 3 ,
f 1 f 2 f 3 = f 1 + f 2 + f 3 + ,
f 1 + sin α 12 = f 3 sin α 23 , f 3 + sin α 13 = f 2 sin α 12 , f 1 sin α 31 = f 2 + sin α 32 .
η 1 η 2 η 3 = 1 ,
t 1 = f 1 + f 2 ,
t 2 = 2 f 2 ( f 1 + f 2 ) f 1 f 2 ,
2 t 1 + t 2 = 2 f 1 ( f 1 + f 2 ) f 1 f 2 .
f 3 f 1 = h 3 h 1 ,
f 4 = f 1 ( 2 f 3 cos β + h 3 sin β sin δ ) f 3 f 1 ,
f 4 = f 1 s 1 Δ y 4 , 3 Δ y 4 , 2 s 4 Δ y 3 , 1 Δ y 1 , 2 + x 0 Δ y 4 , 3 Δ y 4 , 1 s 4 Δ y 3 , 1 , f 3 = f 1 s 1 Δ y 4 , 3 Δ y 3 , 2 s 3 Δ y 4 , 1 Δ y 1 , 2 , f 2 = f 1 s 1 Δ y 4 , 2 Δ y 3 , 2 s 2 Δ y 4 , 1 Δ y 1 , 3 x 0 Δ y 2 , 3 Δ y 2 , 1 s 2 Δ y 3 , 1 ,
Δ y i , j = y i y j and s i = x 0 2 + y i 2 .
z = z f f ( z ) ,
z = z f f ( z e i φ ) .
z i = w i z 0 | z 0 | .
w i + 1 = w i f f ( w i e i φ i z 0 / | z 0 | ) ,
f w i f w i + 1 = ( w i e i φ i z 0 / | z 0 | ) w i .
f w i f w i + 1 = [ e i ( ψ φ i ) ] = cos ( φ i ψ ) ,
f w 0 f w n = i = 0 n 1 cos ( φ i ψ ) = 0 ,
h 2 h 1 f 2 + f 1 = h 1 n f 1
n = f 1 ( h 2 / h 1 ) f 2 + h 2 / h 1 1 .
h 2 h 1 = f 2 f 1 + .
n = f 1 f 2 + f 1 + f 2 f 1 + f 2 .
n = f 1 f 2 + f 1 f 2 f 1 + f 2 = 0.
1 f = 1 f 1 + 1 f 2 , 1 f + = 1 f 1 + + 1 f 2 + .
1 f = i = 1 N 1 f i , 1 f + = i = 1 N 1 f i + .
Q = Q + ( Q N ) a ^ f ( Q N ) a ^ ( Q N ) .
a ^ 1 = z ^ .
a ^ 2 = ( sin α 0 cos α ) .
Q = ( x y z ) ,
Q = f 1 f 2 f 2 ( f 1 z ) f 1 z cos α + f 1 x sin α Q .

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