Abstract

Focusing of an incident light wave through a plano-convex spherical lens is discussed by calculating the light intensity distribution on the lens’s optical axis after the incident wave is multiply scattered inside the lens. It is found that the size and location of the region into which the incident wave is focused are determined by two conditions. It is also found that it is possible for the wave to be focused into two such regions.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Klein and T. Furtak, Optics, 2nd ed. (Wiley, 1986).
  2. L. Levesque, “Close up of monochromatic aberrations using Snell’s law: an undergraduate computational experiment,” Eur. J. Phys. 30, 1201–1215 (2009).
    [CrossRef]
  3. W. D. Furlan, L. Munoz-Escriva, A. Pons, and M. Martinez-Corral, “Optical aberrations measurement with a low cost optometric instrument,” Am. J. Phys. 70, 857–861 (2002).
    [CrossRef]
  4. S. Mak, “Longitudinal spherical aberration of a thick lens,” Am. J. Phys. 55, 247–249 (1987).
    [CrossRef]
  5. P. E. Klingsporn, “Minimum and absolute minimum spherical aberration of a simple, thin lens,” Am. J. Phys. 48, 821–827(1980).
    [CrossRef]
  6. W. Guo, “Multiple scattering of a plane scalar wave from a dielectric slab,” Am. J. Phys. 70, 1039–1043 (2002).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  8. J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).
  9. S. Prasad and W. Guo, “Multiple scattering approach to Mie scattering from a sphere of arbitrary size,” Opt. Commun. 136, 447–460 (1997).
    [CrossRef]
  10. J. D. Jackson, Classical Electrodynamics (Wiley, 1975).
  11. L. E. Helseth, “The almost perfect lens and focusing of evanescent waves,” Opt. Commun. 281, 1981–1985(2008).
    [CrossRef]
  12. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

2009

L. Levesque, “Close up of monochromatic aberrations using Snell’s law: an undergraduate computational experiment,” Eur. J. Phys. 30, 1201–1215 (2009).
[CrossRef]

2008

L. E. Helseth, “The almost perfect lens and focusing of evanescent waves,” Opt. Commun. 281, 1981–1985(2008).
[CrossRef]

2002

W. Guo, “Multiple scattering of a plane scalar wave from a dielectric slab,” Am. J. Phys. 70, 1039–1043 (2002).
[CrossRef]

W. D. Furlan, L. Munoz-Escriva, A. Pons, and M. Martinez-Corral, “Optical aberrations measurement with a low cost optometric instrument,” Am. J. Phys. 70, 857–861 (2002).
[CrossRef]

1997

S. Prasad and W. Guo, “Multiple scattering approach to Mie scattering from a sphere of arbitrary size,” Opt. Commun. 136, 447–460 (1997).
[CrossRef]

1987

S. Mak, “Longitudinal spherical aberration of a thick lens,” Am. J. Phys. 55, 247–249 (1987).
[CrossRef]

1980

P. E. Klingsporn, “Minimum and absolute minimum spherical aberration of a simple, thin lens,” Am. J. Phys. 48, 821–827(1980).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Furlan, W. D.

W. D. Furlan, L. Munoz-Escriva, A. Pons, and M. Martinez-Corral, “Optical aberrations measurement with a low cost optometric instrument,” Am. J. Phys. 70, 857–861 (2002).
[CrossRef]

Furtak, T.

M. Klein and T. Furtak, Optics, 2nd ed. (Wiley, 1986).

Guo, W.

W. Guo, “Multiple scattering of a plane scalar wave from a dielectric slab,” Am. J. Phys. 70, 1039–1043 (2002).
[CrossRef]

S. Prasad and W. Guo, “Multiple scattering approach to Mie scattering from a sphere of arbitrary size,” Opt. Commun. 136, 447–460 (1997).
[CrossRef]

Helseth, L. E.

L. E. Helseth, “The almost perfect lens and focusing of evanescent waves,” Opt. Commun. 281, 1981–1985(2008).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Klein, M.

M. Klein and T. Furtak, Optics, 2nd ed. (Wiley, 1986).

Klingsporn, P. E.

P. E. Klingsporn, “Minimum and absolute minimum spherical aberration of a simple, thin lens,” Am. J. Phys. 48, 821–827(1980).
[CrossRef]

Levesque, L.

L. Levesque, “Close up of monochromatic aberrations using Snell’s law: an undergraduate computational experiment,” Eur. J. Phys. 30, 1201–1215 (2009).
[CrossRef]

Mak, S.

S. Mak, “Longitudinal spherical aberration of a thick lens,” Am. J. Phys. 55, 247–249 (1987).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martinez-Corral, M.

W. D. Furlan, L. Munoz-Escriva, A. Pons, and M. Martinez-Corral, “Optical aberrations measurement with a low cost optometric instrument,” Am. J. Phys. 70, 857–861 (2002).
[CrossRef]

Munoz-Escriva, L.

W. D. Furlan, L. Munoz-Escriva, A. Pons, and M. Martinez-Corral, “Optical aberrations measurement with a low cost optometric instrument,” Am. J. Phys. 70, 857–861 (2002).
[CrossRef]

Pons, A.

W. D. Furlan, L. Munoz-Escriva, A. Pons, and M. Martinez-Corral, “Optical aberrations measurement with a low cost optometric instrument,” Am. J. Phys. 70, 857–861 (2002).
[CrossRef]

Prasad, S.

S. Prasad and W. Guo, “Multiple scattering approach to Mie scattering from a sphere of arbitrary size,” Opt. Commun. 136, 447–460 (1997).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Am. J. Phys.

W. D. Furlan, L. Munoz-Escriva, A. Pons, and M. Martinez-Corral, “Optical aberrations measurement with a low cost optometric instrument,” Am. J. Phys. 70, 857–861 (2002).
[CrossRef]

S. Mak, “Longitudinal spherical aberration of a thick lens,” Am. J. Phys. 55, 247–249 (1987).
[CrossRef]

P. E. Klingsporn, “Minimum and absolute minimum spherical aberration of a simple, thin lens,” Am. J. Phys. 48, 821–827(1980).
[CrossRef]

W. Guo, “Multiple scattering of a plane scalar wave from a dielectric slab,” Am. J. Phys. 70, 1039–1043 (2002).
[CrossRef]

Eur. J. Phys.

L. Levesque, “Close up of monochromatic aberrations using Snell’s law: an undergraduate computational experiment,” Eur. J. Phys. 30, 1201–1215 (2009).
[CrossRef]

Opt. Commun.

S. Prasad and W. Guo, “Multiple scattering approach to Mie scattering from a sphere of arbitrary size,” Opt. Commun. 136, 447–460 (1997).
[CrossRef]

L. E. Helseth, “The almost perfect lens and focusing of evanescent waves,” Opt. Commun. 281, 1981–1985(2008).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

M. Klein and T. Furtak, Optics, 2nd ed. (Wiley, 1986).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Diagram of the plano-convex spherical lens, where z 0 denotes the thickness of the lens, and R is the radius of the spherical surface of the lens.

Fig. 2
Fig. 2

Distribution of light intensity | E | 2 × 10 4 on the optical axis as a function of the normalized position z ˜ = z / z 0 , where g = 2 π × 10 4 and R / z 0 = 20 are used. The expression of E is given in Eq. (6).

Fig. 3
Fig. 3

Distribution of light intensity | E | 2 × 10 4 on the optical axis as a function of the normalized position z ˜ = z / z 0 , where g = 2 π × 10 4 and R / z 0 = 4.1 are used. The expression of E is given in Eq. (6).

Equations (13)

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

E ( z ) = E inc ( z ) + α k 2 ϱ e i k | z z ^ r 1 | | z z ^ r 1 | e i k z 1 d r 1 + ( α k 2 ϱ ) 2 e i k | z z ^ r 1 | | z z ^ r 1 | d r 1 e i k | r 1 r 2 | | r 1 r 2 | e i k z 2 d r 2 + ,
E ( z ) = E inc ( z ) + α k 2 ϱ e i k | z z ^ r 1 | | z z ^ r 1 | d r 1 ( e i k z 1 + α k 2 ϱ e i k | r 1 r 2 | | r 1 r 2 | e i k z 2 d r 2 + ) E inc ( z ) + α k 2 ϱ e i k | z z ^ r 1 | | z z ^ r 1 | M ( r 1 ) d r 1 ,
e i k ( η 1 η 2 ) 2 2 ( z 1 z 2 ) d η 2 2 π i z 1 z 2 k ,
α k 2 ϱ e i k | r 1 r 2 | | r 1 r 2 | e i k z 2 d r 2 = ( α k 2 ϱ ) 2 π i k z 1 e i k z 1 .
M ( r ) = e i k ( 1 + 2 π α ϱ ) z 1 e i k ˜ z 1 ,
E ( z ) = e i k z + α k 2 ϱ e i k | z z ^ r 1 | | z z ^ r 1 | e i k ˜ z 1 d r 1 = e i g ( z ˜ + n 1 ) ( n 1 ) g i 0 1 e i g ( n t + A t + B ) d t ,
0 1 e i g ( n t + A t + B ) d t = i g n e i g ( n b A + A 4 n ) [ e i g n A ( B A 2 n ) 2 e i g n A ( z ˜ 1 A 2 n ) 2 ] + 1 n e i g ( n B A + A 4 n ) z ˜ 1 A 2 n B A 2 n e i g n A t 1 2 d t 1 .
E ( z ) = e i g ( z ˜ + n 1 ) + n 1 n e i g ( n b A + A 4 n ) [ e i g n A ( B A 2 n ) 2 e i g n A ( z ˜ 1 A 2 n ) 2 ] i g n 1 n e i g ( n B A + A 4 n ) z ˜ 1 A 2 n B A 2 n e i g n A t 1 2 d t 1 .
C 1 = i A 2 g n [ e i g n A ( B A 2 n ) 2 ( B A 2 n ) e i g n A ( z ˜ 1 A 2 n ) 2 ( z ˜ 1 A 2 n ) ] ,
C 2 = ( π A g n ) 1 / 2 e i π 4 .
E ( z ) = e i g ( z ˜ + n 1 ) + n 1 n e i g ( n b A + A 4 n ) [ e i g n A ( B A 2 n ) 2 e i g n A ( z ˜ 1 A 2 n ) 2 ] + ( n 1 ) A 2 n 2 e i g ( n B A + A 4 n ) [ e i g n A ( B A 2 n ) 2 ( B A 2 n ) - e i g n A ( z ˜ 1 A 2 n ) 2 ( z ˜ 1 A 2 n ) ] i n 1 n ( π A g n ) 1 / 2 e i π 4 + i g ( n B A + A 4 n ) .
B A 2 n > 0 ,
z ˜ 1 A 2 n < 0.

Metrics