Abstract

The Buchdahl dispersion equation is applied to a system of airspaced thin lenses. The powers, airspaces, and glass dispersion characteristics of a lens system are represented as a series of vectors. The sum of these vectors represents the state of paraxial color correction of the lens system. The vector representation is used as a guide in the selection of glasses for airspaced doublets (dialytes), triplets, and more complex lens systems with apochromatic color correction. Prescriptions and performance data are given.

© 1986 Optical Society of America

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References

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  1. N. v. d. W. Lessing, “Selection of Optical Glasses in Apochromats,” J. Opt. Soc. Am. 47, 955 (1957).
    [CrossRef]
  2. N. v. d. W. Lessing, “Further Considerations on the Selection of Optical Glasses in Aprochromats,” J. Opt. Soc. Am. 48, 269 (1958).
    [CrossRef]
  3. N. v. d. W. Lessing, “Selection of Optical Glasses in Taylor Triplets (Special Method),” J. Opt. Soc. Am. 48, 558 (1958).
    [CrossRef]
  4. N. v. d. W. Lessing, “Selection of Optical Glasses in Taylor Triplets (General Method),” J. Opt. Soc. Am. 49, 31 (1959).
    [CrossRef]
  5. N. v. d. W. Lessing, “Selection of Optical Glasses in Superachromats,” Appl. Opt. 9, 1655 (1970).
  6. R. E. Stephens, “Secondary Chromatic Aberration,” J. Opt. Soc. Am. 47, 1135 (1957).
    [CrossRef]
  7. R. E. Stephens, “Selection of Glasses for Three-Color Achromats,” J. Opt. Soc. Am. 49, 398 (1959).
    [CrossRef]
  8. R. E. Stephens, “Four-Color Achromats and Superachromats,” J. Opt. Soc. Am. 50, 1016 (1960).
    [CrossRef]
  9. R. E. Stephens, “Experimental Verification of Superachromatism,” J. Opt. Soc. Am. 56, 213 (1966).
    [CrossRef]
  10. M. Herzberger, H. Jenkins, “Color Correction in Optical Systems and Types of Glass,” J. Opt. Soc. Am. 39, 984 (1949).
    [CrossRef]
  11. M. Herzberger, “Color Correction in Optical Systems and a New Dispersion Formula,” Opt. Acta 6, 197 (1959).
    [CrossRef]
  12. M. Hetzberger, N. R. McClure, “The Design of Superachromatic Lenses,” Appl. Opt. 2, 553 (1963).
    [CrossRef]
  13. M. Herzberger, H. Pulvermacher, “Die Farbfehlerkorrektion von Multipletts,” Opt. Acta 17, 349 (1970).
    [CrossRef]
  14. M. Herzberger, “Tables of Superachromatic Glasstriples and their Color Correction,” Optik 35, 1 (1972).
  15. H. Drucks, “Bemerkung zur Theorie der Superachromaten,” Optik 23, 523 (1966).
  16. H. Schultz, “Zum Problem der Superachromate,” Optik 25, 203 (1967).
  17. H. Schultz, “Superachromate,” Optik 25, 208 (1967).
  18. H. Pulvermacher, “Theorie der Restspektren von Simpletts,” Optik 30, 297 (1969).
  19. H. Pulvermacher, “New Proofs for the Fundamental Rules for the Correction of Chromatic Aberrations,” Optik 30, 395 (1970).
  20. B. L. Nefedov, “The Design of Apochromats Made from Two and Three Different Glasses,” Sov. J. Opt. Technol. 40, 46 (1973).
  21. A. B. Agurok, “Some Superachromatization Problems,” Sov. J. Opt. Technol. 44, 114 (1977).
  22. M. G. Shpyakin, “Design of Four-Color Thin Apochromats,” Sov. J. Opt. Technol. 45, 81 (1978).
  23. M. G. Shpyakin, “Calculation of the Components of Apochromats Made From Four Types of Glass for a Broad Spectral Region,” Sov. J. Opt. Technol. 45, 219 (1978).
  24. G. A. Mozharov, “Graphical Analysis Method of Choosing the Glasses for the Design of a Four-Lens Three-Color Apochromat,” Sov. J. Opt. Technol. 44, 146 (1977).
  25. G. A. Mozharov, “Two-Component, Four-Color Apochromats,” Sov. J. Opt. Technol. 47, 398 (1980).
  26. T. R. Sloan, “Analysis and Correction of Secondary Color in Optical Systems,” Appl. Opt. 9, 853 (1970).
    [CrossRef] [PubMed]
  27. B. Tatian, “Glass Chart for Analyzing Secondary Color Correction,” Appl. Opt. 24, 544 (1985).
    [CrossRef] [PubMed]
  28. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  29. H. A. Buchdahl, “Many-Color Correction of Thin Doublets,” Appl. Opt. 24, 1878 (1985).
    [CrossRef] [PubMed]
  30. R. I. Mercado, P. N. Robb, “Design of Thick Doublets Corrected at Four and Five Wavelengths,” J. Opt. Soc. Am. 71, 1639 (1981).
  31. P. N. Robb, R. I. Mercado, “Glass Selection for Hyperachromatic Triplets,” J. Opt. Soc. Am. 73, 1882 (1983).
  32. R. I. Mercado, P. N. Robb, “Color-Corrected Optical Systems,” U.S. Patent Application (1982).
  33. P. N. Robb, “Selection of Optical Glasses: 1: Two Materials,” Appl. Opt. 24, 1864 (1985).
    [CrossRef] [PubMed]
  34. P.N. Robb, R. I. Mercado, “Calculation of Refractive Indices Using Buchdahl’s Chromatic Coordinate,” Appl. Opt. 22, 1198 (1983).
    [CrossRef] [PubMed]
  35. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 281.
  36. L. Schupmann, U.S. Patent620,978 (Mar.1899).
  37. accosv is a proprietary product of Scientific Calculations, Inc., Fisher, NY.
  38. R. W. Christen, “A New Approach to Color Correction,” Sky Telesc. 70, 375 (Oct.1985).
  39. R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), pp. 259–268.

1985 (4)

1983 (2)

P. N. Robb, R. I. Mercado, “Glass Selection for Hyperachromatic Triplets,” J. Opt. Soc. Am. 73, 1882 (1983).

P.N. Robb, R. I. Mercado, “Calculation of Refractive Indices Using Buchdahl’s Chromatic Coordinate,” Appl. Opt. 22, 1198 (1983).
[CrossRef] [PubMed]

1981 (1)

R. I. Mercado, P. N. Robb, “Design of Thick Doublets Corrected at Four and Five Wavelengths,” J. Opt. Soc. Am. 71, 1639 (1981).

1980 (1)

G. A. Mozharov, “Two-Component, Four-Color Apochromats,” Sov. J. Opt. Technol. 47, 398 (1980).

1978 (2)

M. G. Shpyakin, “Design of Four-Color Thin Apochromats,” Sov. J. Opt. Technol. 45, 81 (1978).

M. G. Shpyakin, “Calculation of the Components of Apochromats Made From Four Types of Glass for a Broad Spectral Region,” Sov. J. Opt. Technol. 45, 219 (1978).

1977 (2)

G. A. Mozharov, “Graphical Analysis Method of Choosing the Glasses for the Design of a Four-Lens Three-Color Apochromat,” Sov. J. Opt. Technol. 44, 146 (1977).

A. B. Agurok, “Some Superachromatization Problems,” Sov. J. Opt. Technol. 44, 114 (1977).

1973 (1)

B. L. Nefedov, “The Design of Apochromats Made from Two and Three Different Glasses,” Sov. J. Opt. Technol. 40, 46 (1973).

1972 (1)

M. Herzberger, “Tables of Superachromatic Glasstriples and their Color Correction,” Optik 35, 1 (1972).

1970 (4)

H. Pulvermacher, “New Proofs for the Fundamental Rules for the Correction of Chromatic Aberrations,” Optik 30, 395 (1970).

M. Herzberger, H. Pulvermacher, “Die Farbfehlerkorrektion von Multipletts,” Opt. Acta 17, 349 (1970).
[CrossRef]

N. v. d. W. Lessing, “Selection of Optical Glasses in Superachromats,” Appl. Opt. 9, 1655 (1970).

T. R. Sloan, “Analysis and Correction of Secondary Color in Optical Systems,” Appl. Opt. 9, 853 (1970).
[CrossRef] [PubMed]

1969 (1)

H. Pulvermacher, “Theorie der Restspektren von Simpletts,” Optik 30, 297 (1969).

1967 (2)

H. Schultz, “Zum Problem der Superachromate,” Optik 25, 203 (1967).

H. Schultz, “Superachromate,” Optik 25, 208 (1967).

1966 (2)

H. Drucks, “Bemerkung zur Theorie der Superachromaten,” Optik 23, 523 (1966).

R. E. Stephens, “Experimental Verification of Superachromatism,” J. Opt. Soc. Am. 56, 213 (1966).
[CrossRef]

1963 (1)

1960 (1)

1959 (3)

1958 (2)

1957 (2)

1949 (1)

Agurok, A. B.

A. B. Agurok, “Some Superachromatization Problems,” Sov. J. Opt. Technol. 44, 114 (1977).

Buchdahl, H. A.

H. A. Buchdahl, “Many-Color Correction of Thin Doublets,” Appl. Opt. 24, 1878 (1985).
[CrossRef] [PubMed]

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Christen, R. W.

R. W. Christen, “A New Approach to Color Correction,” Sky Telesc. 70, 375 (Oct.1985).

Drucks, H.

H. Drucks, “Bemerkung zur Theorie der Superachromaten,” Optik 23, 523 (1966).

Herzberger, M.

M. Herzberger, “Tables of Superachromatic Glasstriples and their Color Correction,” Optik 35, 1 (1972).

M. Herzberger, H. Pulvermacher, “Die Farbfehlerkorrektion von Multipletts,” Opt. Acta 17, 349 (1970).
[CrossRef]

M. Herzberger, “Color Correction in Optical Systems and a New Dispersion Formula,” Opt. Acta 6, 197 (1959).
[CrossRef]

M. Herzberger, H. Jenkins, “Color Correction in Optical Systems and Types of Glass,” J. Opt. Soc. Am. 39, 984 (1949).
[CrossRef]

Hetzberger, M.

Jenkins, H.

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), pp. 259–268.

Lessing, N. v. d. W.

McClure, N. R.

Mercado, R. I.

P.N. Robb, R. I. Mercado, “Calculation of Refractive Indices Using Buchdahl’s Chromatic Coordinate,” Appl. Opt. 22, 1198 (1983).
[CrossRef] [PubMed]

P. N. Robb, R. I. Mercado, “Glass Selection for Hyperachromatic Triplets,” J. Opt. Soc. Am. 73, 1882 (1983).

R. I. Mercado, P. N. Robb, “Design of Thick Doublets Corrected at Four and Five Wavelengths,” J. Opt. Soc. Am. 71, 1639 (1981).

R. I. Mercado, P. N. Robb, “Color-Corrected Optical Systems,” U.S. Patent Application (1982).

Mozharov, G. A.

G. A. Mozharov, “Two-Component, Four-Color Apochromats,” Sov. J. Opt. Technol. 47, 398 (1980).

G. A. Mozharov, “Graphical Analysis Method of Choosing the Glasses for the Design of a Four-Lens Three-Color Apochromat,” Sov. J. Opt. Technol. 44, 146 (1977).

Nefedov, B. L.

B. L. Nefedov, “The Design of Apochromats Made from Two and Three Different Glasses,” Sov. J. Opt. Technol. 40, 46 (1973).

Pulvermacher, H.

M. Herzberger, H. Pulvermacher, “Die Farbfehlerkorrektion von Multipletts,” Opt. Acta 17, 349 (1970).
[CrossRef]

H. Pulvermacher, “New Proofs for the Fundamental Rules for the Correction of Chromatic Aberrations,” Optik 30, 395 (1970).

H. Pulvermacher, “Theorie der Restspektren von Simpletts,” Optik 30, 297 (1969).

Robb, P. N.

P. N. Robb, “Selection of Optical Glasses: 1: Two Materials,” Appl. Opt. 24, 1864 (1985).
[CrossRef] [PubMed]

P. N. Robb, R. I. Mercado, “Glass Selection for Hyperachromatic Triplets,” J. Opt. Soc. Am. 73, 1882 (1983).

R. I. Mercado, P. N. Robb, “Design of Thick Doublets Corrected at Four and Five Wavelengths,” J. Opt. Soc. Am. 71, 1639 (1981).

R. I. Mercado, P. N. Robb, “Color-Corrected Optical Systems,” U.S. Patent Application (1982).

Robb, P.N.

Schultz, H.

H. Schultz, “Superachromate,” Optik 25, 208 (1967).

H. Schultz, “Zum Problem der Superachromate,” Optik 25, 203 (1967).

Schupmann, L.

L. Schupmann, U.S. Patent620,978 (Mar.1899).

Shpyakin, M. G.

M. G. Shpyakin, “Design of Four-Color Thin Apochromats,” Sov. J. Opt. Technol. 45, 81 (1978).

M. G. Shpyakin, “Calculation of the Components of Apochromats Made From Four Types of Glass for a Broad Spectral Region,” Sov. J. Opt. Technol. 45, 219 (1978).

Sloan, T. R.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 281.

Stephens, R. E.

Tatian, B.

Appl. Opt. (7)

J. Opt. Soc. Am. (11)

Opt. Acta (2)

M. Herzberger, “Color Correction in Optical Systems and a New Dispersion Formula,” Opt. Acta 6, 197 (1959).
[CrossRef]

M. Herzberger, H. Pulvermacher, “Die Farbfehlerkorrektion von Multipletts,” Opt. Acta 17, 349 (1970).
[CrossRef]

Optik (6)

M. Herzberger, “Tables of Superachromatic Glasstriples and their Color Correction,” Optik 35, 1 (1972).

H. Drucks, “Bemerkung zur Theorie der Superachromaten,” Optik 23, 523 (1966).

H. Schultz, “Zum Problem der Superachromate,” Optik 25, 203 (1967).

H. Schultz, “Superachromate,” Optik 25, 208 (1967).

H. Pulvermacher, “Theorie der Restspektren von Simpletts,” Optik 30, 297 (1969).

H. Pulvermacher, “New Proofs for the Fundamental Rules for the Correction of Chromatic Aberrations,” Optik 30, 395 (1970).

Sky Telesc. (1)

R. W. Christen, “A New Approach to Color Correction,” Sky Telesc. 70, 375 (Oct.1985).

Sov. J. Opt. Technol. (6)

B. L. Nefedov, “The Design of Apochromats Made from Two and Three Different Glasses,” Sov. J. Opt. Technol. 40, 46 (1973).

A. B. Agurok, “Some Superachromatization Problems,” Sov. J. Opt. Technol. 44, 114 (1977).

M. G. Shpyakin, “Design of Four-Color Thin Apochromats,” Sov. J. Opt. Technol. 45, 81 (1978).

M. G. Shpyakin, “Calculation of the Components of Apochromats Made From Four Types of Glass for a Broad Spectral Region,” Sov. J. Opt. Technol. 45, 219 (1978).

G. A. Mozharov, “Graphical Analysis Method of Choosing the Glasses for the Design of a Four-Lens Three-Color Apochromat,” Sov. J. Opt. Technol. 44, 146 (1977).

G. A. Mozharov, “Two-Component, Four-Color Apochromats,” Sov. J. Opt. Technol. 47, 398 (1980).

Other (6)

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), pp. 259–268.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

R. I. Mercado, P. N. Robb, “Color-Corrected Optical Systems,” U.S. Patent Application (1982).

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 281.

L. Schupmann, U.S. Patent620,978 (Mar.1899).

accosv is a proprietary product of Scientific Calculations, Inc., Fisher, NY.

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Figures (16)

Fig. 1
Fig. 1

Schott glasses modeled with a quadratic Buchdahl dispersion equation. In this plot, the dispersion coefficients (primary and secondary) of 238 Schott glasses are plotted, although only a selected number of them are identified. The base wavelength for fitting the quadratic dispersion equation was λ0 = 0.574 μm.

Fig. 2
Fig. 2

Scaled glass vectors for an airspaced thin lens achromatic doublet. The scaled glass vectors (shown as bold arrows) for an F/10 doublet composed of the glasses BK7 and F2 are shown. The individual glass vectors, which extend from the origin to the glass position on the plot, are obscured by their scaled counterparts. A copy of the scaled flint glass vector, shown in phantom, is translated for head-to-tail summation with the scaled crown glass vector. The result of this summation is the residual color error vector G0. Note that this combination of glasses, powers, and airspaces yields zero primary color error (i.e., no projection of the color error vector along the vertical axis) but does have considerable secondary color error (projection along the horizontal axis).

Fig. 3
Fig. 3

Color corrected one-glass dialyte objectives. Airspaced dialyte objectives with excellent color correction are possible using only one type of glass. However, a virtual object or image is required. The color error is larger if a more dispersive glass is used due to the chromatic spread of rays on the second lens.

Fig. 4
Fig. 4

Airspaced-induced chromatic aberration in a fixed crown dialyte. The performance of an achromatic doublet made from the common glasses BK7 and F2 is evaluated as a function of central airspace. In this example, only the power of the flint element is allowed to change (to maintain unit focal length) as the separation is varied. As a result of this restraint, considerable primary and secondary chromatic aberration is introduced. This example will be used to explore the requirements and performance of a fictitious flint glass which corrects these errors.

Fig. 5
Fig. 5

Color corrected fixed crown dialyte with Buchdahl modeled flint glass. The performance of dialytes with a crown element of BK7 (whose power is held constant) and a modeled flint glass are shown as a function of central airspace. The flint element is allowed to change in both power and dispersion so that unit focal length and zero color error (as predicted by a quadratic Buchdahl dispersion equation) are maintained. The color error residual, while not zero, is very small (note the scale change from that used in Fig. 4). The dispersion characteristics of this modeled flint glass as a function of airspace is plotted in Fig. 6.

Fig. 6
Fig. 6

Dispersion requirements for the flint element of a dialyte with a fixed BK7 crown element. The flint element glass vector G2 is plotted as a function of airspace. The flint glass was chosen to have a primary dispersion coefficient equivalent to that of F2 when the airspace is zero. If real glasses were available at the positions noted, their use would result in apochromatic correction.

Fig. 7
Fig. 7

Dispersion requirements for the flint element of a dialyte with a fixed LgSK2 crown element. The glass vectors for LgSK2 and a model flint are shown on this expanded region of the dispersion plot. The flint dispersion requirements (as a function of airspace) for apochromatic correction now pass near a number of real glasses. The performance of the four identified flint glasses in combination with the fixed power LgSK2 crown element are evaluated in Fig. 8.

Fig. 8
Fig. 8

Apochromatic dialytes using a fixed LgSK2 crown element. The glass combinations identified in Fig. 7 are evaluated for paraxial color correction. The airspace and power of the flint element were optimized to produce unit focal length and zero primary color error. Each of the resulting F/10 dialytes is an apochromat and has the airspaces indicated in Fig. 7.

Fig. 9
Fig. 9

Scaled glass vectors for an apochromtic airspaced triplet. For an apochromatic airspaced triplet, glasses are selected so that their scaled glass vectors sum to zero. If the sum of the absolute values of the scale factors relative to the net power of the lens system is minimized, the required powers of the lenses will be small. For the glass combination shown, the lenses have quite reasonable powers.

Fig. 10
Fig. 10

Unit focal length thin lens airspaced triplet apochromat. The paraxial color correction of a unit focal length triplet using the glasses identified in Fig. 9 is presented. The airspaces used in this evaluation (3.4% of the net focal length) are equivalent to that used in many Cooke triplets. The performance of this lens when thickened, optimized for axial correction, and scaled in focal length is shown in Fig. 11.

Fig. 11
Fig. 11

Optimized thick lens airspaced apochromatic triplet. The airspaced triplet of Fig. 10 has been thickened and optimized for maximum axial performance. No attempt was made to correct off-axis performance. Note that the scale factors have changed only slightly as a result of these operations. Even at a focal length of 1.5 m, the design is diffraction limited over the entire visible spectrum as shown by the plot of Strehl ratio vs wavelength.

Fig. 12
Fig. 12

Improving the color correction of an achromatic doublet with a zero-power color corrector. The color error Gd of a BK7/F2 achromatic close-spaced doublet can be eliminated if a color correcting doublet with an equal and opposite color error Gc is added. The head-to-tail vector summation is shown in phantom. If the color corrector is itself close-spaced and uses glasses that have the same primary dispersion coefficients (but very different secondary dispersion coefficients), the corrector can have zero net power and can be inserted anywhere after the objective without change in the lens system first-order characteristics. The performance of this apochromatic combination is presented in Fig. 13.

Fig. 13
Fig. 13

Performance of an achromatic doublet objective with a zero-power color corrector. The two-glass zero-power color corrector of Fig. 12 is evaluated for paraxial color error. The addition of the corrector converts the achromatic objective into an apochromat without change to either the objective or the system focal length. In this example, the corrector is airspaced so that its diameter is approximately one-third of the objective.

Fig. 14
Fig. 14

Prescription for a Kingslake telephoto lens and its apochromatic modification. The achromatic telephoto lens design of Kingslake39 has been modified by using different glasses. The glasses were selected so that the lens powers and airspaces (scale factors) were similar, but the design now has apochromatic correction. The scaled glass vectors and performance of these two designs are presented in Figs. 15 and 16.

Fig. 15
Fig. 15

Scaled glass vectors for a Kingslake telephoto lens and its apochromatic modification. The Kingslake design (see Fig. 14 for prescription) used four different glasses but has significant color error residual as indicated by the magnitude of G0. The technique used in selecting glasses for a modified design which will retain the powers and airspaces of the original is to select glasses for the positive and negative doublets that have similar ratios of primary dispersion coefficients. The color error residual of the original design can be reduced by using glasses for the front doublet that have a more similar secondary dispersion coefficient while using glasses for the rear doublet that have a dissimilar and compensating secondary dispersion. The head-to-tail summation of the vectors is shown in phantom.

Fig. 16
Fig. 16

Performance of Kingslake and modified telephoto lens designs. Using the glasses suggested by the vector summation of Fig. 15, the modified telephoto design when optimized shows considerable improvement in color correction over the original design. Only minor changes in lens powers and airspaces were required.

Equations (15)

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N 2 = A 0 + A 1 λ 2 + A 2 λ - 2 + A 3 λ - 4 + A 4 λ - 6 + A 5 λ - 8 .
ω = λ - λ 0 1 + 5 / 2 ( λ - λ 0 ) ,
N = N 0 + ν 1 ω + ν 2 ω 2 + + ν i ω i ,
D ( λ ) = δ N ( λ ) N 0 - 1 = i = 1 n η i ω i ,
D ( λ 1 , λ 2 ) = D ( ω 1 ) - D ( ω 2 ) = η 1 ( ω 1 - ω 2 ) + η 2 ( ω 1 2 - ω 2 2 ) + .
Y = - 1 ϕ 0 Y 1 j = 1 n ϕ j Y j 2 V j ,
t j = y j - y j + 1 y j ϕ j .
Y ( λ ) = - Y 1 D 0 ( λ ) = - 1 ϕ 0 Y 1 j = 1 n ϕ j Y j 2 D j ( λ ) .
η 10 ω = j = 1 n α j η 1 j ω ; η 20 ω 2 = j = 1 n α j η 2 j ω 2 ;
α j = ϕ j Y j 2 ϕ 0 Y 1 2 .
G 0 = j = 1 n α j G j .
j = 1 n α j Y 1 Y j = 1.
α 1 G 1 + α 2 G 2 = 0.
G 2 = - ϕ 1 G 1 ( ϕ 1 - 1 ) ( 1 - t ϕ 1 ) .
α 1 α 2 = - η 12 η 11 .

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