Abstract

Any two-dimensional interpolation scheme which has continuous derivatives may be used to represent an optical surface for ray-tracing purposes. We present bicubic splines in their application to the design of asymmetric surfaces. An as example of a problem requiring an asymmetric system, we analyze the design problems of a color TV lighthouse lens.

© 1971 Optical Society of America

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References

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  1. J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic Press, New York, 1967).
  2. C. de Boor, J. Math. Phys. 41, 212 (1962).
  3. D. S. Grey, J. Opt. Soc. Am. 53, 672, 677 (1963).
    [CrossRef]
  4. R. J. Pegis et al., in Recent Advances in Optimization Techniques, A. Lavi, T. P. Vogl, Eds. (Wiley, New York, 1966), pp. 47–60.
  5. R. Bloomsburgh, R. Jones, J. King, J. Pietrolewicz, IEEE Trans. Broadcasting Television Receivers BTR-11, No. 2, pp. 50–61 (1965).
    [CrossRef]
  6. S. H. Kaplan, “Method of Exposing a Multi-Color Target Structure,” U.S. Pat.3,211,067, 12Oct1965.
  7. D. W. Epstein et al., “Manufacture of Color Kinescopes,” U.S. Pat.2,817,276, 24Dec1957.
  8. D. Thornton, “Light Sources for Shadow Mask Tubes,” U.S. Pat.3,380,354, 30Apr1968.
  9. E. G. Ramberg et al., “Optical System for Use in Making Color Phosphor Mosaic Screens,” U.S. Pat.3,385,184.
  10. J. W. Schwartz, “Lens System for Correcting Effects of Earth’s Magnetic Field in Color Tubes,” U.S. Pat.3,386,354.
  11. G. R. Kautz et al., “Exposure Device,” U.S. Pat.3,395,628, 6Aug1968.
  12. D. W. Epstein et al., “Color Kinescopes,” U.S. Pat.2,885,93512May1969.
  13. F. Hertzfeld et al., “Cathode Ray Tube and Method of Manufacture,” U. S. Pat.3,476,025, 4Nov1969.
  14. J. W. Schwartz, “Method and Means for Producing Color Television Picture Tubes,” U.S. Pat.3,494,267, 10Feb1970.
  15. P. G. J. Barten, “Optical System for Use in Making Color Phosphor Mosaic Screens,” U.S. Pat.3,504,599, 7Apr1970.
  16. G. A. Burdick, “Cathode Ray Tube Exposure Optics,” U.S. Pat.3,509,802, 5May1970.
  17. A. K. Rigler, T. P. Vogl, Appl. Opt. 10, 1648 (1971).
    [CrossRef] [PubMed]

1971

1965

R. Bloomsburgh, R. Jones, J. King, J. Pietrolewicz, IEEE Trans. Broadcasting Television Receivers BTR-11, No. 2, pp. 50–61 (1965).
[CrossRef]

1963

D. S. Grey, J. Opt. Soc. Am. 53, 672, 677 (1963).
[CrossRef]

1962

C. de Boor, J. Math. Phys. 41, 212 (1962).

Ahlberg, J. H.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic Press, New York, 1967).

Barten, P. G. J.

P. G. J. Barten, “Optical System for Use in Making Color Phosphor Mosaic Screens,” U.S. Pat.3,504,599, 7Apr1970.

Bloomsburgh, R.

R. Bloomsburgh, R. Jones, J. King, J. Pietrolewicz, IEEE Trans. Broadcasting Television Receivers BTR-11, No. 2, pp. 50–61 (1965).
[CrossRef]

Burdick, G. A.

G. A. Burdick, “Cathode Ray Tube Exposure Optics,” U.S. Pat.3,509,802, 5May1970.

de Boor, C.

C. de Boor, J. Math. Phys. 41, 212 (1962).

Epstein, D. W.

D. W. Epstein et al., “Manufacture of Color Kinescopes,” U.S. Pat.2,817,276, 24Dec1957.

D. W. Epstein et al., “Color Kinescopes,” U.S. Pat.2,885,93512May1969.

Grey, D. S.

D. S. Grey, J. Opt. Soc. Am. 53, 672, 677 (1963).
[CrossRef]

Hertzfeld, F.

F. Hertzfeld et al., “Cathode Ray Tube and Method of Manufacture,” U. S. Pat.3,476,025, 4Nov1969.

Jones, R.

R. Bloomsburgh, R. Jones, J. King, J. Pietrolewicz, IEEE Trans. Broadcasting Television Receivers BTR-11, No. 2, pp. 50–61 (1965).
[CrossRef]

Kaplan, S. H.

S. H. Kaplan, “Method of Exposing a Multi-Color Target Structure,” U.S. Pat.3,211,067, 12Oct1965.

Kautz, G. R.

G. R. Kautz et al., “Exposure Device,” U.S. Pat.3,395,628, 6Aug1968.

King, J.

R. Bloomsburgh, R. Jones, J. King, J. Pietrolewicz, IEEE Trans. Broadcasting Television Receivers BTR-11, No. 2, pp. 50–61 (1965).
[CrossRef]

Nilson, E. N.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic Press, New York, 1967).

Pegis, R. J.

R. J. Pegis et al., in Recent Advances in Optimization Techniques, A. Lavi, T. P. Vogl, Eds. (Wiley, New York, 1966), pp. 47–60.

Pietrolewicz, J.

R. Bloomsburgh, R. Jones, J. King, J. Pietrolewicz, IEEE Trans. Broadcasting Television Receivers BTR-11, No. 2, pp. 50–61 (1965).
[CrossRef]

Ramberg, E. G.

E. G. Ramberg et al., “Optical System for Use in Making Color Phosphor Mosaic Screens,” U.S. Pat.3,385,184.

Rigler, A. K.

Schwartz, J. W.

J. W. Schwartz, “Lens System for Correcting Effects of Earth’s Magnetic Field in Color Tubes,” U.S. Pat.3,386,354.

J. W. Schwartz, “Method and Means for Producing Color Television Picture Tubes,” U.S. Pat.3,494,267, 10Feb1970.

Thornton, D.

D. Thornton, “Light Sources for Shadow Mask Tubes,” U.S. Pat.3,380,354, 30Apr1968.

Vogl, T. P.

Walsh, J. L.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic Press, New York, 1967).

Appl. Opt.

IEEE Trans. Broadcasting Television Receivers

R. Bloomsburgh, R. Jones, J. King, J. Pietrolewicz, IEEE Trans. Broadcasting Television Receivers BTR-11, No. 2, pp. 50–61 (1965).
[CrossRef]

J. Math. Phys.

C. de Boor, J. Math. Phys. 41, 212 (1962).

J. Opt. Soc. Am.

D. S. Grey, J. Opt. Soc. Am. 53, 672, 677 (1963).
[CrossRef]

Other

R. J. Pegis et al., in Recent Advances in Optimization Techniques, A. Lavi, T. P. Vogl, Eds. (Wiley, New York, 1966), pp. 47–60.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic Press, New York, 1967).

S. H. Kaplan, “Method of Exposing a Multi-Color Target Structure,” U.S. Pat.3,211,067, 12Oct1965.

D. W. Epstein et al., “Manufacture of Color Kinescopes,” U.S. Pat.2,817,276, 24Dec1957.

D. Thornton, “Light Sources for Shadow Mask Tubes,” U.S. Pat.3,380,354, 30Apr1968.

E. G. Ramberg et al., “Optical System for Use in Making Color Phosphor Mosaic Screens,” U.S. Pat.3,385,184.

J. W. Schwartz, “Lens System for Correcting Effects of Earth’s Magnetic Field in Color Tubes,” U.S. Pat.3,386,354.

G. R. Kautz et al., “Exposure Device,” U.S. Pat.3,395,628, 6Aug1968.

D. W. Epstein et al., “Color Kinescopes,” U.S. Pat.2,885,93512May1969.

F. Hertzfeld et al., “Cathode Ray Tube and Method of Manufacture,” U. S. Pat.3,476,025, 4Nov1969.

J. W. Schwartz, “Method and Means for Producing Color Television Picture Tubes,” U.S. Pat.3,494,267, 10Feb1970.

P. G. J. Barten, “Optical System for Use in Making Color Phosphor Mosaic Screens,” U.S. Pat.3,504,599, 7Apr1970.

G. A. Burdick, “Cathode Ray Tube Exposure Optics,” U.S. Pat.3,509,802, 5May1970.

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Figures (5)

Fig. 1
Fig. 1

Ideal arrangement of phosphor dots; B is blue, R is red, and G is green.

Fig. 2
Fig. 2

Registration of electron beam (small circles) and phosphor dots (large circles);(a) correct, (b) incorrect.

Fig. 3
Fig. 3

Phosphor dot tangential correction movement, all values in millimeters.

Fig. 4
Fig. 4

Sketch of lighthouse optics.

Fig. 5
Fig. 5

Isometric view of asymetric correcting surface on lens.

Equations (9)

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c ( x ) = c ( a ) + c ( a ) ( x - a ) + [ 3 c ( b ) - c ( a ) ( b - a ) 2 - c ( b ) + 2 c ( a ) ( b - a ) ] × ( x - a ) 2 + [ - 2 c ( b ) - c ( a ) ( b - a ) 3 + c ( b ) + c ( a ) ( b - a ) 2 ] ( x - a ) 3
h i p i - 1 + 2 ( h i + h i - 1 ) p i + h i - 1 p i + 1 = 3 [ h i - 1 ( z i + 1 - z i h i ) + h i ( z i - z i - 1 h i - 1 ) ] ,             i = 1 , 2 , , N - 1
Z ( x , y ) = m = 0 3 n = 0 3 γ m n ( x - x i - 1 ) m ( y - y j - 1 ) n ,
z i j = z ( x i , y j ) , i = 0 , 1 , , M ; j = 0 , 1 , , N , p i j = z ( x i , y j ) / x , i = 0 , M ; j = 0 , 1 , , N , q i j = z ( x i , y j ) / y , i = 0 , 1 , , M ; j = 0 , N , s i j = 2 z ( x i , y j ) / x y , i = 0 , M ; j = 0 , N .
A ( h ) = [ 1 0 0 0 0 1 0 0 - 3 / h 2 - 2 / h 3 / h 2 - 1 / h 2 / h 3 1 / h 2 - 2 / h 3 1 / h 4 ] D = [ z i - 1 , j - 1 q i - 1 , j - 1 z i - 1 , j q i - 1 , j p i - 1 , j - 1 s i - 1 , j - 1 p i - 1 , j s i - 1 , j z i , j - 1 q i , j - 1 z i j q i j p i , j - 1 s i , j - 1 p i j s i j ] .
h i - 1 p i + 1 , j + 2 ( h i - 1 + h i ) p i j + h i p i - 1 , j = 3 [ h i - 1 h i ( z i + 1 , j - z i j ) + h i h i - 1 ( z i j - z i - 1 , j ) ] ;
h i - 1 s i + 1 , j + 2 ( h i - 1 + h i ) s i j + h i s i - 1 , j = 3 [ h i - 1 h i ( q i + 1 , j - q i j ) + h i h i - 1 ( q i j - q i - 1 , j ) ] ;
k j - 1 q i , j + 1 + 2 ( k j - 1 + k j ) q i j + k j q i , j - 1 = 3 [ k j - 1 k j ( z i , j + 1 - z i j ) + k i k j - 1 ( z i j - z i , j - 1 ) ] ;
k j - 1 s i , j + 1 + 2 ( k j - 1 + k j ) s i j + k j s i , j - 1 = 3 [ k j - 1 k j ( p i , j + 1 - p i j ) + k j k j - 1 ( p i j - p i , j - 1 ) ] .

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