Abstract

Results of wave optics and ray optics are summarized to make a convenient and self-consistent expression for gradient-index imaging. By applying the theory, a limit of the transmitted spatial frequency and focused spot is obtained. An optimum design of a light focuser for a video disk is demonstrated.

© 1980 Optical Society of America

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References

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  1. M. Kawazu, Y. Ogura, in Digest of Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WC1.
  2. M. Mino, in Ref. 1, paper WC3.
  3. T. Miyazawa, K. Okada, T. Kubo, K. Nishizawa, I. Kitano, K. Iga, in Ref. 1, paper WA3.
  4. K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), p. 367.
  5. K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber Integrated Opt. 2, 1 (1979).
    [CrossRef]
  6. W. J. Tomlinson, in Ref. 1, paper WC6.
  7. R. W. Wood, Physical Optics (Macmillan, New York), (1905).
  8. A. L. Mikaeliyan, Dokl.Akad. Nauk SSSR 81, 569 (1951).
  9. D. W. Berreman, Bell Syst. Tech. J. 43, 1469 (1964).
  10. Y. Suematsu, K. Iga, S. Ito, IEEE Trans. Microwave Theory Tech. MTT-14, 657 (1966).
    [CrossRef]
  11. P. Kaiser, Bell Syst. Tech. J. 49, 137 (1970).
  12. E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
  13. Y. Suematsu, H. Fukinuki, J. IECE Jpn. 48, 1684 (1965).
  14. Y. Suematsu, K. Iga, J. IECE Jpn. 49, 1645 (1966).
  15. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
    [CrossRef]
  16. A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
    [CrossRef]
  17. S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
    [CrossRef]
  18. D. Gloge, Appl.Opt. 10, 2442 (1971).
    [CrossRef] [PubMed]
  19. R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
    [CrossRef] [PubMed]
  20. K. Furuya, Y. Suematsu, J. Nayyer, S. Ishikawa, Appl. Phys. Lett. 27, 456 (1975).
    [CrossRef]
  21. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).
  22. A. K. Gathak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
    [CrossRef]
  23. K. Iga, N. Yamamoto, Appl. Opt. 16, 1305 (1977).
    [CrossRef] [PubMed]
  24. Y. Yamamoto, K. Iga, in Ref. 1, paper WB3.
  25. K. Iga, Y. Yokubun, Appl. Opt. 17, 1972 (1978);paper presented at 1979 CLEA 13.7.
    [CrossRef] [PubMed]
  26. K. Iga, in Ref. 1, paper TuA2.
  27. Y. Suematsu, K. Furuya, Trans. IECE Jpn. 54-B, 325 (1971).
  28. K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
    [CrossRef]
  29. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  30. A. Yariv, J. Opt. Soc. Am. 66, 301 (1976).
    [CrossRef]
  31. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).
  32. W. Streifer, K. B. Paxton, Appl. Opt. 10, 769 (1971).
    [CrossRef] [PubMed]
  33. K. Iga, S. Ohshima, unpublished work;S. Ohshima, M.A. Thesis, Tokyo Institute of Technology (1978).

1979

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

1978

1977

1976

1975

K. Furuya, Y. Suematsu, J. Nayyer, S. Ishikawa, Appl. Phys. Lett. 27, 456 (1975).
[CrossRef]

1974

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

1971

W. Streifer, K. B. Paxton, Appl. Opt. 10, 769 (1971).
[CrossRef] [PubMed]

D. Gloge, Appl.Opt. 10, 2442 (1971).
[CrossRef] [PubMed]

Y. Suematsu, K. Furuya, Trans. IECE Jpn. 54-B, 325 (1971).

1970

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

P. Kaiser, Bell Syst. Tech. J. 49, 137 (1970).

1969

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

1968

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

1966

Y. Suematsu, K. Iga, J. IECE Jpn. 49, 1645 (1966).

Y. Suematsu, K. Iga, S. Ito, IEEE Trans. Microwave Theory Tech. MTT-14, 657 (1966).
[CrossRef]

1965

Y. Suematsu, H. Fukinuki, J. IECE Jpn. 48, 1684 (1965).

1964

E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).

D. W. Berreman, Bell Syst. Tech. J. 43, 1469 (1964).

1951

A. L. Mikaeliyan, Dokl.Akad. Nauk SSSR 81, 569 (1951).

Berreman, D. W.

D. W. Berreman, Bell Syst. Tech. J. 43, 1469 (1964).

French, W. G.

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Fukinuki, H.

Y. Suematsu, H. Fukinuki, J. IECE Jpn. 48, 1684 (1965).

Fukuyo, H.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Furuya, K.

K. Furuya, Y. Suematsu, J. Nayyer, S. Ishikawa, Appl. Phys. Lett. 27, 456 (1975).
[CrossRef]

Y. Suematsu, K. Furuya, Trans. IECE Jpn. 54-B, 325 (1971).

Gathak, A. K.

A. K. Gathak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Gloge, D.

D. Gloge, Appl.Opt. 10, 2442 (1971).
[CrossRef] [PubMed]

Hata, S.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Iga, K.

K. Iga, Y. Yokubun, Appl. Opt. 17, 1972 (1978);paper presented at 1979 CLEA 13.7.
[CrossRef] [PubMed]

K. Iga, N. Yamamoto, Appl. Opt. 16, 1305 (1977).
[CrossRef] [PubMed]

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Y. Suematsu, K. Iga, J. IECE Jpn. 49, 1645 (1966).

Y. Suematsu, K. Iga, S. Ito, IEEE Trans. Microwave Theory Tech. MTT-14, 657 (1966).
[CrossRef]

T. Miyazawa, K. Okada, T. Kubo, K. Nishizawa, I. Kitano, K. Iga, in Ref. 1, paper WA3.

K. Iga, in Ref. 1, paper TuA2.

K. Iga, S. Ohshima, unpublished work;S. Ohshima, M.A. Thesis, Tokyo Institute of Technology (1978).

Y. Yamamoto, K. Iga, in Ref. 1, paper WB3.

Ishikawa, R.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), p. 367.

Ishikawa, S.

K. Furuya, Y. Suematsu, J. Nayyer, S. Ishikawa, Appl. Phys. Lett. 27, 456 (1975).
[CrossRef]

Ito, S.

Y. Suematsu, K. Iga, S. Ito, IEEE Trans. Microwave Theory Tech. MTT-14, 657 (1966).
[CrossRef]

Kaiser, P.

P. Kaiser, Bell Syst. Tech. J. 49, 137 (1970).

Kato, Y.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Kawakami, S.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Kawazu, M.

M. Kawazu, Y. Ogura, in Digest of Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WC1.

Keck, D. B.

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

T. Miyazawa, K. Okada, T. Kubo, K. Nishizawa, I. Kitano, K. Iga, in Ref. 1, paper WA3.

Kobayashi, K.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), p. 367.

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Kubo, T.

T. Miyazawa, K. Okada, T. Kubo, K. Nishizawa, I. Kitano, K. Iga, in Ref. 1, paper WA3.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Mikaeliyan, A. L.

A. L. Mikaeliyan, Dokl.Akad. Nauk SSSR 81, 569 (1951).

Minemura, K.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), p. 367.

Mino, M.

M. Mino, in Ref. 1, paper WC3.

Miyazawa, T.

T. Miyazawa, K. Okada, T. Kubo, K. Nishizawa, I. Kitano, K. Iga, in Ref. 1, paper WA3.

Nayyer, J.

K. Furuya, Y. Suematsu, J. Nayyer, S. Ishikawa, Appl. Phys. Lett. 27, 456 (1975).
[CrossRef]

Nishizawa, J.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Nishizawa, K.

T. Miyazawa, K. Okada, T. Kubo, K. Nishizawa, I. Kitano, K. Iga, in Ref. 1, paper WA3.

Ogura, Y.

M. Kawazu, Y. Ogura, in Digest of Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WC1.

Ohshima, S.

K. Iga, S. Ohshima, unpublished work;S. Ohshima, M.A. Thesis, Tokyo Institute of Technology (1978).

Okada, K.

T. Miyazawa, K. Okada, T. Kubo, K. Nishizawa, I. Kitano, K. Iga, in Ref. 1, paper WA3.

Olshansky, R.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Paxton, K. B.

Pearson, A. D.

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Rawson, E. G.

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Streifer, W.

Suematsu, Y.

K. Furuya, Y. Suematsu, J. Nayyer, S. Ishikawa, Appl. Phys. Lett. 27, 456 (1975).
[CrossRef]

Y. Suematsu, K. Furuya, Trans. IECE Jpn. 54-B, 325 (1971).

Y. Suematsu, K. Iga, J. IECE Jpn. 49, 1645 (1966).

Y. Suematsu, K. Iga, S. Ito, IEEE Trans. Microwave Theory Tech. MTT-14, 657 (1966).
[CrossRef]

Y. Suematsu, H. Fukinuki, J. IECE Jpn. 48, 1684 (1965).

Sugimoto, S.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), p. 367.

Thyagarajan, K.

A. K. Gathak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Tomlinson, W. J.

W. J. Tomlinson, in Ref. 1, paper WC6.

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, New York), (1905).

Yamamoto, N.

Yamamoto, Y.

Y. Yamamoto, K. Iga, in Ref. 1, paper WB3.

Yariv, A.

Yokubun, Y.

Appl. Opt.

Appl. Phys. Lett.

K. Furuya, Y. Suematsu, J. Nayyer, S. Ishikawa, Appl. Phys. Lett. 27, 456 (1975).
[CrossRef]

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Appl.Opt.

D. Gloge, Appl.Opt. 10, 2442 (1971).
[CrossRef] [PubMed]

Bell Syst. Tech. J.

P. Kaiser, Bell Syst. Tech. J. 49, 137 (1970).

E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).

D. W. Berreman, Bell Syst. Tech. J. 43, 1469 (1964).

Dokl.Akad. Nauk SSSR

A. L. Mikaeliyan, Dokl.Akad. Nauk SSSR 81, 569 (1951).

Fiber Integrated Opt.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

IEEE J. Quantum Electron.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Y. Suematsu, K. Iga, S. Ito, IEEE Trans. Microwave Theory Tech. MTT-14, 657 (1966).
[CrossRef]

J. IECE Jpn.

Y. Suematsu, H. Fukinuki, J. IECE Jpn. 48, 1684 (1965).

Y. Suematsu, K. Iga, J. IECE Jpn. 49, 1645 (1966).

J. Opt. Soc. Am.

Jpn. J. Appl. Phys.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Trans. IECE Jpn.

Y. Suematsu, K. Furuya, Trans. IECE Jpn. 54-B, 325 (1971).

Other

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

K. Iga, S. Ohshima, unpublished work;S. Ohshima, M.A. Thesis, Tokyo Institute of Technology (1978).

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).

A. K. Gathak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

K. Iga, in Ref. 1, paper TuA2.

Y. Yamamoto, K. Iga, in Ref. 1, paper WB3.

W. J. Tomlinson, in Ref. 1, paper WC6.

R. W. Wood, Physical Optics (Macmillan, New York), (1905).

M. Kawazu, Y. Ogura, in Digest of Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper WC1.

M. Mino, in Ref. 1, paper WC3.

T. Miyazawa, K. Okada, T. Kubo, K. Nishizawa, I. Kitano, K. Iga, in Ref. 1, paper WA3.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimoto, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo (1977), p. 367.

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Figures (9)

Fig. 1
Fig. 1

Index profile of a GRIN medium. The broken line is the extended parabolic.

Fig. 2
Fig. 2

Optical imaging system with a GRIN medium and free spaces.

Fig. 3
Fig. 3

Configuration for making a spot diagram.

Fig. 4
Fig. 4

One-dimensional spot diagram. The horizontal axis denotes the relative ray position given by gn3(0) (xx0)/[−3/4(1 + gb) (h4 − ⅔)].

Fig. 5
Fig. 5

Amplitude of OTF.

Fig. 6
Fig. 6

Intensity distribution of a focused spot (ray optics).

Fig. 7
Fig. 7

A limit of the focused spot by a one-quarter-pitch lens.

Fig. 8
Fig. 8

Rays focused by a lens shorter than one-quarter pitch. (a) h4 − ⅔ = 10; (b) h4 − ⅔ = 2. The horizontal axis denotes z axis normalized by g. The planes CG and Cs mean the Gaussian image plane and the circle of least confusion, respectively.

Fig. 9
Fig. 9

Optimum h4 for focusing.

Equations (19)

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

n 2 ( r ) = n 2 ( 0 ) [ 1 ( gr ) 2 + h 4 ( gr ) 4 + h 6 ( gr ) 6 + . . . ] , = n 2 2 , ( r < a ) ( r > a )
1 r d dr ( r d ψ dr ) + d 2 ψ d θ 2 + 0 2 k n 2 ( r ) ψ = β 2 ψ
ω 0 = a / ( V ) 1 / 2 ,
V = k 0 n ( 0 ) a ( 2 Δ ) 1 / 2 ,
β nm k ( 0 ) = 1 ( 2 n + m + 1 ) [ g k ( 0 ) ] + 1 2 { h 4 [ 3 2 ( 2 n + m + 1 ) 2 + 1 2 ( 1 m 2 ) ] ( 2 n + m + 1 ) 2 } [ g k ( 0 ) ] 2 + O { [ g k ( 0 ) ] 3 } .
ϕ 2 ( r 2 , θ 2 ) = j λ b exp [ jk ( 0 ) b ] × gb sin gb × 0 0 2 π r 1 dr 1 d θ 1 ϕ 1 ( r 1 , θ 1 ) K ( r 1 , θ 1 ; r 2 , θ 2 ) ,
K ( r 1 , θ 1 ; r 2 , θ 2 ) = exp { j k ( 0 ) g tan gb [ r 1 2 2 r 1 r 2 cos ( θ 1 + θ 2 ) cos gb + r 2 2 ] } .
R 3 4 [ g k ( 0 ) ] ( h 4 2 3 ) ( N + 1 ) .
N max + 1 = Δ [ g k ( 0 ) ] 1 .
d 2 x dz 2 = 1 2 n 2 ( x i , y i ) cos 2 γ i x n 2 ( x , y ) ,
d 2 y dz 2 = 1 2 n 2 ( x i , y i ) cos 2 γ i y n 2 ( x , y ) ,
Ω / g = 1 3 4 ( h 4 2 3 ) [ ( g x i ) 2 + x ̇ i 2 ] + ( h 4 2 3 ) O ( x i 4 ) + ( h 6 + 17 45 ) O ( x i 4 ) ,
x = x i cos Ω z + x ̇ i g sin Ω z ,
u = u gn ( 0 ) [ 3 4 ( 1 + gb ) | h 4 2 3 | n 2 ( 0 ) ] .
B = 460 gn 3 ( 0 ) / [ 3 4 ( 1 + gb ) | h 4 2 3 | ] .
ϕ 1 ( r 1 ) = { 1 0 < r 1 < A 0 r 1 > A
ϕ 2 ( r 2 ) = C A 2 2 2 J 1 [ k ( 0 ) gr 2 A ] k ( 0 ) gr 2 A .
Δ D d 3 k ( 0 ) gA = 1 2 λ NA * .
Δ D a = 0.084 1 g | h 4 2 3 | ( gb ) ( gA ) 3 .

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