Abstract

In a recent publication (T. Ergin et al., Science 328, 337 (2010)), three-dimensional broadband dielectric carpet cloaks have been fabricated and experimentally characterized by optical bright-field and dark-field microscopy using unpolarized light from an incandescent lamp. A direct comparison with theory has not been provided so far. In the present work, we treat the carpet cloak as well as the entire optical microscope within the ray-optics approximation and the cloak within the effective-medium approximation. We find good qualitative agreement between experimental results and our calculations.

© 2010 OSA

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
    [CrossRef] [PubMed]
  3. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
    [CrossRef] [PubMed]
  4. V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
    [CrossRef] [PubMed]
  5. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
    [CrossRef] [PubMed]
  6. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
    [CrossRef] [PubMed]
  7. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
    [CrossRef] [PubMed]
  8. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
    [CrossRef] [PubMed]
  9. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
    [CrossRef]
  10. J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17(15), 12922–12928 (2009).
    [CrossRef] [PubMed]
  11. I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: application to optical cloaking,” Phys. Rev. Lett. 102(21), 213901 (2009).
    [CrossRef] [PubMed]
  12. J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express 17(22), 19328–19336 (2009).
    [CrossRef] [PubMed]
  13. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
    [CrossRef] [PubMed]
  14. H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1(3), 1–6 (2010).
    [CrossRef]
  15. B. Zhang, T. Chan, and B.-I. Wu, “Lateral Shift Makes a Ground-Plane Cloak Detectable,” Phys. Rev. Lett. 104(23), 233903 (2010).
    [CrossRef] [PubMed]

2010 (4)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1(3), 1–6 (2010).
[CrossRef]

B. Zhang, T. Chan, and B.-I. Wu, “Lateral Shift Makes a Ground-Plane Cloak Detectable,” Phys. Rev. Lett. 104(23), 233903 (2010).
[CrossRef] [PubMed]

2009 (6)

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17(15), 12922–12928 (2009).
[CrossRef] [PubMed]

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: application to optical cloaking,” Phys. Rev. Lett. 102(21), 213901 (2009).
[CrossRef] [PubMed]

J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express 17(22), 19328–19336 (2009).
[CrossRef] [PubMed]

2008 (2)

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
[CrossRef] [PubMed]

2006 (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Blair, J.

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

Chan, T.

B. Zhang, T. Chan, and B.-I. Wu, “Lateral Shift Makes a Ground-Plane Cloak Detectable,” Phys. Rev. Lett. 104(23), 233903 (2010).
[CrossRef] [PubMed]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Cui, T. J.

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1(3), 1–6 (2010).
[CrossRef]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Ergin, T.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express 17(22), 19328–19336 (2009).
[CrossRef] [PubMed]

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Halimeh, J. C.

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kildishev, A. V.

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: application to optical cloaking,” Phys. Rev. Lett. 102(21), 213901 (2009).
[CrossRef] [PubMed]

Lee, J. H.

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Ma, H. F.

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1(3), 1–6 (2010).
[CrossRef]

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Mueller, J.

Park, W.

Pendry, J. B.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Rhee, S. J.

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: application to optical cloaking,” Phys. Rev. Lett. 102(21), 213901 (2009).
[CrossRef] [PubMed]

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
[CrossRef] [PubMed]

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Smolyaninov, I. I.

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: application to optical cloaking,” Phys. Rev. Lett. 102(21), 213901 (2009).
[CrossRef] [PubMed]

Smolyaninova, V. N.

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: application to optical cloaking,” Phys. Rev. Lett. 102(21), 213901 (2009).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express 17(22), 19328–19336 (2009).
[CrossRef] [PubMed]

Summers, C. J.

Tamma, V. A.

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Wegener, M.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express 17(22), 19328–19336 (2009).
[CrossRef] [PubMed]

Wu, B.-I.

B. Zhang, T. Chan, and B.-I. Wu, “Lateral Shift Makes a Ground-Plane Cloak Detectable,” Phys. Rev. Lett. 104(23), 233903 (2010).
[CrossRef] [PubMed]

Wu, Q.

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Zhang, B.

B. Zhang, T. Chan, and B.-I. Wu, “Lateral Shift Makes a Ground-Plane Cloak Detectable,” Phys. Rev. Lett. 104(23), 233903 (2010).
[CrossRef] [PubMed]

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Nat. Mater. (2)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

Nat. Photonics (1)

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Nature Commun. (1)

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1(3), 1–6 (2010).
[CrossRef]

Opt. Express (2)

Phys. Rev. Lett. (3)

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: application to optical cloaking,” Phys. Rev. Lett. 102(21), 213901 (2009).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

B. Zhang, T. Chan, and B.-I. Wu, “Lateral Shift Makes a Ground-Plane Cloak Detectable,” Phys. Rev. Lett. 104(23), 233903 (2010).
[CrossRef] [PubMed]

Science (6)

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

(a) False-color representation of the refractive-index distribution of the reference structure within the xy-plane. Within the structure, the refractive index is constant along the z-direction. The structure is located on a thick glass substrate. The sample rotation axis is parallel to the z-direction and located at the depicted black cross. (b) Same as (a), but for the carpet cloak structure. On the left- and right-hand side, the cloaking structure is surrounded by regions of constant refractive index equal to the reference refractive index of n = 1.2. (c) Three-dimensional illustration of the overall structure in (b).

Fig. 2
Fig. 2

Sketch of the microscope geometry (not to scale) used in our ray-tracing calculations of the 3D carpet cloak. The illumination light source is either a point source or a homogeneous disk with diameter d. For all images shown in this work, we focus the microscope onto the flat metal film on top of the polymer structure on the glass substrate. A magnified view of the sample is shown in Fig. 1.

Fig. 3
Fig. 3

Microscope images rendered by ray tracing for the geometry outlined in Fig. 2 and for the case of a bump without cloak (see Fig. 1(a)) and for point-source illumination. Images are shown for a cos2-shaped bump (left column) and for a triangular bump (right column). The five rows show different sample tilt angles increasing from top to bottom. Zero-degree tilt angle corresponds to the bright-field mode, large angles to the dark-field mode. The local image intensity is false-color coded. The values quoted on the right-hand side correspond to the total number of rays that have hit a square pixel element in the image plane, the area of which corresponds to 0.5 µm × 0.5 µm in the sample plane. The total number of rays that have illuminated the sample through the objective lens is given by the white number in the individual panels. The white scale bar shown in all panels corresponds to 50 µm length in the sample plane. The white arrows point to the scattering of light off the bump. Within the ray-optics approximation, the depicted images do not depend on wavelength at all.

Fig. 4
Fig. 4

Rendered microscope images as in Fig. 3, but for a finite disk-shaped light source with diameter d (see Fig. 2) rather than for point-source illumination. The dashed white rectangles outline the depicted areas in Figs. 5 and 6.

Fig. 5
Fig. 5

Microscope images for the bright-field mode (left column) and the dark-field mode with 35-degree sample tilt (right column) without (first row) and with carpet cloak (second row), all rendered by ray tracing. The structure dimensions are given in Sections 2 and 3, the objective lens numerical aperture is NA = 0.5 (see Fig. 2). The refractive-index distribution underlying the carpet cloak is shown in Fig. 1. A total number of 108 rays have illuminated each sample. The scale bar corresponds to a length of 10 µm in the sample plane. The area of the image shown here is outlined in Fig. 4. The dashed white line marks the position of the bump reflection in both dark-field images.

Fig. 6
Fig. 6

Microscope bright-field images as in the left column of Fig. 5 (with NA = 0.5), but for a numerical aperture of the objective lens of NA = 0.7 and NA = 0.9 as indicated (see Fig. 2). For these cases, the light samples a much larger fraction of all possible directions in three-dimensional space.

Equations (6)

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

1 a + 1 a ' = 1 f .
α = arctan ( b f tan( α ) ) .
m = 1 a ' f = 80.
N A = sin( φ )=0 .5 .
a s = t s ( 1 1 n s ) ,
a c = t c ( 1 1 n c ) ,

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