Abstract

We study the ray optics of generalized lenses (glenses), which are ideal thin lenses generalized to have different object- and image-sided focal lengths, and the most general light-ray-direction-changing surfaces that stigmatically image any point in object space to a corresponding point in image space. Gabor superlenses [UK patent 541,753 (1940); J. Opt. A 1, 94 (1999) [CrossRef]  ] can be seen as pixelated realizations of glenses. Our analysis is centered on the nodal point. Whereas the nodal point of a thin lens always resides in the lens plane, that of a glens can reside anywhere on the optical axis. Utilizing the nodal point, we derive simple equations that describe the mapping between object and image space and the light-ray-direction change. We demonstrate our findings with the help of ray-tracing simulations. Glenses allow novel optical instruments to be realized, at least theoretically, and our results facilitate the design and analysis of such devices.

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References

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  1. J. Courtial, “Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets,” Opt. Commun. 282, 2480–2483 (2009).
    [Crossref]
  2. J. Courtial, S. Oxburgh, and T. Tyc, “Direct, stigmatic, imaging with curved surfaces,” J. Opt. Soc. Am. A 32, 478–481 (2015).
    [Crossref]
  3. D. Gabor, “Improvements in or relating to optical systems composed of lenticules,” UK patent541,753 (December10, 1941).
  4. C. Hembd-Sölner, R. F. Stevens, and M. C. Hutley, “Imaging properties of the Gabor superlens,” J. Opt. A 1, 94–102 (1999).
    [Crossref]
  5. D. S. Goodman, “General principles of geometric optics,” in Handbook of Optics, M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed., Fundamentals, Techniques, and Design (McGraw-Hill, 1995), Chap. 1.15, Vol. I, pp. 1.60–1.68.
  6. J. Courtial and T. Tyc, “Generalised laws of refraction that can lead to wave-optically forbidden light-ray fields,” J. Opt. Soc. Am. A 29, 1407–1411 (2012).
    [Crossref]
  7. E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Resolution limit of pixellated optical components” (in preparation).
  8. 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]
  9. G. Lippmann, “La photographie intégrale,” C. R. Hebd. Acad. Sci. 146, 446–451 (1908).
  10. T. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99–106 (1992).
    [Crossref]
  11. R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (2005).
  12. R. Ng, “Fourier slice photography,” ACM Trans. Graph. 24, 735–744 (2005).
    [Crossref]
  13. R. F. Stevens, “Integral images formed by lens arrays,” poster presented at Conference on Microlens Arrays, Teddington, London, May11–12, 1995, Vol. 5.
  14. J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008).
    [Crossref]
  15. A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
    [Crossref]
  16. S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
    [Crossref]
  17. 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. (in press).
  18. M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3, 133–142 (1994).
    [Crossref]
  19. J. Courtial, “Standard and non-standard metarefraction with confocal lenslet arrays,” Opt. Commun. 282, 2634–2641 (2009).
    [Crossref]
  20. T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
    [Crossref]
  21. E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Optical design of generalised confocal lenslet arrays” (in preparation).
  22. E. N. Cowie and J. Courtial, “Engineering the field of view of generalised confocal lenslet arrays (GCLAs)” (in preparation).
  23. W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), Chap. 2.2.
  24. S. Oxburgh and J. Courtial, “Perfect imaging with planar interfaces,” J. Opt. Soc. Am. A 30, 2334–2338 (2013).
    [Crossref]
  25. J. C. R. Wylie, Introduction to Projective Geometry (Dover, 2008), pp. 186–190.
  26. A. Heyting, N. G. D. Bruijn, J. D. Groot, and A. C. Zaanen, Axiomatic Projective Geometry, 2nd ed. (North-Holland, 1980), Chap. 2, p. 41.
  27. H. S. M. Coxeter, Projective Geometry, 2nd ed. (University of Toronto, 1974), pp. 49–57.
  28. S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
    [Crossref]

2015 (1)

2014 (2)

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

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

2013 (1)

2012 (1)

2011 (1)

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

2009 (4)

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

J. Courtial, “Standard and non-standard metarefraction with confocal lenslet arrays,” Opt. Commun. 282, 2634–2641 (2009).
[Crossref]

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

2008 (1)

J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008).
[Crossref]

2005 (1)

R. Ng, “Fourier slice photography,” ACM Trans. Graph. 24, 735–744 (2005).
[Crossref]

1999 (1)

C. Hembd-Sölner, R. F. Stevens, and M. C. Hutley, “Imaging properties of the Gabor superlens,” J. Opt. A 1, 94–102 (1999).
[Crossref]

1994 (1)

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3, 133–142 (1994).
[Crossref]

1992 (1)

T. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99–106 (1992).
[Crossref]

1908 (1)

G. Lippmann, “La photographie intégrale,” C. R. Hebd. Acad. Sci. 146, 446–451 (1908).

Adelson, T.

T. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99–106 (1992).
[Crossref]

Antoniou, G.

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

Bourgenot, C.

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Optical design of generalised confocal lenslet arrays” (in preparation).

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Resolution limit of pixellated optical components” (in preparation).

Brédif, M.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (2005).

Bruijn, N. G. D.

A. Heyting, N. G. D. Bruijn, J. D. Groot, and A. C. Zaanen, Axiomatic Projective Geometry, 2nd ed. (North-Holland, 1980), Chap. 2, p. 41.

Courtial, J.

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]

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

S. Oxburgh and J. Courtial, “Perfect imaging with planar interfaces,” J. Opt. Soc. Am. A 30, 2334–2338 (2013).
[Crossref]

J. Courtial and T. Tyc, “Generalised laws of refraction that can lead to wave-optically forbidden light-ray fields,” J. Opt. Soc. Am. A 29, 1407–1411 (2012).
[Crossref]

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

J. Courtial, “Standard and non-standard metarefraction with confocal lenslet arrays,” Opt. Commun. 282, 2634–2641 (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]

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

J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008).
[Crossref]

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Resolution limit of pixellated optical components” (in preparation).

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. (in press).

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Optical design of generalised confocal lenslet arrays” (in preparation).

E. N. Cowie and J. Courtial, “Engineering the field of view of generalised confocal lenslet arrays (GCLAs)” (in preparation).

Cowie, E. N.

E. N. Cowie and J. Courtial, “Engineering the field of view of generalised confocal lenslet arrays (GCLAs)” (in preparation).

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Optical design of generalised confocal lenslet arrays” (in preparation).

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Resolution limit of pixellated optical components” (in preparation).

Coxeter, H. S. M.

H. S. M. Coxeter, Projective Geometry, 2nd ed. (University of Toronto, 1974), pp. 49–57.

Duval, G.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (2005).

Gabor, D.

D. Gabor, “Improvements in or relating to optical systems composed of lenticules,” UK patent541,753 (December10, 1941).

Goodman, D. S.

D. S. Goodman, “General principles of geometric optics,” in Handbook of Optics, M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed., Fundamentals, Techniques, and Design (McGraw-Hill, 1995), Chap. 1.15, Vol. I, pp. 1.60–1.68.

Groot, J. D.

A. Heyting, N. G. D. Bruijn, J. D. Groot, and A. C. Zaanen, Axiomatic Projective Geometry, 2nd ed. (North-Holland, 1980), Chap. 2, p. 41.

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]

Hanrahan, P.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (2005).

Hembd-Sölner, C.

C. Hembd-Sölner, R. F. Stevens, and M. C. Hutley, “Imaging properties of the Gabor superlens,” J. Opt. A 1, 94–102 (1999).
[Crossref]

Heyting, A.

A. Heyting, N. G. D. Bruijn, J. D. Groot, and A. C. Zaanen, Axiomatic Projective Geometry, 2nd ed. (North-Holland, 1980), Chap. 2, p. 41.

Horowitz, M.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (2005).

Hunt, R.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3, 133–142 (1994).
[Crossref]

Hutley, M. C.

C. Hembd-Sölner, R. F. Stevens, and M. C. Hutley, “Imaging properties of the Gabor superlens,” J. Opt. A 1, 94–102 (1999).
[Crossref]

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3, 133–142 (1994).
[Crossref]

Juzeliunas, G.

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

Levoy, M.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (2005).

Lippmann, G.

G. Lippmann, “La photographie intégrale,” C. R. Hebd. Acad. Sci. 146, 446–451 (1908).

Maceina, T.

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

Ng, R.

R. Ng, “Fourier slice photography,” ACM Trans. Graph. 24, 735–744 (2005).
[Crossref]

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (2005).

Orife, E.

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

Oxburgh, S.

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]

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

S. Oxburgh and J. Courtial, “Perfect imaging with planar interfaces,” J. Opt. Soc. Am. A 30, 2334–2338 (2013).
[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. (in press).

Robertson, D.

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Resolution limit of pixellated optical components” (in preparation).

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Optical design of generalised confocal lenslet arrays” (in preparation).

Savander, P.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3, 133–142 (1994).
[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. (in press).

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), Chap. 2.2.

Stevens, R. F.

C. Hembd-Sölner, R. F. Stevens, and M. C. Hutley, “Imaging properties of the Gabor superlens,” J. Opt. A 1, 94–102 (1999).
[Crossref]

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3, 133–142 (1994).
[Crossref]

R. F. Stevens, “Integral images formed by lens arrays,” poster presented at Conference on Microlens Arrays, Teddington, London, May11–12, 1995, Vol. 5.

Tyc, T.

Wang, J. Y. A.

T. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99–106 (1992).
[Crossref]

White, C. D.

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

Wylie, J. C. R.

J. C. R. Wylie, Introduction to Projective Geometry (Dover, 2008), pp. 186–190.

Zaanen, A. C.

A. Heyting, N. G. D. Bruijn, J. D. Groot, and A. C. Zaanen, Axiomatic Projective Geometry, 2nd ed. (North-Holland, 1980), Chap. 2, p. 41.

ACM Trans. Graph. (1)

R. Ng, “Fourier slice photography,” ACM Trans. Graph. 24, 735–744 (2005).
[Crossref]

C. R. Hebd. Acad. Sci. (1)

G. Lippmann, “La photographie intégrale,” C. R. Hebd. Acad. Sci. 146, 446–451 (1908).

Comp. Phys. Commun. (1)

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

T. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99–106 (1992).
[Crossref]

J. Opt. A (2)

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

C. Hembd-Sölner, R. F. Stevens, and M. C. Hutley, “Imaging properties of the Gabor superlens,” J. Opt. A 1, 94–102 (1999).
[Crossref]

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

New J. Phys. (2)

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. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008).
[Crossref]

Opt. Commun. (3)

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

J. Courtial, “Standard and non-standard metarefraction with confocal lenslet arrays,” Opt. Commun. 282, 2634–2641 (2009).
[Crossref]

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

Pure Appl. Opt. (1)

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3, 133–142 (1994).
[Crossref]

Other (12)

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. (in press).

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (2005).

R. F. Stevens, “Integral images formed by lens arrays,” poster presented at Conference on Microlens Arrays, Teddington, London, May11–12, 1995, Vol. 5.

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Resolution limit of pixellated optical components” (in preparation).

D. Gabor, “Improvements in or relating to optical systems composed of lenticules,” UK patent541,753 (December10, 1941).

D. S. Goodman, “General principles of geometric optics,” in Handbook of Optics, M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds., 2nd ed., Fundamentals, Techniques, and Design (McGraw-Hill, 1995), Chap. 1.15, Vol. I, pp. 1.60–1.68.

E. N. Cowie, C. Bourgenot, D. Robertson, and J. Courtial, “Optical design of generalised confocal lenslet arrays” (in preparation).

E. N. Cowie and J. Courtial, “Engineering the field of view of generalised confocal lenslet arrays (GCLAs)” (in preparation).

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), Chap. 2.2.

J. C. R. Wylie, Introduction to Projective Geometry (Dover, 2008), pp. 186–190.

A. Heyting, N. G. D. Bruijn, J. D. Groot, and A. C. Zaanen, Axiomatic Projective Geometry, 2nd ed. (North-Holland, 1980), Chap. 2, p. 41.

H. S. M. Coxeter, Projective Geometry, 2nd ed. (University of Toronto, 1974), pp. 49–57.

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Figures (8)

Fig. 1.
Fig. 1.

(a) Rays passing through a generalized thin lens (glens). It can be seen that the focal points, F and F+, are located at different distances from the glens. The glens is indicated by a gray line with asymmetric triangles on both ends, a modification of the traditional symmetric triangles at the ends of a thin lens to reflect the asymmetry between the spaces on either side. The optical axis is shown as a dash–dotted line. It coincides with the axis of a Cartesian coordinate system, with corresponding coordinate a, such that the glens lies in the plane a=0. We use the same coordinate system for the spaces on both sides of the glens. The triangles at the ends of the line indicating the glens are drawn on the side of the glens facing in the positive a direction. Another way to indicate the sense of the optical axis is to draw next to the glens a “−” on the side of negative space and a “+” on the side of positive space. F and F+ are the focal points in the glens’s positive and negative space; f and f+ are the corresponding a coordinates, the focal lengths; the focal planes are shown as dotted vertical lines. A number of rays (solid red lines) are shown, namely, the principal rays through the positive and negative focal points, the ray through the glens center, P, where the optical axis intersects the glens, and the undeviated ray between the object–image pair Q and Q+, which passes through the nodal point, N. (b) Glen Clova, one of the Scottish valleys also known as glens.

Fig. 2.
Fig. 2.

Location of the nodal point of a glens. Two parallel light rays are incident on the glens with direction d, one (marked “1”) through the negative focal point, F, and intersecting the glens at position S, the other (marked “2”) through the point I where the first light ray intersects the image-sided focal plane and intersecting the optical axis at N, the nodal point. The triangles FSP and NIF+ are congruent.

Fig. 3.
Fig. 3.

Glens mapping. Here shown for light rays incident from negative space.

Fig. 4.
Fig. 4.

Light-ray-direction change upon transmission through a glens. A light ray (solid red line) is incident from object space, with direction d, at position S. If the light ray is incident from positive (negative) space, then object space is positive (negative) space and image space is negative (positive) space. The auxiliary light ray (dotted red line) that passes through the nodal point N with the same direction intersects the image-sided focal plane at the same position, I, as the refracted light ray.

Fig. 5.
Fig. 5.

Ray-tracing simulations of objects located near the nodal plane of a glens (b), (d) seen through that glens and (a), (c) for comparison without the glens. (a) and (b) have been calculated for one position of the virtual camera, (c) and (d) for another. The sphere, the cone, and the cylinder touch the nodal plane from behind; the vertex of the cone is located at the nodal point, N. The glens present in (b) and (d) is located a distance 2 (in units of floor-tile widths) in front of the nodal plane, with its a axis pointing away from the camera in (b). Its focal lengths are f=2 and f+=4. In all frames, the virtual camera is focused on the nodal plane.

Fig. 6.
Fig. 6.

Ray-tracing simulation of the view through a glens taken with a camera located at its nodal point. (a) Scene without glens (pinhole camera); (b) with glens (pinhole camera); (c) with glens, focused on the plane touching the closest parts of the object; (d) with glens, focused on the image of this plane (which is located in a plane behind the camera).

Fig. 7.
Fig. 7.

Glens telescope comprising a pair of glenses that share a common focal point, namely, the first glens’s positive-space focal point, F1+, and the second glens’s negative-space focal point, F2, and a common nodal point, N. The trajectories of a bundle of initially parallel rays (red lines) is shown. As it passes through the telescope from left to right, the ray bundle gets expanded without reducing its angle with the optical axis.

Fig. 8.
Fig. 8.

Simulated view through the glens telescope shown in Fig. 7, which multiplies the distance of any light ray from N by a factor M=3 without altering the ray’s direction. The camera is positioned at the glenses’ common nodal point, N. In (a), the camera is focused on the plane where the objects actually are (here, z=10). In (b), it is focused on a plane M times further away (z=30). In both frames, the central, circular, part of the view is seen through both glenses, which provide a sharp image in (b).

Equations (22)

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n=f++f
QQa=QNfa,
Q=Q+afa(QN).
(a,b,c)=1af[a(nf),bf,cf],
(a,b,c)=1af(af,bf,cf).
a=afaf,
fa+fa=1.
MT=bb=faf.
ML=aa=f(nf)(af)2=ff(af)2=ffMT2.
MT=ML=ff.
a=a(f1n1)(f2n2)f1f2a(f1+f2)+an1.
Q=QaNf.
Nf=(δxδy1η),
dIS,
da=d·a^
d=daf(IS).
I=N+fdda.
d=ffd+daf(NS).
Q=Q+af/ga/g(QgNg).
Q=Qaf/gNg.
d=f/gf/gd+daf/g(NgSg).
d=f/gf/gd+daf/gNg.

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