Abstract

The recent need in space astronomy for wide-field telescopes having the best transmission and exempt from the chromatic aberration in uv, the severe conditions of the aspheric surface quality for the shortest wavelength, leads to new developments both in the theory and design of those surfaces. A two-surface all-reflective Schmidt telescope is analyzed (1) with Fermat’s principle and (2) with Abbe sine condition. A dioptric elasticity method is presented to aspherize the primary mirror which is a nonaxisymmetric surface. Experimental results are given for an f/1.5, 180 mm focal length all-reflective Schmidt.

© 1976 Optical Society of America

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References

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  1. C. R. Burch, Proc. Phys. Soc. London 55, 6, 433 (1943).
  2. E. H. Linfoot, Recent Advances in Optics (Clarendon, Oxford, England, 1955), p. 277.
  3. C. G. Wynne, J. Opt. Soc. Am. 59, 5, 572 (1969).
    [Crossref]
  4. K. Schwarzschild, “Untersuchung zur geometrischen Optik,” Abhandlungen der Königl, Gessellschaft der Wissenschaften zu Göttingen, Math. -Phys. Kl. 4, (1905–1906), Nos. 1, 2, 3. Other references in Wynne paper (Ref. 3).
  5. B. Schmidt, Zentral Zeit. f. Opt. Mech. 52, 2 (1931).
  6. C. R. Burch, Proc. Phys. Soc. London 59, 41 (1947).
    [Crossref]
  7. F. E. Ross, Astrophys. J. 92, 400 (1940).
    [Crossref]
  8. J. G. Baker, Proc. Am. Philos. Soc. 82, 323 (1940).
  9. R. D. Sigler, Appl. Opt,  14, 2302 (1975).
    [Crossref] [PubMed]
  10. G. Lemaître, Astron. Astrophys. 44, 305 (1975).
  11. D. O. Hendrix and W. H. Christie, Sci. Am. 8,(1939).
  12. M. Migeotte, Ciel Terre 64, 145 (1948).
  13. L. C. Epstein, Publ. Astron. Soc. Pacific 79, 132 (1967).
    [Crossref]
  14. L. C. Epstein, Sky Telesc. 33, 204 (1967).
  15. L. C. Epstein, Appl. Opt. 12, 4, 926 (1973).
  16. G. Courtès, “New Techniques in Space Astronomy,” Symposium IAU N° 41, pp. 273–301, Eds. Labuhn and Lüst (1971).
  17. K. Henize, “The Role of Surveys in Space Astronomy,” in Optical Telescope Technology, (U. S. Government Printing Office, Washington D. C.1970).
  18. G. Monnet, R. Zaharia, and G. Lemaitre, “Programme FAUST,” (1970).
  19. D. E. Oinen, “A Correcting Surface for Two Surface All Reflecting Schmidt Lens,” Publ. Eastman Kodak Company, 901 Elmgrove Road, Rochester, N. Y. 14650.
  20. D. Korsh, Appl. Opt. 13, 9, 2005 (1974).
  21. G. Lemaître, C. R. Acad. Sci. Ser. B 276, 145 (1973).
  22. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 210.
  23. H. Chrétien, in Calcul des Combinaisons Optiques (J. and R. Sennac, Paris, 1958), p. 377.
  24. A. Baranne, “Le Télescope Ritchey-Chrétien de 3, 60m.” Publ. Obs. Hte-Provence 8, 22 (1966).
  25. B. Schmidt, Mitt. Hamburg Sternw. Bergedorf 7, 36, 15 (1932).
  26. R. Schorr, Mitt. Hamburg Sternw. Bergedorf 7, 42, 175 (1936).
  27. R. Schorr, Z. Instrum. Kde,  56, 336 (1936). Another paper (Ref. 28) gives details on the elasticity method that was used by Schmidt.
  28. R. Schorr, Astron. Nachr. 258, 45 (1936). This paper was translated into English by N. U. Mayall (Ref. 29).
    [Crossref]
  29. N. U. Mayall, Publ. Astron. Soc. Pacific 58, 282 (1946).
    [Crossref]
  30. A. Couder, C. R. Acad. Sci. Paris 210, 327 (1940).
  31. B. A. J. Clark, J. Astron. Soc. Victoria 6, 76 (1964).
  32. E. Everhart, Appl. Opt. 5, 713 (1966).
    [Crossref] [PubMed]
  33. F. Cooke, Appl. Opt. 11, 222 (1972).
    [Crossref]
  34. G. Lemaître, Appl. Opt. 7, 1630 (1972).
    [Crossref]
  35. G. Lemaître, Astron. Astrophys. 44, 305 (1975).
  36. G. Lemaître, Nouv. Rev. Opt. 5, 361 (1974).
    [Crossref]
  37. S. Timoshenko and S. Woinowsky-Krieger, “Théorie des Plaques et Coques,” Edt. Béranger, Paris, 1961), p. 282.
  38. A. Maréchal, Rev. Opt. 26, 257 (1947).
  39. M. Cohendet, , (1972).
  40. R. K. Dakin and E. G. Loewen, Opt. Spectra, October, p. 29 (1975).

1975 (4)

R. D. Sigler, Appl. Opt,  14, 2302 (1975).
[Crossref] [PubMed]

G. Lemaître, Astron. Astrophys. 44, 305 (1975).

G. Lemaître, Astron. Astrophys. 44, 305 (1975).

R. K. Dakin and E. G. Loewen, Opt. Spectra, October, p. 29 (1975).

1974 (2)

G. Lemaître, Nouv. Rev. Opt. 5, 361 (1974).
[Crossref]

D. Korsh, Appl. Opt. 13, 9, 2005 (1974).

1973 (2)

G. Lemaître, C. R. Acad. Sci. Ser. B 276, 145 (1973).

L. C. Epstein, Appl. Opt. 12, 4, 926 (1973).

1972 (2)

G. Lemaître, Appl. Opt. 7, 1630 (1972).
[Crossref]

F. Cooke, Appl. Opt. 11, 222 (1972).
[Crossref]

1969 (1)

C. G. Wynne, J. Opt. Soc. Am. 59, 5, 572 (1969).
[Crossref]

1967 (2)

L. C. Epstein, Publ. Astron. Soc. Pacific 79, 132 (1967).
[Crossref]

L. C. Epstein, Sky Telesc. 33, 204 (1967).

1966 (2)

A. Baranne, “Le Télescope Ritchey-Chrétien de 3, 60m.” Publ. Obs. Hte-Provence 8, 22 (1966).

E. Everhart, Appl. Opt. 5, 713 (1966).
[Crossref] [PubMed]

1964 (1)

B. A. J. Clark, J. Astron. Soc. Victoria 6, 76 (1964).

1948 (1)

M. Migeotte, Ciel Terre 64, 145 (1948).

1947 (2)

C. R. Burch, Proc. Phys. Soc. London 59, 41 (1947).
[Crossref]

A. Maréchal, Rev. Opt. 26, 257 (1947).

1946 (1)

N. U. Mayall, Publ. Astron. Soc. Pacific 58, 282 (1946).
[Crossref]

1943 (1)

C. R. Burch, Proc. Phys. Soc. London 55, 6, 433 (1943).

1940 (3)

F. E. Ross, Astrophys. J. 92, 400 (1940).
[Crossref]

J. G. Baker, Proc. Am. Philos. Soc. 82, 323 (1940).

A. Couder, C. R. Acad. Sci. Paris 210, 327 (1940).

1939 (1)

D. O. Hendrix and W. H. Christie, Sci. Am. 8,(1939).

1936 (3)

R. Schorr, Mitt. Hamburg Sternw. Bergedorf 7, 42, 175 (1936).

R. Schorr, Z. Instrum. Kde,  56, 336 (1936). Another paper (Ref. 28) gives details on the elasticity method that was used by Schmidt.

R. Schorr, Astron. Nachr. 258, 45 (1936). This paper was translated into English by N. U. Mayall (Ref. 29).
[Crossref]

1932 (1)

B. Schmidt, Mitt. Hamburg Sternw. Bergedorf 7, 36, 15 (1932).

1931 (1)

B. Schmidt, Zentral Zeit. f. Opt. Mech. 52, 2 (1931).

Baker, J. G.

J. G. Baker, Proc. Am. Philos. Soc. 82, 323 (1940).

Baranne, A.

A. Baranne, “Le Télescope Ritchey-Chrétien de 3, 60m.” Publ. Obs. Hte-Provence 8, 22 (1966).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 210.

Burch, C. R.

C. R. Burch, Proc. Phys. Soc. London 59, 41 (1947).
[Crossref]

C. R. Burch, Proc. Phys. Soc. London 55, 6, 433 (1943).

Chrétien, H.

H. Chrétien, in Calcul des Combinaisons Optiques (J. and R. Sennac, Paris, 1958), p. 377.

Christie, W. H.

D. O. Hendrix and W. H. Christie, Sci. Am. 8,(1939).

Clark, B. A. J.

B. A. J. Clark, J. Astron. Soc. Victoria 6, 76 (1964).

Cohendet, M.

M. Cohendet, , (1972).

Cooke, F.

Couder, A.

A. Couder, C. R. Acad. Sci. Paris 210, 327 (1940).

Courtès, G.

G. Courtès, “New Techniques in Space Astronomy,” Symposium IAU N° 41, pp. 273–301, Eds. Labuhn and Lüst (1971).

Dakin, R. K.

R. K. Dakin and E. G. Loewen, Opt. Spectra, October, p. 29 (1975).

Epstein, L. C.

L. C. Epstein, Appl. Opt. 12, 4, 926 (1973).

L. C. Epstein, Publ. Astron. Soc. Pacific 79, 132 (1967).
[Crossref]

L. C. Epstein, Sky Telesc. 33, 204 (1967).

Everhart, E.

Hendrix, D. O.

D. O. Hendrix and W. H. Christie, Sci. Am. 8,(1939).

Henize, K.

K. Henize, “The Role of Surveys in Space Astronomy,” in Optical Telescope Technology, (U. S. Government Printing Office, Washington D. C.1970).

Korsh, D.

D. Korsh, Appl. Opt. 13, 9, 2005 (1974).

Lemaitre, G.

G. Monnet, R. Zaharia, and G. Lemaitre, “Programme FAUST,” (1970).

Lemaître, G.

G. Lemaître, Astron. Astrophys. 44, 305 (1975).

G. Lemaître, Astron. Astrophys. 44, 305 (1975).

G. Lemaître, Nouv. Rev. Opt. 5, 361 (1974).
[Crossref]

G. Lemaître, C. R. Acad. Sci. Ser. B 276, 145 (1973).

G. Lemaître, Appl. Opt. 7, 1630 (1972).
[Crossref]

Linfoot, E. H.

E. H. Linfoot, Recent Advances in Optics (Clarendon, Oxford, England, 1955), p. 277.

Loewen, E. G.

R. K. Dakin and E. G. Loewen, Opt. Spectra, October, p. 29 (1975).

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

Mayall, N. U.

N. U. Mayall, Publ. Astron. Soc. Pacific 58, 282 (1946).
[Crossref]

Migeotte, M.

M. Migeotte, Ciel Terre 64, 145 (1948).

Monnet, G.

G. Monnet, R. Zaharia, and G. Lemaitre, “Programme FAUST,” (1970).

Oinen, D. E.

D. E. Oinen, “A Correcting Surface for Two Surface All Reflecting Schmidt Lens,” Publ. Eastman Kodak Company, 901 Elmgrove Road, Rochester, N. Y. 14650.

Ross, F. E.

F. E. Ross, Astrophys. J. 92, 400 (1940).
[Crossref]

Schmidt, B.

B. Schmidt, Mitt. Hamburg Sternw. Bergedorf 7, 36, 15 (1932).

B. Schmidt, Zentral Zeit. f. Opt. Mech. 52, 2 (1931).

Schorr, R.

R. Schorr, Mitt. Hamburg Sternw. Bergedorf 7, 42, 175 (1936).

R. Schorr, Z. Instrum. Kde,  56, 336 (1936). Another paper (Ref. 28) gives details on the elasticity method that was used by Schmidt.

R. Schorr, Astron. Nachr. 258, 45 (1936). This paper was translated into English by N. U. Mayall (Ref. 29).
[Crossref]

Schwarzschild, K.

K. Schwarzschild, “Untersuchung zur geometrischen Optik,” Abhandlungen der Königl, Gessellschaft der Wissenschaften zu Göttingen, Math. -Phys. Kl. 4, (1905–1906), Nos. 1, 2, 3. Other references in Wynne paper (Ref. 3).

Sigler, R. D.

R. D. Sigler, Appl. Opt,  14, 2302 (1975).
[Crossref] [PubMed]

Timoshenko, S.

S. Timoshenko and S. Woinowsky-Krieger, “Théorie des Plaques et Coques,” Edt. Béranger, Paris, 1961), p. 282.

Woinowsky-Krieger, S.

S. Timoshenko and S. Woinowsky-Krieger, “Théorie des Plaques et Coques,” Edt. Béranger, Paris, 1961), p. 282.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 210.

Wynne, C. G.

C. G. Wynne, J. Opt. Soc. Am. 59, 5, 572 (1969).
[Crossref]

Zaharia, R.

G. Monnet, R. Zaharia, and G. Lemaitre, “Programme FAUST,” (1970).

Abhandlungen der Königl, Gessellschaft der Wissenschaften zu Göttingen, Math. -Phys. Kl. (1)

K. Schwarzschild, “Untersuchung zur geometrischen Optik,” Abhandlungen der Königl, Gessellschaft der Wissenschaften zu Göttingen, Math. -Phys. Kl. 4, (1905–1906), Nos. 1, 2, 3. Other references in Wynne paper (Ref. 3).

Appl. Opt (1)

R. D. Sigler, Appl. Opt,  14, 2302 (1975).
[Crossref] [PubMed]

Appl. Opt. (5)

L. C. Epstein, Appl. Opt. 12, 4, 926 (1973).

D. Korsh, Appl. Opt. 13, 9, 2005 (1974).

E. Everhart, Appl. Opt. 5, 713 (1966).
[Crossref] [PubMed]

F. Cooke, Appl. Opt. 11, 222 (1972).
[Crossref]

G. Lemaître, Appl. Opt. 7, 1630 (1972).
[Crossref]

Astron. Astrophys. (2)

G. Lemaître, Astron. Astrophys. 44, 305 (1975).

G. Lemaître, Astron. Astrophys. 44, 305 (1975).

Astron. Nachr. (1)

R. Schorr, Astron. Nachr. 258, 45 (1936). This paper was translated into English by N. U. Mayall (Ref. 29).
[Crossref]

Astrophys. J. (1)

F. E. Ross, Astrophys. J. 92, 400 (1940).
[Crossref]

C. R. Acad. Sci. Paris (1)

A. Couder, C. R. Acad. Sci. Paris 210, 327 (1940).

C. R. Acad. Sci. Ser. B (1)

G. Lemaître, C. R. Acad. Sci. Ser. B 276, 145 (1973).

Ciel Terre (1)

M. Migeotte, Ciel Terre 64, 145 (1948).

J. Astron. Soc. Victoria (1)

B. A. J. Clark, J. Astron. Soc. Victoria 6, 76 (1964).

J. Opt. Soc. Am. (1)

C. G. Wynne, J. Opt. Soc. Am. 59, 5, 572 (1969).
[Crossref]

Mitt. Hamburg Sternw. Bergedorf (2)

B. Schmidt, Mitt. Hamburg Sternw. Bergedorf 7, 36, 15 (1932).

R. Schorr, Mitt. Hamburg Sternw. Bergedorf 7, 42, 175 (1936).

Nouv. Rev. Opt. (1)

G. Lemaître, Nouv. Rev. Opt. 5, 361 (1974).
[Crossref]

Opt. Spectra (1)

R. K. Dakin and E. G. Loewen, Opt. Spectra, October, p. 29 (1975).

Proc. Am. Philos. Soc. (1)

J. G. Baker, Proc. Am. Philos. Soc. 82, 323 (1940).

Proc. Phys. Soc. London (2)

C. R. Burch, Proc. Phys. Soc. London 59, 41 (1947).
[Crossref]

C. R. Burch, Proc. Phys. Soc. London 55, 6, 433 (1943).

Publ. Astron. Soc. Pacific (2)

L. C. Epstein, Publ. Astron. Soc. Pacific 79, 132 (1967).
[Crossref]

N. U. Mayall, Publ. Astron. Soc. Pacific 58, 282 (1946).
[Crossref]

Publ. Obs. Hte-Provence (1)

A. Baranne, “Le Télescope Ritchey-Chrétien de 3, 60m.” Publ. Obs. Hte-Provence 8, 22 (1966).

Rev. Opt. (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

Sci. Am. (1)

D. O. Hendrix and W. H. Christie, Sci. Am. 8,(1939).

Sky Telesc. (1)

L. C. Epstein, Sky Telesc. 33, 204 (1967).

Z. Instrum. Kde (1)

R. Schorr, Z. Instrum. Kde,  56, 336 (1936). Another paper (Ref. 28) gives details on the elasticity method that was used by Schmidt.

Zentral Zeit. f. Opt. Mech. (1)

B. Schmidt, Zentral Zeit. f. Opt. Mech. 52, 2 (1931).

Other (9)

E. H. Linfoot, Recent Advances in Optics (Clarendon, Oxford, England, 1955), p. 277.

G. Courtès, “New Techniques in Space Astronomy,” Symposium IAU N° 41, pp. 273–301, Eds. Labuhn and Lüst (1971).

K. Henize, “The Role of Surveys in Space Astronomy,” in Optical Telescope Technology, (U. S. Government Printing Office, Washington D. C.1970).

G. Monnet, R. Zaharia, and G. Lemaitre, “Programme FAUST,” (1970).

D. E. Oinen, “A Correcting Surface for Two Surface All Reflecting Schmidt Lens,” Publ. Eastman Kodak Company, 901 Elmgrove Road, Rochester, N. Y. 14650.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 210.

H. Chrétien, in Calcul des Combinaisons Optiques (J. and R. Sennac, Paris, 1958), p. 377.

M. Cohendet, , (1972).

S. Timoshenko and S. Woinowsky-Krieger, “Théorie des Plaques et Coques,” Edt. Béranger, Paris, 1961), p. 282.

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Figures (10)

FIG. 1
FIG. 1

Notation and basic geometry of an all-reflecting Schmidt system.

FIG. 2
FIG. 2

Approximately true forms of the geometrical ray traces and mirrors, first (C1) with the stigmatic condition and second (C2) with the aplanatic condition. Abbe’s sphere ∑ is representated for an aperture angle of 2π sr.

FIG. 3
FIG. 3

Notations used in the ϕ rotation of the primary mirror.

FIG. 4
FIG. 4

Equilibrium of the elementary slice with respect (1) to the tangential axis ωt giving Q ρ, (2) to the radial axis ωρ giving Q θ, and (3) to the axial axis ωz giving q.

FIG. 5
FIG. 5

Configurations of thickness H 04 and loads F0 giving the correction of the third-order spherical aberration.

FIG. 6
FIG. 6

Configurations of thickness H 24 and loads F2 cos 2θ giving the correction of the fifth-order elliptical astigmatism.

FIG. 7
FIG. 7

Interferograms at 632.8 nm of the elastic aspherization with respect to a plane, obtained (1) in the case of Fig. 5, (2) in the case of Fig. 6 with a thickness H 04 instead of H 24, and (3) with superposition of (1) and (2).

FIG. 8
FIG. 8

Ray traces evolution for f/1.5 and ϕ = 15° of the on–axis image when adding successive coefficients Am,n. With the seven coefficients given in Table I for the primary mirror, the angular image sizes are Δx/f = 3.8 10−6 and Δy/f = 5.7 10−6 (i.e., ≃1 arc sec).

FIG. 9
FIG. 9

Ray traces of four angular sections along the focal surface. The spot diagrams are given for 1.25° and 2.5° from the axis. The image sizes in the field must be considered as maximum values. For a given field, the optimization ray tracing process leads to better performance at the edge of the field.

FIG. 10
FIG. 10

Reduction of the on-axis aberrations of an f/1.5 all-reflective Schmidt telescope (f = 180 nm) with a 30° beam deviation. Starting from the paraxial image i0 (which is reduced by a factor 2 in the figure), one can see various stages the reduction of spherical aberration (i1, i3, i5) and astigmatism (i2, i4).

Tables (1)

Tables Icon

TABLE I Values of the Am,n coefficients for the primary mirror.

Equations (52)

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ζ = A m , n ρ n cos m θ ,
sin α = 1 2 sin u .
l 1 + l 2 + ζ = constant = 3 ,
ζ = l 2 cos ( u - 2 α ) - l 1 cos u - 1 ,
ρ = - l 2 sin ( u - 2 α ) + l 1 sin u .
l 1 = cos 3 u [ ( 1 + 3 sec 2 u ) 1 / 2 - 1 ] ,
l 2 = [ 1 + tan 2 ( 1 2 u - α ) ] { 2 + [ 1 2 - cos ( u - α ) ] ( sec u - 1 ) } ,
l 1 = 1 + 1 2 2 u 2 + 5 3.2 6 u 4 - 29 5.3 2 .2 9 u 6 - 17.43 7.3 2 .2 14 u 8 ,
l 2 = 2 - 1 2 2 u 2 - 1 3.2 3 u 4 + 67 5.3 2 .2 8 u 6 + 449 7.5.3 2 .2 11 u 8 .
ζ = - 2 + 1 4 ρ 2 + i = 2 1 2 4 i - 3 ( 2 i - 3 ) ! i ! ( i - 2 ) ! ρ 2 i .
ζ ( 0 ; 0 ; 1 / 2 6 ; 3 / 2 9 ; 5 × 3 2 / 2 14 ) ,
ζ ( - 2 ; 1 / 2 2 ; 1 / 2 6 ; 1 / 2 9 ; 5 / 2 14 ) .
ζ = 11 3 - 19 , ρ = 7 3 - 11 ,             and             ζ = - 1 , ρ = 3 .
ρ / sin u = constant = 1 ,
ζ = 1 - 1 2 t ( 1 - t ) - ( 1 - 1 2 t ) 3 ( 1 - t ) - 1 .
1 / l 1 = 1 2 t + ( 1 - t ) 2 ( 1 - 1 2 t ) - 1 ,
ζ = - 1 - ( 1 - 2 t ) ( t 2 + 2 ( 1 - t ) 2 2 - t ) - 1 ,
ζ ( 0 ; 0 ; 1 / 2 6 ; 3 / 2 9 ; 3 / 2 10 ) ,
ζ ( - 2 ; 1 / 2 2 ; 1 / 2 6 ; 1 / 2 9 ; 1 / 2 11 ) .
ζ = 1 / 32 , ρ = 1 ,             and             ζ = - 1 , ρ = 12 / 7.
ζ M * * = n = 0 , 4 A 0 , 2 n * ρ * 2 n ,
ζ M * = tan ϕ y M * * + ζ M * * / cos 2 ϕ .
x * = x , y * = y cos ϕ - ζ sin ϕ , ζ * = y sin ϕ + ζ cos ϕ ,
ζ M = ζ M * * / cos ϕ .
ρ * 2 = ρ 2 ( 1 - cos 2 θ sin 2 ϕ ) ρ 2 ( 1 - a - a cos 2 θ ) ,
ρ * 4 = ρ 4 [ 1 - 2 a + 3 2 a 2 - 2 a ( 1 - a ) cos 2 θ + 1 2 a 2 cos 4 θ ] ,
ρ * 6 ρ 6 [ 1 - 3 a - 3 a cos 2 θ ] ,
ρ * 8 ρ 8 .
A 04 = A 04 * ( 1 - sin 2 ϕ + 3 8 sin 4 ϕ ) / cos ϕ ,
A 24 = - A 04 * sin 2 ϕ ( 1 - 1 2 sin 2 ϕ ) / cos ϕ ,
A 44 = A 04 * 1 8 sin 4 ϕ / cos ϕ ,
A 06 = A 06 * ( 1 - 3 2 sin 2 ϕ ) / cos ϕ ,
A 26 = - A 06 * 3 2 sin 2 ϕ / cos ϕ ,
A 08 = A 08 * / cos ϕ .
y M * = y M * * - 4 A 04 * tan ϕ y M * * 4 .
A 17 = A 04 * 2 16 tan ϕ .
D = E H 3 / 12 ( 1 - ν 2 ) ,
M ρ = - D [ 2 ζ ρ 2 + ν ( 1 ρ ζ ρ + 1 ρ 2 2 ζ θ 2 ) ] ,
M θ = - D [ 1 ρ ζ ρ + 1 ρ 2 2 ζ θ 2 + ν 2 ζ ρ 2 ] ,
M ρ θ = ( 1 - ν ) D [ 1 ρ 2 ζ ρ θ - 1 ρ 2 ζ θ ] .
Q ρ = - D ρ ( 2 ζ ) ,
Q θ = - D ρ θ ( 2 ζ ) .
Q ρ = M ρ ρ + 1 ρ ( M ρ - M θ - M ρ θ ρ ) ,
Q θ = - M ρ θ ρ + 1 ρ ( M θ θ - 2 M ρ θ ) ,
q = - 1 ρ [ ρ ( ρ Q ρ ) + Q θ θ ] .
D = i = 1 j A i ρ - α i ,
Q ρ = - i = 1 j { ( n - 2 ) ( n 2 - m 2 ) - α i [ n ( n - 1 ) + ν ( n - m 2 ) ] } × A i m n ρ n - 3 - α i cos m θ ,
Q θ = - i = 1 j m [ n 2 - m 2 - α i ( n - 1 ) ( 1 - ν ) ] × A i m n ρ n - 3 - α i sin m θ .
q = i = 1 j { α i 2 [ n ( n - 1 ) + ν ( n - m 2 ) ] - α i [ n ( n - 2 ) ( 2 n - 1 + ν ) - m 2 ( 2 n - 3 - ν ) ] + ( n 2 - m 2 ) [ ( n - 2 ) 2 - m 2 ] } × A i m n ρ n - 4 - α i cos m θ .
H 04 = ( 1 / ρ 8 / ( 3 + ν ) - 1 / ρ 2 ) 1 / 3 .
H 24 = ( 1 / ρ 3 + ν - 1 ) 1 / 3 ,
ζ * = - 1 + 1 2 ρ * 2 + 1 2 8 ρ * cos θ * ,             with ϕ = 15° .