Abstract

The theory of the concave grating has been examined in considerable detail by methods of geometrical optics. The results derived by means of geometrical optics were checked by comparing them with those based on physical optics in special cases. The conditions for image formation and aberrations in the image may be expressed by equations somewhat different from those given by Beutler which are shown to be in error. The astigmatism and other aberrations are treated with respect to finite length of slit illumination, finite grating size and deviations of the optical components from the Rowland plane but still lying on the Rowland cylinder. Within certain limitations, it is shown that the aberrations present in an off-plane Eagle mounting may be corrected by a very small rotation of the slit in a plane perpendicular to the optic axis. The optimum width of the grating, instrumental line half-width, and resolving power of the grating are also discussed. Finally, the Beutler treatment of the concave grating is examined in detail and the important errors are pointed out.

© 1959 Optical Society of America

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References

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  1. H. A. Rowland, Phil. Mag. 13, 469 (1882);Phil. Mag. 16, 197 and 210 (1883).
  2. R. T. Glazebrook, Phil. Mag. 15, 414 (1883).
  3. E. Mascart, J. phys. 2, 5 (1883).
  4. W. Baily, Phil. Mag. 22, 47 (1886).
  5. H. Kayser, Handbuch der Spectroscopic (Hirzel, Leipzig, 1900), Vol. I, pp. 450–470.
  6. A. Eagle, Astrophys. J. 31, 120 (1910);Proc. Phys. Soc. (London) 23, 233 (1911).
    [Crossref]
  7. C. Runge and K. W. Meissner, Handbuch der Astrophysik (Julius Springer, Berlin, 1933), Vol. 1, pp. 235–257.
  8. Mack, Stehn, and Edlén, J. Opt. Soc. Am. 22, 245 (1932);J. E. Mack and J. R. Stehn, J. Opt. Soc. Am. 23, 184 (1933).
    [Crossref]
  9. D. L. MacAdam, J. Opt. Soc. Am. 23, 178 (1933).
    [Crossref]
  10. G. H. Dieke, J. Opt. Soc. Am. 23, 274 (1933).
    [Crossref]
  11. I. S. Bowen, J. Opt. Soc. Am. 23, 313 (1933).
    [Crossref]
  12. R. A. Sawyer, Experimental Spectroscopy (Prentice Hall, Inc., New York, 1944), 2nd Ed., pp. 132–146 and pp. 165–178.
  13. H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945).Also see F. Zernike, Pieter Zeeman (Martinus Nijhoff, Hague, Netherlands, 1935), pp. 323–335. Beutler’s treatment is almost the same as Zernike’s.
    [Crossref]
  14. P. G. Wilkinson, J. Mol. Spectroscopy 1, 288 (1957).
    [Crossref]
  15. Beutler’s paper was nearly completed at the time of his death and was prepared for publication by Dr. R. A. Sawyer. Since Dr. Beutler had intended to enlarge some parts of the paper, he would have probably corrected some errors which appeared in his paper.
  16. F. L. O. Wadsworth, Astrophys. J. 3, 47 (1896);C. Runge and F. Paschen, Ann. Physik 61, 641 (1897);W. F. Meggers and K. Burns, Sci. Papers Natl. Bur. Standards (U. S.) [411] 18, 185 (1922);R. F. Jarrell, J. Opt. Soc. Am. 32, 666 (1942).
    [Crossref]
  17. C. Runge and F. Paschen, Abhandl. deut. K. Akad. Wiss-Berlin, Anhang.1 (1902);F. S. Tomkins and M. Fred, Spectrochim. Acta 6, 139 (1954).
    [Crossref]
  18. W. deW, Abney, Phil. Trans. Roy. Soc. (London) 177, 11 and 457 (1886).
  19. T. Lyman, Spectroscopy of the Extreme Ultraviolet (Longmans, Green and Company, New York, 1928).
  20. J. B. Hoag, Astrophys. J. 66, 225 (1927);M. Siegbahn and T. Magnusson, Z. Physik,  95, 133 (1935);P. G. Kruger, Rev. Sci. Instr. 4, 128 (1933).
    [Crossref]
  21. When the slit is not parallel to the grating rulings, it (except the midpoint) deviates a very small amount from the Rowland cylinder. Since the terms involving (cos2α/r)−(cosα/R) are no longer zero for all points (except the center of the slit) we must take into account this fact in our treatment. The variation of 1/r, Δ(1/r), is fortunately very small and Δ(1/r) = φΔz·tanα0/(R2 cos2α0)∼O(z0/R3). Therefore, only one term, −wφΔz· tanα0/R2 comes into (10) as the correction [note that we retain the terms down to O(1/R3)].
  22. For the 21-ft off-plane Eagle mounting vacuum spectrograph at the University of Chicago, W = 12.7 cm, z0= 12.7 cm. We have the ratio z0/R= 0.02 and (W2/16)/z02= 0.0625. Therefore, this assumption is not far from a practical case.
  23. We are now considering a point in the image where the m th-order line due to light of wavelength (λ−Δλ) would have its central maximum intensity. In other words, Δλ is not a change in the wavelength λ but means displacement, which is measured in units of wavelength, of an image point from the position where the m th-order line of wavelength λ would have its central maximum intensity. It has the same sign as Δβ.
  24. But if we assume uniform illumination over the grating and a perfect grating, then, for simplicity, all the δn’s may be treated as equal.
  25. In the case of the in-plane mounting, R asymptotically approaches 0.75Woptm/σ as W≫Wopt (see Mack, Stehn, and Edlén, reference 8).
  26. The word“slit” used here is equivalent to the illuminated part of the slit.
  27. W. R. Hamilton, “Geometrical Optics,” Mathematical Papers (Cambridge University Press, London, 1931), Vol. 1, p. 17;J. L. Synge, “Geometrical optics (an introduction to Hamilton’s method),” Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge University Press, London, 1937), No. 37, p. 17;J. L. Synge, J. Opt. Soc. Am. 27, 75 (1937).
    [Crossref]
  28. Private communication with Dr. D. L. MacAdam. This comment resulted from examination of the manuscript of this paper by Dr. Herzberger and himself.
  29. Strictly speaking g is not independent of l and such dependence becomes appreciable for grazing incidence cases. In that case (45) can give the correct answer only under certain restrictions, i.e., w, l⩽12Wopt.
  30. See (83) in his paper (reference 13). Equation (83) is misprinted and the exponent 2 should be put on the sines within the round bracket.
  31. B. Edlén, Nova Acta Regiae Soc. Sci. Upsaliensis 9, No. 6 (1933).

1957 (1)

P. G. Wilkinson, J. Mol. Spectroscopy 1, 288 (1957).
[Crossref]

1945 (1)

1933 (4)

1932 (1)

1927 (1)

J. B. Hoag, Astrophys. J. 66, 225 (1927);M. Siegbahn and T. Magnusson, Z. Physik,  95, 133 (1935);P. G. Kruger, Rev. Sci. Instr. 4, 128 (1933).
[Crossref]

1910 (1)

A. Eagle, Astrophys. J. 31, 120 (1910);Proc. Phys. Soc. (London) 23, 233 (1911).
[Crossref]

1902 (1)

C. Runge and F. Paschen, Abhandl. deut. K. Akad. Wiss-Berlin, Anhang.1 (1902);F. S. Tomkins and M. Fred, Spectrochim. Acta 6, 139 (1954).
[Crossref]

1896 (1)

F. L. O. Wadsworth, Astrophys. J. 3, 47 (1896);C. Runge and F. Paschen, Ann. Physik 61, 641 (1897);W. F. Meggers and K. Burns, Sci. Papers Natl. Bur. Standards (U. S.) [411] 18, 185 (1922);R. F. Jarrell, J. Opt. Soc. Am. 32, 666 (1942).
[Crossref]

1886 (2)

W. deW, Abney, Phil. Trans. Roy. Soc. (London) 177, 11 and 457 (1886).

W. Baily, Phil. Mag. 22, 47 (1886).

1883 (2)

R. T. Glazebrook, Phil. Mag. 15, 414 (1883).

E. Mascart, J. phys. 2, 5 (1883).

1882 (1)

H. A. Rowland, Phil. Mag. 13, 469 (1882);Phil. Mag. 16, 197 and 210 (1883).

Baily, W.

W. Baily, Phil. Mag. 22, 47 (1886).

Beutler, H. G.

Bowen, I. S.

deW, W.

W. deW, Abney, Phil. Trans. Roy. Soc. (London) 177, 11 and 457 (1886).

Dieke, G. H.

Eagle, A.

A. Eagle, Astrophys. J. 31, 120 (1910);Proc. Phys. Soc. (London) 23, 233 (1911).
[Crossref]

Edlén,

Edlén, B.

B. Edlén, Nova Acta Regiae Soc. Sci. Upsaliensis 9, No. 6 (1933).

Glazebrook, R. T.

R. T. Glazebrook, Phil. Mag. 15, 414 (1883).

Hamilton, W. R.

W. R. Hamilton, “Geometrical Optics,” Mathematical Papers (Cambridge University Press, London, 1931), Vol. 1, p. 17;J. L. Synge, “Geometrical optics (an introduction to Hamilton’s method),” Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge University Press, London, 1937), No. 37, p. 17;J. L. Synge, J. Opt. Soc. Am. 27, 75 (1937).
[Crossref]

Hoag, J. B.

J. B. Hoag, Astrophys. J. 66, 225 (1927);M. Siegbahn and T. Magnusson, Z. Physik,  95, 133 (1935);P. G. Kruger, Rev. Sci. Instr. 4, 128 (1933).
[Crossref]

Kayser, H.

H. Kayser, Handbuch der Spectroscopic (Hirzel, Leipzig, 1900), Vol. I, pp. 450–470.

Lyman, T.

T. Lyman, Spectroscopy of the Extreme Ultraviolet (Longmans, Green and Company, New York, 1928).

MacAdam, D. L.

Mack,

Mascart, E.

E. Mascart, J. phys. 2, 5 (1883).

Meissner, K. W.

C. Runge and K. W. Meissner, Handbuch der Astrophysik (Julius Springer, Berlin, 1933), Vol. 1, pp. 235–257.

Paschen, F.

C. Runge and F. Paschen, Abhandl. deut. K. Akad. Wiss-Berlin, Anhang.1 (1902);F. S. Tomkins and M. Fred, Spectrochim. Acta 6, 139 (1954).
[Crossref]

Rowland, H. A.

H. A. Rowland, Phil. Mag. 13, 469 (1882);Phil. Mag. 16, 197 and 210 (1883).

Runge, C.

C. Runge and F. Paschen, Abhandl. deut. K. Akad. Wiss-Berlin, Anhang.1 (1902);F. S. Tomkins and M. Fred, Spectrochim. Acta 6, 139 (1954).
[Crossref]

C. Runge and K. W. Meissner, Handbuch der Astrophysik (Julius Springer, Berlin, 1933), Vol. 1, pp. 235–257.

Sawyer, R. A.

R. A. Sawyer, Experimental Spectroscopy (Prentice Hall, Inc., New York, 1944), 2nd Ed., pp. 132–146 and pp. 165–178.

Stehn,

Wadsworth, F. L. O.

F. L. O. Wadsworth, Astrophys. J. 3, 47 (1896);C. Runge and F. Paschen, Ann. Physik 61, 641 (1897);W. F. Meggers and K. Burns, Sci. Papers Natl. Bur. Standards (U. S.) [411] 18, 185 (1922);R. F. Jarrell, J. Opt. Soc. Am. 32, 666 (1942).
[Crossref]

Wilkinson, P. G.

P. G. Wilkinson, J. Mol. Spectroscopy 1, 288 (1957).
[Crossref]

Abhandl. deut. K. Akad. Wiss-Berlin, Anhang. (1)

C. Runge and F. Paschen, Abhandl. deut. K. Akad. Wiss-Berlin, Anhang.1 (1902);F. S. Tomkins and M. Fred, Spectrochim. Acta 6, 139 (1954).
[Crossref]

Abney, Phil. Trans. Roy. Soc. (London) (1)

W. deW, Abney, Phil. Trans. Roy. Soc. (London) 177, 11 and 457 (1886).

Astrophys. J. (3)

F. L. O. Wadsworth, Astrophys. J. 3, 47 (1896);C. Runge and F. Paschen, Ann. Physik 61, 641 (1897);W. F. Meggers and K. Burns, Sci. Papers Natl. Bur. Standards (U. S.) [411] 18, 185 (1922);R. F. Jarrell, J. Opt. Soc. Am. 32, 666 (1942).
[Crossref]

A. Eagle, Astrophys. J. 31, 120 (1910);Proc. Phys. Soc. (London) 23, 233 (1911).
[Crossref]

J. B. Hoag, Astrophys. J. 66, 225 (1927);M. Siegbahn and T. Magnusson, Z. Physik,  95, 133 (1935);P. G. Kruger, Rev. Sci. Instr. 4, 128 (1933).
[Crossref]

J. Mol. Spectroscopy (1)

P. G. Wilkinson, J. Mol. Spectroscopy 1, 288 (1957).
[Crossref]

J. Opt. Soc. Am. (5)

J. phys. (1)

E. Mascart, J. phys. 2, 5 (1883).

Nova Acta Regiae Soc. Sci. Upsaliensis (1)

B. Edlén, Nova Acta Regiae Soc. Sci. Upsaliensis 9, No. 6 (1933).

Phil. Mag. (3)

W. Baily, Phil. Mag. 22, 47 (1886).

H. A. Rowland, Phil. Mag. 13, 469 (1882);Phil. Mag. 16, 197 and 210 (1883).

R. T. Glazebrook, Phil. Mag. 15, 414 (1883).

Other (15)

H. Kayser, Handbuch der Spectroscopic (Hirzel, Leipzig, 1900), Vol. I, pp. 450–470.

R. A. Sawyer, Experimental Spectroscopy (Prentice Hall, Inc., New York, 1944), 2nd Ed., pp. 132–146 and pp. 165–178.

T. Lyman, Spectroscopy of the Extreme Ultraviolet (Longmans, Green and Company, New York, 1928).

Beutler’s paper was nearly completed at the time of his death and was prepared for publication by Dr. R. A. Sawyer. Since Dr. Beutler had intended to enlarge some parts of the paper, he would have probably corrected some errors which appeared in his paper.

C. Runge and K. W. Meissner, Handbuch der Astrophysik (Julius Springer, Berlin, 1933), Vol. 1, pp. 235–257.

When the slit is not parallel to the grating rulings, it (except the midpoint) deviates a very small amount from the Rowland cylinder. Since the terms involving (cos2α/r)−(cosα/R) are no longer zero for all points (except the center of the slit) we must take into account this fact in our treatment. The variation of 1/r, Δ(1/r), is fortunately very small and Δ(1/r) = φΔz·tanα0/(R2 cos2α0)∼O(z0/R3). Therefore, only one term, −wφΔz· tanα0/R2 comes into (10) as the correction [note that we retain the terms down to O(1/R3)].

For the 21-ft off-plane Eagle mounting vacuum spectrograph at the University of Chicago, W = 12.7 cm, z0= 12.7 cm. We have the ratio z0/R= 0.02 and (W2/16)/z02= 0.0625. Therefore, this assumption is not far from a practical case.

We are now considering a point in the image where the m th-order line due to light of wavelength (λ−Δλ) would have its central maximum intensity. In other words, Δλ is not a change in the wavelength λ but means displacement, which is measured in units of wavelength, of an image point from the position where the m th-order line of wavelength λ would have its central maximum intensity. It has the same sign as Δβ.

But if we assume uniform illumination over the grating and a perfect grating, then, for simplicity, all the δn’s may be treated as equal.

In the case of the in-plane mounting, R asymptotically approaches 0.75Woptm/σ as W≫Wopt (see Mack, Stehn, and Edlén, reference 8).

The word“slit” used here is equivalent to the illuminated part of the slit.

W. R. Hamilton, “Geometrical Optics,” Mathematical Papers (Cambridge University Press, London, 1931), Vol. 1, p. 17;J. L. Synge, “Geometrical optics (an introduction to Hamilton’s method),” Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge University Press, London, 1937), No. 37, p. 17;J. L. Synge, J. Opt. Soc. Am. 27, 75 (1937).
[Crossref]

Private communication with Dr. D. L. MacAdam. This comment resulted from examination of the manuscript of this paper by Dr. Herzberger and himself.

Strictly speaking g is not independent of l and such dependence becomes appreciable for grazing incidence cases. In that case (45) can give the correct answer only under certain restrictions, i.e., w, l⩽12Wopt.

See (83) in his paper (reference 13). Equation (83) is misprinted and the exponent 2 should be put on the sines within the round bracket.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the optical system. A(x,y,z) is an illuminated point on the entrance slit, B(x′,y′,z′) is a focal point, and P(u,w,l) is a point on the grating rulings. OC is the normal to the center of the grating, O, and C(R,0,0) is the center of curvature of the grating, α and β are the angles of incidence and diffraction, respectively, both measured in the xy plane. r and r′ are the distances between the center of the grating and the projected points of A and B on the xy plane, respectively. R is the radius of curvature of the grating.

Fig. 2
Fig. 2

Relative intensity curves. The two crosses on each curve show the values of 1 2 r and r. r is the value of 2m(Δλ)rw1/(σλ) which would be necessary to separate two lines just resolved, whose wavelength difference is (Δλ)r in the sense of Mack, Stehn, and Edlén’s new definition, m, σ, w1, and λ are the spectral order, the grating constant, the quantity related to the width of the first Huyghens zone on the grating and the wavelength under consideration, respectively. I and I0 are the intensity at the center of the image and at an image point displaced by (−Δλ) in the direction of the run of the spectrum from the center of the image, respectively.

Fig. 3
Fig. 3

The resolving power R plotted against the grating width W. The straight line, dashed curve and solid curve correspond to the plane grating, concave grating in an in-plane mounting, and the concave grating in an off-plane mounting R2z02 ≫ (W/4)2, respectively. For the solid curve the upper scale of the abscissa and the scale of the left ordinate are used and for the dashed curve the lower scale of the abscissa and the scale of the right ordinate are used. Both scales can be used for the straight line, m, R, z0, and W, are the spectral order, the radius of curvature of the grating, the distance between the center of the slit and the Rowland plane, and the width of the grating, respectively. w1, and η are the quantities related to the width of the first Huyghens zone on the grating in an off-plane mounting and in an in-plane mounting, respectively.

Equations (97)

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F = A P + B P + w m λ σ ,
[ A P ] 2 = ( x u ) 2 + ( y w ) 2 + ( z l ) 2
[ P B ] 2 = ( x u ) 2 + ( y w ) 2 + ( z l ) 2 .
[ P B ] 2 = r 2 + z 2 + ( u 2 + w 2 + l 2 ) 2 u r cos β 2 w r sin β 2 z l ,
( u R ) 2 + w 2 + l 2 = R 2 , u = R ± [ R 2 ( w 2 + l 2 ) ] 1 2 .
u = w 2 + l 2 2 R + ( w 2 + l 2 ) 2 8 R 3 + ( w 2 + l 2 ) 3 16 R 5 + .
[ A P ] 2 = ( r w sin α ) 2 + w 2 ( cos 2 α r cos α R ) + l 2 ( 1 r cos α R ) 2 z l + z 2 + ( w 2 + l 2 ) 2 4 R 2 ( 1 r cos α R ) ( 1 + w 2 + l 2 2 R 2 + ) .
A P = r ( 1 + z 2 r 2 ) 1 2 w sin α ( 1 + z 2 r 2 ) 1 2 l z r ( 1 + z 2 r 2 ) 1 2 + 1 2 w 2 ( cos 2 α r cos α R ) n = 0 ( w sin α r ) n + 1 2 l 2 ( 1 r cos α R ) + 1 2 l 2 w sin α r ( 1 r cos α R ) + l w z sin α r 2 w 2 8 r ( cos 2 α r cos α R ) [ 1 2 w 2 ( cos 2 α r cos α R ) + l 2 ( 1 r cos α R ) + z 2 2 2 z l r ] n = 0 ( n + 1 ) ( n + 2 ) ( w sin α r ) n + 1 2 l 2 w 2 sin 2 α r 2 ( 1 r cos α R ) + 1 2 w 2 sin 2 α r 3 ( z 2 2 z l ) + ( w 2 + l 2 ) 2 8 R 2 ( 1 r cos α R ) l 2 8 r [ { l ( 1 r cos α R ) 2 z r } 2 + 2 z 2 r ( 1 r cos α R ) ] + O ( w 5 R 4 ) ,
F = A P + P B + w m λ σ = r ( 1 + z 2 r 2 ) 1 2 + r ( 1 + z 2 r 2 ) 1 2 + w [ m λ σ ( 1 + z 2 r 2 ) 1 2 sin α ( 1 + z 2 r 2 ) 1 2 sin β ] l [ z r ( 1 + z 2 r 2 ) 1 2 + z r ( 1 + z 2 r 2 ) 1 2 ] + 1 2 w 2 n = 0 w n [ ( sin α r ) n ( cos 2 α r cos α R ) + ( sin β r ) n ( cos 2 β r cos β R ) ] + 1 2 l 2 ( 1 r cos α R + 1 r cos β R ) + [ 1 2 l 2 w { sin α r ( 1 r cos α R ) + sin β r ( 1 r cos β R ) } l w ( z sin α r 2 + z sin β r 2 ) ] w 2 8 [ 1 r ( cos 2 α r cos α R ) { 1 2 w 2 ( cos 2 α r cos α R ) + l 2 ( 1 r cos α R ) + z 2 r 2 z l r } · n = 0 ( n + 1 ) ( n + 2 ) ( w sin α r ) n + 1 r ( cos 2 β r cos β R ) { 1 2 w 2 ( cos 2 β r cos β R ) + l 2 ( 1 r cos β R ) + z 2 r 2 z l r } · n = 0 ( n + 1 ) ( n + 2 ) ( w sin β r ) n } + [ 1 2 l 2 w 2 { sin 2 α r 2 ( 1 r cos α R ) + sin 2 β r 2 ( 1 r cos β R ) } l w 2 ( z sin 2 α r 3 + z sin 2 β r 3 ) + 1 2 w 2 ( z 2 sin 2 α r 3 + z 2 sin 2 β r 3 ) + ( w 2 + l 2 ) 2 8 R 2 ( 1 r cos α R + 1 r cos β R ) ] l 2 8 [ 1 r { l ( 1 r cos α R ) 2 z r } 2 + 2 z 2 r 2 ( 1 r cos α R ) + 1 r { l ( 1 r cos β R ) 2 z r } 2 + 2 z 2 r 2 ( 1 r cos β R ) ] + O ( w 5 R 4 ) .
F w = 0
F l = 0 .
F w = 0 = [ m λ σ ( 1 + z 2 r 2 ) 1 2 sin α ( 1 + z 2 r 2 ) 1 2 sin β ] + 1 2 n = 0 ( n + 2 ) w n + 1 [ ( sin α r ) n ( cos 2 α r cos α R ) + ( sin β r ) n ( cos 2 β r cos β R ) ] + [ 1 2 l 2 { sin α r ( 1 r cos α R ) + sin β r ( 1 r cos β R ) } l ( z sin α r 2 + z sin β r 2 ) ] 1 16 r ( cos 2 α r cos α R ) [ ( cos 2 α r cos α R ) n = 0 ( n + 1 ) ( n + 2 ) ( n + 4 ) w n + 3 ( sin α r ) n + 2 { l 2 ( 1 r cos α R ) + z 2 r 2 z l r } n = 0 ( n + 1 ) ( n + 2 ) 2 w n + 1 ( sin α r ) n ] 1 16 r ( cos 2 β r cos β R ) [ ( cos 2 β r cos β R ) n = 0 ( n + 1 ) ( n + 2 ) ( n + 4 ) w n + 3 ( sin β r ) n + 2 { l 2 ( 1 r cos β R ) + z 2 r 2 z l r } n = 0 ( n + 1 ) ( n + 2 ) 2 w n + 1 ( sin β r ) n ] + w [ l 2 { sin 2 α r 2 ( 1 r cos α R ) + sin 2 β r 2 ( 1 r cos β R ) } 2 l ( z sin 2 α r 3 + z sin 2 β r 3 ) + z 2 sin 2 α r 3 + z 2 sin 2 β r 3 + w 2 + l 2 2 R 2 ( 1 r cos α R + 1 r cos β R ) ] + O ( w 4 R 4 ) ,
F l = 0 = [ z r ( 1 + z 2 r 2 ) 1 2 + z r ( 1 + z 2 r 2 ) 1 2 ] + l ( 1 r cos α R + 1 r cos β R ) + w [ l { sin α r ( 1 r cos α R ) + sin β r ( 1 r cos β R ) } ( z sin α r 2 + z sin β r 2 ) ] w 2 4 [ 1 r ( cos 2 α r cos α R ) { l ( 1 r cos α R ) z r } n = 0 ( n + 1 ) ( n + 2 ) ( sin α r ) n + 1 r ( cos 2 β r cos β R ) { l ( 1 r cos β R ) z r } n = 0 ( n + 1 ) ( n + 2 ) ( w sin β r ) n ] + [ l w 2 { sin 2 α r 2 ( 1 r cos α R ) + sin 2 β r 2 ( 1 r cos β R ) } w 2 ( z sin 2 α r 3 + z sin 2 β r 3 ) + l ( w 2 + l 2 ) 2 R 2 ( 1 r cos α R + 1 r cos β R ) ] l 2 [ 1 r { l ( 1 r cos α R ) 2 z r } { l ( 1 r cos α R ) z r } + z 2 r 2 ( 1 r cos α R ) + 1 r { l ( 1 r cos β R ) 2 z r } { l ( 1 r cos β R ) z r } + z 2 r 2 ( 1 r cos β R ) ] + O ( w 4 R 4 ) .
m λ σ ( 1 + z 2 r 2 ) 1 2 sin α ( 1 + z 0 2 r 0 2 ) 1 2 sin β 0 = 0 ,
z r ( 1 + z 2 r 2 ) 1 2 + z 0 r 0 ( 1 + z 0 2 r 0 2 ) 1 2 = 0 ,
( 1 + z 2 r 2 ) 1 2 ( sin α + sin β 0 ) = m λ σ
z r = z 0 r 0 .
cos β 0 · Δ β + 1 2 n = 0 ( n + 2 ) w n + 1 [ ( sin α r ) n ( cos 2 α r cos α R ) + ( sin β r ) n ( cos 2 β r cos β R ) ] + O ( w 2 R 2 ) = 0
Δ ( z r ) + l ( 1 r cos α R + 1 r cos β R ) + O ( w 2 R 2 ) = 0 .
cos 2 α r cos α R + cos 2 β r cos β R = 0 .
r = R cos α and r = R cos β ,
r = and r = R cos 2 β cos α + cos β .
F = R [ ( 1 + z 2 R 2 cos 2 α ) 1 2 cos α + ( 1 + z 2 R 2 cos 2 β ) 1 2 cos β ] + w [ m λ σ ( 1 + z 2 R 2 cos 2 α ) 1 2 sin α ( 1 + z 2 R 2 cos 2 β ) 1 2 sin β ] l R [ z cos α ( 1 + z 2 R 2 cos 2 α ) 1 2 + z cos β ( 1 + z 2 R 2 cos 2 β ) 1 2 ] + l 2 2 R ( sin 2 α cos α + sin 2 β cos β ) + w 2 R 2 [ l 2 ( sin 3 α cos 2 α + sin 3 β cos 2 β ) 2 l ( z sin α cos 2 α + z sin β cos 2 β ) ] + 1 2 R 3 [ l 2 w 2 ( sin 4 α cos 3 α + sin 4 β cos 3 β ) 2 l w 2 ( z sin 2 α cos 3 α + z sin 2 β cos 3 β ) + w 2 ( z 2 sin 2 α cos 3 α + z 2 sin 2 β cos 3 β ) + ( w 2 + l 2 ) 2 4 ( sin 2 α cos α + sin 2 β cos β ) l 2 4 cos 3 α { ( l sin 2 α 2 z ) 2 + 2 z 2 sin 2 α } l 2 4 cos 3 β { ( l sin 2 β 2 z ) 2 + 2 z 2 sin 2 β } ] + O ( w 5 R 4 ) ,
F w = 0 = [ m λ σ ( 1 + z 2 R 2 cos 2 α ) 1 2 sin α ( 1 + z 2 R 2 cos 2 β ) 1 2 sin β ] + 1 2 R 2 [ l 2 ( sin 3 α cos 2 α + sin 3 β cos 2 β ) 2 l ( z sin α cos 2 α + z sin β cos 2 β ) ] + w R 3 [ l 2 ( sin 4 α cos 3 α + sin 4 β cos 3 β ) 2 l ( z sin 2 α cos 3 α + z sin 2 β cos 3 β ) + ( z 2 sin 2 α cos 3 α + z 2 sin 2 β cos 3 β ) + 1 2 ( w 2 + l 2 ) ( sin 2 α cos α + sin 2 β cos β ) ] + O ( w 4 R 4 ) ,
F l = 0 = 1 R [ z cos α ( 1 + z 2 R 2 cos 2 α ) 1 2 + z cos β ( 1 + z 2 R 2 cos 2 β ) 1 2 ] + l R ( sin 2 α cos α + sin 2 β cos β ) + w R 2 [ l ( sin 3 α cos 2 α + sin 3 β cos 2 β ) ( z sin α cos 2 α + z sin β cos 2 β ) ] + 1 R 3 [ l w 2 ( sin 4 α cos 3 α + sin 4 β cos 3 β ) w 2 ( z sin 2 α cos 3 α + z sin 2 β cos 3 β ) + 1 2 l ( w 2 + l 2 ) ( sin 2 α cos α + sin 2 β cos β ) l cos 3 α { ( l sin 2 α 2 z ) ( l sin 2 α z ) + z 2 sin 2 α } l cos 3 β { ( l sin 2 β 2 z ) ( l sin 2 β z ) + z 2 sin 2 β } ] + O ( w 4 R 4 ) ,
( 1 + z 2 R 2 cos 2 α ) 1 2 ( sin α + sin β 0 ) = m λ σ
z cos α = z 0 cos β 0 .
Δ α = Δ z r tan φ = Δ z · φ R cos α 0 .
l = ( sin 2 α cos α + sin 2 β cos β ) 1 [ 1 + w R · sin 3 α cos 2 β + sin 3 β cos 2 α cos α cos β ( sin 2 α cos β + sin 2 β cos α ) ] 1 · [ z cos α + z cos β + w R ( z sin α cos 2 α + z sin β cos 2 β ) + O ( w 3 R 2 ) ] = 1 A [ z cos β + z cos α + w R A sin ( α β ) ( z sin 2 β z sin 2 α ) + O ( w 3 R 2 ) ] ,
A = sin 2 α cos β + cos α sin 2 β
w ( sin 3 α cos 2 β + sin 3 β cos 2 α ) R A cos α cos β < 1
l = 1 A 0 [ Δ z cos β 0 + Δ z cos α 0 + w R A 0 sin ( α 0 β 0 ) × ( z 0 A 0 cos α 0 + Δ z sin 2 β 0 Δ z sin 2 α 0 ) + O ( w 3 R 2 ) ] ,
A 0 = sin 2 α 0 cos β 0 + cos α 0 sin 2 β 0 .
m λ σ ( 1 + Z 0 2 R 2 cos 2 α 0 ) 1 2 ( sin α 0 + sin β 0 ) ( cos α 0 · Δ α + cos β 0 · Δ β ) + 1 2 R 2 [ l 2 ( sin 3 α 0 cos 2 α 0 + sin 3 β 0 cos 2 β 0 ) + Z 0 2 cos 2 α 0 ( sin α 0 + sin β 0 ) + 2 z 0 cos α 0 ( Δ z · tan α Δ z · tan β 0 ) + ( Δ z ) 2 sin α 0 cos 2 α 0 + ( Δ z ) 2 sin β 0 cos 2 β 0 2 z 0 l sin ( α 0 β 0 ) cos 2 α 0 cos β 0 2 l ( Δ z sin α 0 cos 2 α 0 + Δ z sin β 0 cos 2 β 0 ) ] + w R 3 [ l 2 ( sin 4 α 0 cos 3 α 0 + sin 4 β 0 cos 3 β 0 ) + Z 0 2 A 0 cos 3 α 0 cos β 0 + 2 z 0 Δ z sin 2 α 0 cos 3 α 0 2 z 0 Δ z sin 2 β 0 cos α 0 cos 2 β 0 + ( Δ z ) 2 sin 2 α 0 cos 3 α 0 + ( Δ z ) 2 sin 2 β 0 cos 3 β 0 2 l Z 0 cos 3 α 0 cos 2 β 0 sin ( α 0 + β 0 ) sin ( α 0 β 0 ) 2 l ( Δ z sin 2 α 0 cos 3 α 0 + Δ z sin 2 β 0 cos 3 β 0 ) R φ tan α 0 · Δ z ] + w ( w 2 + l 2 ) A 0 2 R 3 cos α 0 cos β 0 + O ( w 4 R 4 ) = 0 .
Δ β · cos β 0 = 1 2 R 2 [ 2 z 0 Δ z { sin α 0 cos 2 α 0 R φ z 0 sin ( α 0 β 0 ) A 0 cos 2 α 0 } B z 0 Δ z + C ( Δ z ) 2 + D ( Δ z ) 2 E Δ z Δ z ] + w R 3 [ z 0 Δ z { tan α 0 ( 2 sin α 0 cos 2 α 0 R φ z 0 ) G } H z 0 Δ z + J ( Δ z ) 2 + K ( Δ z ) 2 M Δ z Δ z + 1 2 sec 3 α 0 sec β 0 { w 2 A 0 cos 2 α 0 + 4 z 0 2 A 0 4 z 0 2 A 0 sin 2 ( α 0 β 0 ) } ] ,
B = 2 cos α 0 cos β [ sin β 0 + sin ( α 0 β 0 ) A 0 ] , C = sec 2 α 0 [ sin α 0 sin α 0 cos β 0 A 0 sin 2 β 0 cos α 0 sin ( α 0 β 0 ) A 0 2 ] , D = sec 2 β 0 [ sin β 0 sin β 0 cos α 0 A 0 + sin 2 α 0 cos β 0 sin ( α 0 β 0 ) A 0 2 ] , E = 2 A 0 2 sin α 0 sin β 0 ( sin α 0 sin β 0 ) , G = 1 A 0 sec 3 α 0 sec β 0 sin ( α 0 β 0 ) [ 2 sin ( α 0 + β 0 ) + sin α 0 cos β 0 cos 2 α 0 cos 2 β 0 A 0 + 1 A 0 sin 2 β 0 cos α 0 sin ( α 0 β 0 ) ] , H = sec 2 α 0 sec 2 β 0 [ 2 sin 2 β 0 cos α 0 + 1 A 0 sin ( α 0 β 0 ) { 2 sin ( α 0 + β 0 ) + sin β 0 cos α 0 cos 2 α 0 cos 2 β 0 A 0 } 1 A 0 2 sin 2 α 0 cos β 0 sin 2 ( α 0 β 0 ) ] , J = cos 2 β 0 ( sin 4 α 0 cos 3 α 0 + sin 4 β 0 cos 3 β 0 ) + sin 2 α 0 cos 3 α 0 + cos β 0 2 A 0 cos 3 α 0 ( 1 5 sin 2 α 0 ) 1 A 0 2 sin ( α 0 β 0 ) sin 2 β 0 ( sin α 0 cos 2 α 0 cos β 0 A 0 ) , K = cos 2 α 0 ( sin 4 α 0 cos 3 α 0 + sin 4 β 0 cos 3 β 0 ) + sin 2 β 0 cos 3 β 0 + cos α 0 2 A 0 cos 3 β 0 ( 1 5 sin 2 β 0 ) + 1 A 0 2 sin ( α 0 β 0 ) sin 2 α 0 ( sin β 0 cos 2 β 0 cos β 0 A 0 ) , M = 2 cos α 0 cos β 0 ( sin 4 α 0 cos 3 α 0 + sin 4 β 0 cos 3 β 0 ) + 1 A 0 ( 1 2 tan 2 α 0 2 tan 2 β 0 ) + sin ( α 0 β 0 ) A 0 2 ( sin 3 α 0 cos 2 α 0 sin 3 β 0 cos 2 β 0 sin 2 α 0 cos β 0 cos α 0 sin 2 β 0 A 0 ) ,
a 0 Δ z a 0 , A 0 L 2 cos α 0 Δ z cos β 0 cos α 0 Δ z A 0 L 2 cos α 0 Δ z cos β 0 cos α 0 ,
φ = z 0 R tan α 0 sec α 0
Δ β 0 · cos β 0 = 1 2 R 2 [ C ( Δ z ) 2 + D ( Δ z ) 2 E Δ z Δ z ] + w R 3 [ J ( Δ z ) 2 + K ( Δ z ) 2 + M Δ z Δ z + w 2 A 0 2 cos α 0 cos β 0 ] ,
Δ β 0 · cos β 0 = 1 2 R 2 [ 2 z 0 Δ z sin ( α 0 β 0 ) A 0 cos 2 α 0 B z 0 Δ z + C ( Δ z ) 2 + D ( Δ z ) 2 E Δ z Δ z ] + w R 3 [ { tan 2 α 0 sec α 0 G ) } z 0 Δ z H z 0 Δ z + J ( Δ z ) 2 + K ( Δ z ) 2 + M Δ z Δ z + 1 2 cos 3 α 0 cos β 0 { w 2 A 0 cos 2 α 0 + 4 z 0 2 A 0 4 z 0 2 A 0 sin 2 ( α 0 β 0 ) } ] ,
a 0 Δ z a 0 , A 0 L 2 cos α 0 Δ z cos β 0 cos α 0 Δ z A 0 L 2 cos α 0 Δ z cos β 0 cos α 0
Δ β 0 · cos β 0 = C 2 R 2 ( Δ z ) 2 + w R 3 [ J ( Δ z ) 2 + w 2 A 0 2 cos α 0 cos β 0 ] ,
Δ β 0 · cos β 0 = 1 2 R 2 [ C ( Δ z ) 2 2 z 0 Δ z sin ( α 0 β 0 ) A 0 cos 2 α 0 ] + w R 3 [ ( tan 2 α 0 sec α 0 G ) z 0 Δ z + J ( Δ z ) 2 + 1 2 cos 3 α 0 cos β 0 { w 2 A 0 cos 2 α 0 + 4 z 0 2 A 0 4 z 0 2 A 0 sin 2 ( α 0 β 0 ) } ] ,
| Δ z | A 0 L 2 cos β 0 if A 0 L 2 cos α 0 a 0 , | Δ z | a 0 if A 0 L 2 cos α 0 a 0
Δ z = l A 0 cos α 0 Δ z cos β 0 cos α 0
[ Δ z ] ast = ( L A 0 2 cos α 0 Δ z cos β 0 cos α 0 ) ( L A 0 2 cos α 0 Δ z cos β 0 cos α 0 ) = L A 0 cos α 0 = L ( sin 2 α 0 cos β 0 cos α 0 + sin 2 β 0 ) .
2 a 0 cos β 0 cos α 0 + L A 0 cos α 0 .
1 2 W 1 2 W d w 1 2 L 1 2 L d l δ n e 2 π i F / λ
λ 4 ( 2 j 3 ) < [ F ] w = 0 [ F ] w = l = 0 < λ 4 ( 2 j 1 )
l j 2 2 R ( sin 2 α 0 cos α 0 + sin 2 β 0 cos β 0 ) + O ( w 4 R 3 ) = λ 4 ( 2 j 1 ) , l j = ± [ ( j 1 2 ) R λ cos α 0 cos β 0 A 0 ] 1 2 ,
L 1 = 2 | l 1 | = [ 2 R λ cos α 0 cos β 0 A 0 ] 1 2 ,
Case ( 1 ) , z 0 = 0
W opt = 2.36 η ,
η = [ 4 λ R 3 cos α 0 cos β 0 π A 0 ] 1 4 ;
R opt = 0.92 W opt m / σ ,
R = 0.75 W opt m / σ .
Case ( 2 ) , R 2 z 0 2 ( W / 4 ) 2
F = C + w m Δ λ σ + w 2 z 0 2 2 R 3 cos α 0 ( sin 2 α 0 cos α 0 + sin 2 β 0 cos β 0 ) + w 4 8 R 3 ( sin 2 α 0 cos α 0 + sin 2 β 0 cos β 0 ) + O ( w 5 R 4 ) C + w m Δ λ σ + w 2 z 0 2 A 0 2 R 3 cos 3 α 0 cos β 0 ,
C = R [ ( 1 + z 0 2 R 2 cos 2 α 0 ) 1 2 cos α 0 + ( 1 + z 0 2 R 2 cos 2 β ) 1 2 cos β ] , m λ σ ( 1 + z 0 2 R 2 cos 2 α 0 ) 1 2 sin α 0 ( 1 + z 0 R 2 cos 2 β ) 1 2 sin β = m λ σ ( 1 + z 0 2 R 2 cos 2 α 0 ) 1 2 ( sin α 0 + sin β 0 ) cos β 0 · Δ β + O ( w 4 R 4 ) = m Δ λ σ + O ( w 4 R 4 ) . 23
I = | n = 1 2 N 1 2 N δ n e 2 π i F / λ | 2 = | n = 1 2 N 1 2 N δ n exp ( 2 π i C / λ + 1 2 i π ξ 2 1 2 i π 2 ) | 2 ,
I = | ( W / 2 w 1 ) + ( W / 2 w 1 ) + d ξ w 1 δ n exp ( 2 π i C / λ + 1 2 i π ξ 2 1 2 i π 2 ) | 2 .
I = | w 1 δ n exp ( 2 π i C / λ 1 2 i π 2 ) ( W / 2 w 1 ) + ( W / 2 w 1 ) + exp ( 1 2 i π ξ 2 ) d ξ | 2 = ( w 1 δ n ) 2 | ( W / 2 w 1 ) + ( W / 2 w 1 ) + exp ( 1 2 i π ξ 2 ) d ξ | 2 = ( w 1 δ n ) 2 [ { 0 ( W / 2 w 1 ) + cos ( 1 2 π ξ 2 ) d ξ + 0 ( W / 2 w 1 ) cos ( 1 2 π ξ 2 ) d ξ } 2 + { 0 ( W / 2 w 1 ) + sin ( 1 2 π ξ 2 ) d ξ + 0 ( W / 2 w ) 1 sin ( 1 2 π ξ 2 ) d ξ } 2 ] .
r = 2 m σ λ ( Δ λ ) r 2 w 1 = m ( Δ λ ) r σ λ w 1 .
R λ ( Δ λ ) r = m w 1 σ r = m N ( w 1 / r ) W .
W opt = 2.2 w 1 ,
R opt = ( 2.1 w 1 / 2.2 w 1 ) m N opt = 0.95 W opt m / σ .
Δ β cos β 0 = ( m / σ ) Δ λ = w 3 A 0 / ( 2 R 3 cos α 0 cos β 0 ) .
m ( Δ λ ) r σ = w 0 3 R 3 A 0 cos α 0 cos β 0 = n 3 W opt 3 8 R 3 A 0 cos α 0 cos β 0 .
R opt = 0.92 W opt m σ = 8 m R 3 λ σ n 3 W opt 3 cos α 0 cos β 0 A 0 .
n 3 = 0.22 or n = 0.604 .
n = 0.217
Δ β cos β 0 = m Δ λ σ = 2 w z 0 2 R 3 A 0 cos 3 α 0 cos β 0 , w 0 = 1 2 n W opt , R opt = 0.95 W opt m σ = m R 3 λ 2 n W opt z 0 2 σ cos 3 α 0 cos β 0 A 0 ,
( Δ λ ) h ( Δ λ ) r = 0.22 [ σ w 3 A 0 m R 3 cos α 0 cos β 0 ] w = 1 2 W opt [ from ( 17 a ) ] = 0.275 σ W opt 3 A 0 m R 3 cos α 0 cos β 0 ,
( Δ λ ) h ( Δ λ ) r = 0.22 [ σ w m R 3 cos 3 α 0 cos β 0 { w 2 A 0 cos 2 α 0 + 4 z 0 2 A 0 4 z 0 2 A 0 sin 2 ( α 0 β 0 ) } ] w = 1 2 W opt [ from ( 17 b ) ] = 0.0275 σ W opt A 0 m R 3 cos α 0 cos β 0 × [ W opt 2 + 16 z 0 2 cos 2 α 0 { 1 sin 2 ( α 0 β 0 ) A 0 2 } ] .
( Δ λ ) ± = [ σ m cos β 0 Δ β ] w = 0 , Δ z = ± a = σ a 2 m R 2 cos 2 α 0 [ 2 z 0 sin ( α 0 β 0 ) A 0 + a { sin α 0 sin α 0 cos β 0 A 0 sin 2 β 0 cos α 0 sin ( α 0 β 0 ) A 0 2 } ] .
( Δ λ ) H [ ( Δ λ ) h 2 + ( Δ λ ) a 2 / 4 ] 1 2 .
R opt λ ( Δ λ ) H .
W 1 2 z 0 2 A 0 8 R 3 cos 3 α 0 cos β 0 + W 1 4 A 0 128 R 3 cos α 0 cos β 0 = λ 4 ,
W 1 = 2 [ { 4 z 0 4 cos 4 α 0 + 2 R 3 λ cos α 0 cos β 0 A 0 } 1 2 2 z 0 2 cos 3 α 0 ] 1 2 ,
W 1 = 2.38 [ R 3 λ cos α 0 cos β 0 A 0 ] 1 4
W 1 2 [ R 3 λ 2 z 0 2 cos 3 α 0 cos β 0 A 0 ] 1 2
W opt = 1.06 W 1 = 2.12 [ { 4 z 0 4 cos 4 α 0 + 2 R 3 λ cos α 0 cos β 0 A 0 } 1 2 2 z 0 2 cos 2 α 0 ] 1 2 .
V / x = μ , V / y = τ , V / z = ν , V / x = μ , V / y = τ , V / z = ν ,
μ 2 + τ 2 + ν 2 = 1 , μ 2 + τ 2 + ν 2 = 1 ,
V 1 / w = τ 0 , V 1 / l = ν 0 ,
V 2 / w = τ , V 2 / l = ν ,
V / w = ( V 1 + V 2 ) / w = τ 0 + τ , V / l = ( V 1 + V 2 ) / l = ν 0 + ν .
V / w = Δ τ , V / l = Δ ν ,
Δ s = R Δ τ = R ( V / w ) , Δ z = R cos β Δ ν = R cos β ( V / l ) . }
V 1 = A P = [ ( x u ) 2 + ( y w ) 2 + ( z l ) 2 ] 1 2 ,
V 2 = P B + f ( w , l ) = [ ( x u ) 2 + ( y w ) 2 + ( z l ) 2 ] 1 2 + f ( w , l ) ,
f ( w , l ) ( m λ / σ ) w + g ( w ) ,
h ( w , l , f ) h ( w , g ) = 0
Δ s = R [ ( A P + P B + m λ w / σ ) / w + g ( w ) / w ] = R [ F / w + g ( w ) / w ] ,
Δ z = R cos β ( A P + P B ) / w = R cos β F / w ,
( Δ λ ) h = σ cos β 4 m R 3 [ 4 w ( w 2 + l 2 ) ] ( sin 2 α cos α + sin 2 β cos β ) .
R opt = 8 m λ R 3 σ W opt 3 cos β ( sin 2 α cos α + sin 2 β cos β ) 1 = π m 15.51 σ cos β W opt ,