Abstract

The first-order imaging properties of dielectric rods with refractive index decreasing quadratically with distance from the axis are analyzed from the viewpoint of gaussian ray optics. Cardinal-point locations vary with rod length, but the usual object–image relationships are then still applicable. Several ray diagrams are shown; one of the examples is a contact magnifier.

© 1970 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
  2. A. Fletcher, T. Murphy, and A. Young, Proc. Roy. Soc. (London) 223, 216 (1954).
    [Crossref]
  3. E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229 (1967).
  4. S. Kawakami and J. Nizhizawa, IEEE Trans. MTT-16, 814 (1968).
  5. E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 753 (1970).
    [Crossref] [PubMed]
  6. C. N. Kurtz and W. Streifer, IEEE Trans. MTT-17, 360 (1969) and references therein.
  7. P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129 (1965).
    [Crossref]
  8. S. Kawakami and J. Nishizawa, J. Appl. Phys. 38, 4807 (1967).
    [Crossref]
  9. S. E. Miller, Bell System Tech. J. 44, 2017 (1965).
    [Crossref]
  10. J. P. Gordon, Bell System Tech. J. 45, 321 (1966).
    [Crossref]
  11. E. A. J. Marcatili, Bell System Tech. J. 46, 149 (1967).
    [Crossref]
  12. H. Kita, I. Kitano, T. Uchida, and M. Furukawa, paper presented at the 72nd Annual Meeting of the Am. Ceram. Soc., May 1970, Philadelphia, Pa.
  13. A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 76 (1969).
    [Crossref]
  14. E. T. Kornhauser and G. S. Heller, in Proceedings of the Symposium on Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, New York, 1963), p. 891.

1970 (1)

1969 (2)

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 76 (1969).
[Crossref]

C. N. Kurtz and W. Streifer, IEEE Trans. MTT-17, 360 (1969) and references therein.

1968 (1)

S. Kawakami and J. Nizhizawa, IEEE Trans. MTT-16, 814 (1968).

1967 (3)

S. Kawakami and J. Nishizawa, J. Appl. Phys. 38, 4807 (1967).
[Crossref]

E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229 (1967).

E. A. J. Marcatili, Bell System Tech. J. 46, 149 (1967).
[Crossref]

1966 (1)

J. P. Gordon, Bell System Tech. J. 45, 321 (1966).
[Crossref]

1965 (2)

S. E. Miller, Bell System Tech. J. 44, 2017 (1965).
[Crossref]

P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[Crossref]

1954 (1)

A. Fletcher, T. Murphy, and A. Young, Proc. Roy. Soc. (London) 223, 216 (1954).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Fletcher, A.

A. Fletcher, T. Murphy, and A. Young, Proc. Roy. Soc. (London) 223, 216 (1954).
[Crossref]

French, W. G.

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 76 (1969).
[Crossref]

Furukawa, M.

H. Kita, I. Kitano, T. Uchida, and M. Furukawa, paper presented at the 72nd Annual Meeting of the Am. Ceram. Soc., May 1970, Philadelphia, Pa.

Gordon, J. P.

J. P. Gordon, Bell System Tech. J. 45, 321 (1966).
[Crossref]

P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[Crossref]

Heller, G. S.

E. T. Kornhauser and G. S. Heller, in Proceedings of the Symposium on Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, New York, 1963), p. 891.

Herriott, D. R.

Kawakami, S.

S. Kawakami and J. Nizhizawa, IEEE Trans. MTT-16, 814 (1968).

S. Kawakami and J. Nishizawa, J. Appl. Phys. 38, 4807 (1967).
[Crossref]

Kita, H.

H. Kita, I. Kitano, T. Uchida, and M. Furukawa, paper presented at the 72nd Annual Meeting of the Am. Ceram. Soc., May 1970, Philadelphia, Pa.

Kitano, I.

H. Kita, I. Kitano, T. Uchida, and M. Furukawa, paper presented at the 72nd Annual Meeting of the Am. Ceram. Soc., May 1970, Philadelphia, Pa.

Kornhauser, E. T.

E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229 (1967).

E. T. Kornhauser and G. S. Heller, in Proceedings of the Symposium on Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, New York, 1963), p. 891.

Kurtz, C. N.

C. N. Kurtz and W. Streifer, IEEE Trans. MTT-17, 360 (1969) and references therein.

Marcatili, E. A. J.

E. A. J. Marcatili, Bell System Tech. J. 46, 149 (1967).
[Crossref]

McKenna, J.

Miller, S. E.

S. E. Miller, Bell System Tech. J. 44, 2017 (1965).
[Crossref]

Murphy, T.

A. Fletcher, T. Murphy, and A. Young, Proc. Roy. Soc. (London) 223, 216 (1954).
[Crossref]

Nishizawa, J.

S. Kawakami and J. Nishizawa, J. Appl. Phys. 38, 4807 (1967).
[Crossref]

Nizhizawa, J.

S. Kawakami and J. Nizhizawa, IEEE Trans. MTT-16, 814 (1968).

Pearson, A. D.

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 76 (1969).
[Crossref]

Rawson, E. G.

E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9, 753 (1970).
[Crossref] [PubMed]

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 76 (1969).
[Crossref]

Streifer, W.

C. N. Kurtz and W. Streifer, IEEE Trans. MTT-17, 360 (1969) and references therein.

Tien, P. K.

P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[Crossref]

Uchida, T.

H. Kita, I. Kitano, T. Uchida, and M. Furukawa, paper presented at the 72nd Annual Meeting of the Am. Ceram. Soc., May 1970, Philadelphia, Pa.

Whinnery, J. R.

P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Yaghjian, A. D.

E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229 (1967).

Young, A.

A. Fletcher, T. Murphy, and A. Young, Proc. Roy. Soc. (London) 223, 216 (1954).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Letters (1)

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Letters 15, 76 (1969).
[Crossref]

Bell System Tech. J. (3)

S. E. Miller, Bell System Tech. J. 44, 2017 (1965).
[Crossref]

J. P. Gordon, Bell System Tech. J. 45, 321 (1966).
[Crossref]

E. A. J. Marcatili, Bell System Tech. J. 46, 149 (1967).
[Crossref]

IEEE Trans. (2)

S. Kawakami and J. Nizhizawa, IEEE Trans. MTT-16, 814 (1968).

C. N. Kurtz and W. Streifer, IEEE Trans. MTT-17, 360 (1969) and references therein.

J. Appl. Phys. (1)

S. Kawakami and J. Nishizawa, J. Appl. Phys. 38, 4807 (1967).
[Crossref]

Proc. IEEE (1)

P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[Crossref]

Proc. Roy. Soc. (London) (1)

A. Fletcher, T. Murphy, and A. Young, Proc. Roy. Soc. (London) 223, 216 (1954).
[Crossref]

Radio Sci. (1)

E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229 (1967).

Other (3)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

H. Kita, I. Kitano, T. Uchida, and M. Furukawa, paper presented at the 72nd Annual Meeting of the Am. Ceram. Soc., May 1970, Philadelphia, Pa.

E. T. Kornhauser and G. S. Heller, in Proceedings of the Symposium on Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, New York, 1963), p. 891.

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

The paraxial imaging of an object point by a quadratic-gradient cylindrical lens. The lens is extended by the dashed line to one full period, for which the corresponding ray lines are also indicated.

Fig. 2
Fig. 2

Normalized object distances Z1 = (n0/n1)(z1/L) vs normalized image distances Z2 = (n0/n2)(z2/L) for several fractional lens periods Δ = D/L.

Fig. 3
Fig. 3

Image positions for fixed object positions and varying lens lengths. Parameters used are n1 = 1 = n2, n0 = 1.5, L = 6/π, z2 = −2. The left and right axial solid dots indicate the first and second focal points, respectively; these are at infinity for Δ = 180° and 360°. The left and right axial crosses indicate the first and second principal points, respectively; an exception is Δ = 135° for which they are interchanged.

Tables (1)

Tables Icon

Table I Ray equations and slopes for the three regions of Fig. 1.

Equations (23)

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

n ( x ) = n 0 sech ( x / L ) ,
n ( x ) = n 0 ( 1 + x 2 / L 2 ) - 1 2 .
n ( x ) = n 0 ( 1 - x 2 / 2 L 2 ) ,             L x ,
[ 1 + ( d x d z ) 2 ] ( n x - d x d z n z ) = n ( x , z ) d 2 x d z 2 .
z 2 ( z 1 ) = n 2 z 1 - L ( n 1 / n 0 ) tan Δ n 1 + n 0 ( z 1 / L ) tan Δ , x 2 ( z 1 ) = x 1 cos Δ + ( n 0 / n 1 ) ( z 1 / L ) sin Δ . ( Δ = D L )
z 1 ( z 2 ) = n 1 z 2 + L ( n 2 / n 0 ) tan Δ n 2 - n 0 ( z 2 / L ) tan Δ , x 1 ( z 2 ) = x 2 cos Δ - ( n 0 / n 2 ) ( z 2 / L ) sin Δ .
M t = x 2 x 1 = cos Δ - n 0 n 2 z 2 L sin Δ = ( cos Δ + n 0 n 1 z 1 L sin Δ ) - 1 ,
M l = d z 2 / d z 1 = ( n 2 / n 1 ) M t 2 .
M a = ( m A - m B ) z > D ( m A - m B ) z < 0 = M t M l = n 1 n 2 1 M t .
z 1 ( z 2 = ) F 1 = - ( n 1 / n 0 ) L cot Δ .
z 2 ( z 1 = - ) F 2 = ( n 2 / n 0 ) L cot Δ ,
x 2 ( z 2 = F 2 ) = 0.
M t ( z 1 = P 1 ) = i.e. P 1 = - n 1 n 0 L cos Δ - 1 sin Δ ,
M t ( z 2 = P 2 ) = i.e. P 2 = n 2 n 0 L cos Δ - 1 sin Δ .
M a ( z 1 = N 1 ) = i.e. N 1 = - n 1 n 0 L cos Δ - n 2 / n 1 sin Δ ,
M a ( z 2 = N 2 ) = i.e. N 2 = n 2 n 0 L cos Δ - n 1 / n 2 sin Δ .
F 1 - P 1 f 1 = - ( n 1 / n 0 ) L csc Δ ,
F 2 - P 2 f 2 = ( n 2 / n 0 ) L csc Δ .
s 1 = z 1 - P 1 ,             s 2 = z 2 - P 2 ,
f 1 / s 1 + f 2 / s 2 = 1 ,             M t = s 2 / s 1 .
d 1 d 2 = f 1 f 2 ,             M t = f 1 / d 2 = d 1 / f 2 ,
d 1 = z 1 - F 1 ,             d 2 = z 2 - F 2 .
θ / φ = n 0 a / L = a / f 2 ,