Abstract

Geometrical optic techniques are used to analyze and compare the symmetrical spherical mirror to the unsymmetrical spherical-mirror Czerny–Turner spectrometer. The aberration problems due to diffraction from the grating are analyzed and methods of partial correction of the aberrations are derived. The flux and resolution advantage of gratings with high blaze angles used in the unsymmetrical spherical-mirror Czerny–Turner is shown. A design and ray tracing routine employing a digital computer is utilized to illustrate the geometric effects of the diffraction grating and the partial corrective measures. Slit curvature is analyzed numerically and some general results are abstracted from the numerical data. It may be inferred from the results of theory and numerical calculations that the unsymmetrical Czerny–Turner spectrometer using two spherical mirrors can be made superior to a similar symmetrical Czerny–Turner spectrometer. A comparison of luminosity is made between the Czerny–Turner spectrometer, utilizing a high blaze grating, and various interf erometric and modulating spectrometers and it is shown that the luminosity of the Czerny–Turner spectrometer is comparable or superior.

© 1964 Optical Society of America

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References

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  1. M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
    [Crossref]
  2. W. G. Fastie, J. Opt. Soc. Am. 42, 641, 647 (1952).
    [Crossref]
  3. H. Ebert, Wiedemann Ann. 38, 489 (1889).
    [Crossref]
  4. K. Kudo, Sci. Light (Tokyo) 9, 1 (1960).
  5. W. Leo, Z. Angew. Phys. 8, 196 (1956).
  6. W. Leo, Z. Instrumentenk. 66, 240 (1958);Z. Instrumentenk,  709 (1962).
  7. W. R. Hamilton, Mathematical Papers, Geometrical Optics (Cambridge University Press, London, 1931);Mathematical Papers, Geometrical Optics Vol.  I, Chap. 17, p. 168;J. L. Synge, Geometrical Optics, Cambridge University Press (1937);J. L. Synge, Hamilton’s Method in Geometrical Optics (The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland, 1951);J. L. Synge, J. Opt. Soc. Am. 27, 75 (1937);M. Herzberger, ibid., p. 133.
    [Crossref]
  8. P. Jacquinot and C. Dufour, J. Rech. Centre, Natl. Rech. Sci. Lab, Bellevue (Paris) No. 6, 91 (1948).
  9. P. Jacquinot, J. Opt. Soc. Am. 44, 761 (1954).
    [Crossref]
  10. G. W. Stroke, J. Opt. Soc. Am. 51, 1321 (1961). Stroke has stressed with the help of numerical values, the advantages in using high-blazed gratings in spectrometers and spectrographs.See also, G. W. Stroke, Progr. Opt. 2, 3 (1963).
    [Crossref]
  11. V. I. Malyshev and S. G. Rautian, Opt. i Spectroskopiya 6, 550 (1959)[English transl.: Opt. Spectry 6, 351 (1959)].
  12. R. Chabbal, Rev. Opt. 37, 49, 336, 501 (1958).
  13. P. Jacquinot, Rept. Progr. Phys.23, 267 (1960).
    [Crossref]
  14. H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945).
    [Crossref]
  15. G. Rosendahl, J. Opt. Soc. Am. 52, 412 (1962), has derived Eq. (15) without the dependence upon the radii of curvature, by utilizing a different geometrical optic technique.W. G. Fastie (private communication and U. S. Patent3,011,391) has derived an equation similar to (17) but without the dependence upon the radii of curvature.
    [Crossref]
  16. W. G. Fastie experimentally noted that the effect of coma may be corrected by tilting the second mirror [Symp. Interferometry, No. 11, 243 (1960), Natl. Phys. Lab., G. Brit.].The large, high-resolution Ebert spectrometer, which has a focal length of 183 cm [W. G. Fastie, H. M. Crosswhite, and P. Gloersen, J. Opt. Soc. Am. 48, 106 (1958)], achieved a resolving power in excess of 500,000. This was done at an f number >20, i.e., a projected grating width ⪝9 cm. A re-examination of the lines revealed an asymmetry and a coma-like flare (private communication). After a discussion with one of the authors, R. Brower (private communication) of Brower Laboratories, Wellesley Hills, Massachusetts set up a point source, two spherical mirrors, and grating. By using the grating in the zeroth order and aligning the instrument, the effect of coma is eliminated with a symmetrical arrangement; to eliminate the effect of coma for the first or higher orders of diffraction of an echellette grating, the second mirror had to be tilted. When R2is made <R1, Brower found that the image quality is superior to that with R1=R2.
    [Crossref]
  17. K. Nienhuis, thesis, University of Groningen, 1948.
  18. N. G. Van Kampen, Physica 15, 575 (1949).
    [Crossref]
  19. K. Nienhuis and B. R. A. Nijborn, Physica 15, 590 (1949).
    [Crossref]
  20. M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), p. 477.
  21. M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin, 1963).
  22. E. Wolf, Rept. Progr. Phys.14, 95 (1951).
    [Crossref]
  23. To illustrate the sensitivity of the slit curvature upon the distance from the entrance slit to the collimating mirror: for one of the instruments analyzed, a shift of 0.3 mm distorted the slit curvature from a circle to a smooth curve which deviates from a circle by ≈6 μ at the slit center to ≈0.2 μ, 2.75 cm above or below the slit center.
  24. J. E. Mack, D. P. McNutt, F. L. Roesler, and R. Chabbal, Appl. Opt. 2, 855 (1963).
    [Crossref]
  25. R. G. Greenler, J. Opt. Soc. Am. 47, 642 (1957).
    [Crossref] [PubMed]
  26. We shall be interested only in resolution ≤ 1 cm−1 in the above stated wavelength region. This value of resolution, to the author’s knowledge, has not been achieved by interferometers employing the Fourier transformation techniques. In the spectral region where photomultiplying tubes can be used as detectors (⪝1μ) the signal/noise advantage of these types of interferometers is nullified. The argument shall proceed under the assumption that higher resolution will be obtained in the future.
  27. J. Connes, J. Phys Radium19, 197 (1958); Symposium on Interferometry, National Physics Laboratory, Great Britian No. 11, 409 (1960).
  28. A. Girard, Opt. Acta,  7, 81 (1960).
    [Crossref]
  29. A. Girard, Appl. Opt. 2, 79 (1963).
    [Crossref]
  30. G. Stroke (private communication) informed us that he gave Girard a grating blazed at 45°. Evidently Girard did not use it in the previous referenced papers since the scanning range was 37°<θ<22° in a Littrow system. If the grille spectrometer is used with a grating of 45° blaze angle, then the luminosity advantage of the grille spectrometer over the Czerny–Turner monochromator would be ≈30 to 1.3.

1963 (2)

1962 (1)

1961 (1)

1960 (3)

W. G. Fastie experimentally noted that the effect of coma may be corrected by tilting the second mirror [Symp. Interferometry, No. 11, 243 (1960), Natl. Phys. Lab., G. Brit.].The large, high-resolution Ebert spectrometer, which has a focal length of 183 cm [W. G. Fastie, H. M. Crosswhite, and P. Gloersen, J. Opt. Soc. Am. 48, 106 (1958)], achieved a resolving power in excess of 500,000. This was done at an f number >20, i.e., a projected grating width ⪝9 cm. A re-examination of the lines revealed an asymmetry and a coma-like flare (private communication). After a discussion with one of the authors, R. Brower (private communication) of Brower Laboratories, Wellesley Hills, Massachusetts set up a point source, two spherical mirrors, and grating. By using the grating in the zeroth order and aligning the instrument, the effect of coma is eliminated with a symmetrical arrangement; to eliminate the effect of coma for the first or higher orders of diffraction of an echellette grating, the second mirror had to be tilted. When R2is made <R1, Brower found that the image quality is superior to that with R1=R2.
[Crossref]

K. Kudo, Sci. Light (Tokyo) 9, 1 (1960).

A. Girard, Opt. Acta,  7, 81 (1960).
[Crossref]

1959 (1)

V. I. Malyshev and S. G. Rautian, Opt. i Spectroskopiya 6, 550 (1959)[English transl.: Opt. Spectry 6, 351 (1959)].

1958 (2)

R. Chabbal, Rev. Opt. 37, 49, 336, 501 (1958).

W. Leo, Z. Instrumentenk. 66, 240 (1958);Z. Instrumentenk,  709 (1962).

1957 (1)

1956 (1)

W. Leo, Z. Angew. Phys. 8, 196 (1956).

1954 (1)

1952 (1)

1949 (2)

N. G. Van Kampen, Physica 15, 575 (1949).
[Crossref]

K. Nienhuis and B. R. A. Nijborn, Physica 15, 590 (1949).
[Crossref]

1948 (1)

P. Jacquinot and C. Dufour, J. Rech. Centre, Natl. Rech. Sci. Lab, Bellevue (Paris) No. 6, 91 (1948).

1945 (1)

1930 (1)

M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
[Crossref]

1889 (1)

H. Ebert, Wiedemann Ann. 38, 489 (1889).
[Crossref]

Beutler, H. G.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), p. 477.

Cagnet, M.

M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin, 1963).

Chabbal, R.

Connes, J.

J. Connes, J. Phys Radium19, 197 (1958); Symposium on Interferometry, National Physics Laboratory, Great Britian No. 11, 409 (1960).

Czerny, M.

M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
[Crossref]

Dufour, C.

P. Jacquinot and C. Dufour, J. Rech. Centre, Natl. Rech. Sci. Lab, Bellevue (Paris) No. 6, 91 (1948).

Ebert, H.

H. Ebert, Wiedemann Ann. 38, 489 (1889).
[Crossref]

Fastie, W. G.

W. G. Fastie experimentally noted that the effect of coma may be corrected by tilting the second mirror [Symp. Interferometry, No. 11, 243 (1960), Natl. Phys. Lab., G. Brit.].The large, high-resolution Ebert spectrometer, which has a focal length of 183 cm [W. G. Fastie, H. M. Crosswhite, and P. Gloersen, J. Opt. Soc. Am. 48, 106 (1958)], achieved a resolving power in excess of 500,000. This was done at an f number >20, i.e., a projected grating width ⪝9 cm. A re-examination of the lines revealed an asymmetry and a coma-like flare (private communication). After a discussion with one of the authors, R. Brower (private communication) of Brower Laboratories, Wellesley Hills, Massachusetts set up a point source, two spherical mirrors, and grating. By using the grating in the zeroth order and aligning the instrument, the effect of coma is eliminated with a symmetrical arrangement; to eliminate the effect of coma for the first or higher orders of diffraction of an echellette grating, the second mirror had to be tilted. When R2is made <R1, Brower found that the image quality is superior to that with R1=R2.
[Crossref]

W. G. Fastie, J. Opt. Soc. Am. 42, 641, 647 (1952).
[Crossref]

Francon, M.

M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin, 1963).

Girard, A.

A. Girard, Appl. Opt. 2, 79 (1963).
[Crossref]

A. Girard, Opt. Acta,  7, 81 (1960).
[Crossref]

Greenler, R. G.

Hamilton, W. R.

W. R. Hamilton, Mathematical Papers, Geometrical Optics (Cambridge University Press, London, 1931);Mathematical Papers, Geometrical Optics Vol.  I, Chap. 17, p. 168;J. L. Synge, Geometrical Optics, Cambridge University Press (1937);J. L. Synge, Hamilton’s Method in Geometrical Optics (The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland, 1951);J. L. Synge, J. Opt. Soc. Am. 27, 75 (1937);M. Herzberger, ibid., p. 133.
[Crossref]

Jacquinot, P.

P. Jacquinot, J. Opt. Soc. Am. 44, 761 (1954).
[Crossref]

P. Jacquinot and C. Dufour, J. Rech. Centre, Natl. Rech. Sci. Lab, Bellevue (Paris) No. 6, 91 (1948).

P. Jacquinot, Rept. Progr. Phys.23, 267 (1960).
[Crossref]

Kudo, K.

K. Kudo, Sci. Light (Tokyo) 9, 1 (1960).

Leo, W.

W. Leo, Z. Instrumentenk. 66, 240 (1958);Z. Instrumentenk,  709 (1962).

W. Leo, Z. Angew. Phys. 8, 196 (1956).

Mack, J. E.

Malyshev, V. I.

V. I. Malyshev and S. G. Rautian, Opt. i Spectroskopiya 6, 550 (1959)[English transl.: Opt. Spectry 6, 351 (1959)].

McNutt, D. P.

Nienhuis, K.

K. Nienhuis and B. R. A. Nijborn, Physica 15, 590 (1949).
[Crossref]

K. Nienhuis, thesis, University of Groningen, 1948.

Nijborn, B. R. A.

K. Nienhuis and B. R. A. Nijborn, Physica 15, 590 (1949).
[Crossref]

Rautian, S. G.

V. I. Malyshev and S. G. Rautian, Opt. i Spectroskopiya 6, 550 (1959)[English transl.: Opt. Spectry 6, 351 (1959)].

Roesler, F. L.

Rosendahl, G.

Stroke, G. W.

Thrierr, J. C.

M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin, 1963).

Turner, A. F.

M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
[Crossref]

Van Kampen, N. G.

N. G. Van Kampen, Physica 15, 575 (1949).
[Crossref]

Wolf, E.

E. Wolf, Rept. Progr. Phys.14, 95 (1951).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), p. 477.

Appl. Opt. (2)

J. Opt. Soc. Am. (6)

J. Rech. Centre, Natl. Rech. Sci. Lab, Bellevue (Paris) (1)

P. Jacquinot and C. Dufour, J. Rech. Centre, Natl. Rech. Sci. Lab, Bellevue (Paris) No. 6, 91 (1948).

Opt. Acta (1)

A. Girard, Opt. Acta,  7, 81 (1960).
[Crossref]

Opt. i Spectroskopiya (1)

V. I. Malyshev and S. G. Rautian, Opt. i Spectroskopiya 6, 550 (1959)[English transl.: Opt. Spectry 6, 351 (1959)].

Physica (2)

N. G. Van Kampen, Physica 15, 575 (1949).
[Crossref]

K. Nienhuis and B. R. A. Nijborn, Physica 15, 590 (1949).
[Crossref]

Rev. Opt. (1)

R. Chabbal, Rev. Opt. 37, 49, 336, 501 (1958).

Sci. Light (Tokyo) (1)

K. Kudo, Sci. Light (Tokyo) 9, 1 (1960).

Symp. Interferometry (1)

W. G. Fastie experimentally noted that the effect of coma may be corrected by tilting the second mirror [Symp. Interferometry, No. 11, 243 (1960), Natl. Phys. Lab., G. Brit.].The large, high-resolution Ebert spectrometer, which has a focal length of 183 cm [W. G. Fastie, H. M. Crosswhite, and P. Gloersen, J. Opt. Soc. Am. 48, 106 (1958)], achieved a resolving power in excess of 500,000. This was done at an f number >20, i.e., a projected grating width ⪝9 cm. A re-examination of the lines revealed an asymmetry and a coma-like flare (private communication). After a discussion with one of the authors, R. Brower (private communication) of Brower Laboratories, Wellesley Hills, Massachusetts set up a point source, two spherical mirrors, and grating. By using the grating in the zeroth order and aligning the instrument, the effect of coma is eliminated with a symmetrical arrangement; to eliminate the effect of coma for the first or higher orders of diffraction of an echellette grating, the second mirror had to be tilted. When R2is made <R1, Brower found that the image quality is superior to that with R1=R2.
[Crossref]

Wiedemann Ann. (1)

H. Ebert, Wiedemann Ann. 38, 489 (1889).
[Crossref]

Z. Angew. Phys. (1)

W. Leo, Z. Angew. Phys. 8, 196 (1956).

Z. Instrumentenk. (1)

W. Leo, Z. Instrumentenk. 66, 240 (1958);Z. Instrumentenk,  709 (1962).

Z. Physik (1)

M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
[Crossref]

Other (10)

W. R. Hamilton, Mathematical Papers, Geometrical Optics (Cambridge University Press, London, 1931);Mathematical Papers, Geometrical Optics Vol.  I, Chap. 17, p. 168;J. L. Synge, Geometrical Optics, Cambridge University Press (1937);J. L. Synge, Hamilton’s Method in Geometrical Optics (The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland, 1951);J. L. Synge, J. Opt. Soc. Am. 27, 75 (1937);M. Herzberger, ibid., p. 133.
[Crossref]

K. Nienhuis, thesis, University of Groningen, 1948.

P. Jacquinot, Rept. Progr. Phys.23, 267 (1960).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), p. 477.

M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin, 1963).

E. Wolf, Rept. Progr. Phys.14, 95 (1951).
[Crossref]

To illustrate the sensitivity of the slit curvature upon the distance from the entrance slit to the collimating mirror: for one of the instruments analyzed, a shift of 0.3 mm distorted the slit curvature from a circle to a smooth curve which deviates from a circle by ≈6 μ at the slit center to ≈0.2 μ, 2.75 cm above or below the slit center.

We shall be interested only in resolution ≤ 1 cm−1 in the above stated wavelength region. This value of resolution, to the author’s knowledge, has not been achieved by interferometers employing the Fourier transformation techniques. In the spectral region where photomultiplying tubes can be used as detectors (⪝1μ) the signal/noise advantage of these types of interferometers is nullified. The argument shall proceed under the assumption that higher resolution will be obtained in the future.

J. Connes, J. Phys Radium19, 197 (1958); Symposium on Interferometry, National Physics Laboratory, Great Britian No. 11, 409 (1960).

G. Stroke (private communication) informed us that he gave Girard a grating blazed at 45°. Evidently Girard did not use it in the previous referenced papers since the scanning range was 37°<θ<22° in a Littrow system. If the grille spectrometer is used with a grating of 45° blaze angle, then the luminosity advantage of the grille spectrometer over the Czerny–Turner monochromator would be ≈30 to 1.3.

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Figures (7)

F. 1
F. 1

A two-dimensional schematic of a generalized spherical mirror Czerny–Turner arrangement. A is the entrance slit, B the exit slit. M1 and M2 are the collimating mirror and condensing mirror, respectively. G is the grating. This is not the type of instrument analyzed by M. Czerny and A. F. Turner, since it is asymmetric.

F. 2
F. 2

The mirror surface M1 is tangent to the OY axis at O, hence its center of curvature is at C1,2 on the OX axis. For an Ebert system, i.e., a concentric Czerny–Turner spectrometer, the mirror surface M2 is tangent to the OY axis at O and its center is at C1,2. For a corrected Czerny–Turner spectrometer by making R2 < R1 such that αmαm a central ray will trace a path similar to the path from A to B; or if one has the condition αm < αm and R 1 R 2 then a central ray will follow a path similar to the one from A to B′. Under this condition, the mirror surface, M2′ has its center-of-curvature at some point C2′. The central ray shown is parallel to the OX axis; this is not necessary and any path of the central ray from A to M1 to the grating is valid.

F. 3
F. 3

The above spot diagrams at the exit slit are of a low-blaze Ebert spectrometer (a); a low-blaze Czerny–Turner spectrometer where R1 = R2 = 200 cm and coma correction is made by increasing the angle αm (b); and a Czerny–Turner spectrometer where R1 = 200 cm and R2 = 180 cm (c). The coma shown in (a) is reduced in (b) although there is a slight increase of the astigmatism. In both patterns a slight curvature of the astigmatic pattern is noted. The asymmetric Czerny–Turner (c) shows a reduction of both coma and astigmatism. The numerical scale is in microns.

F. 4
F. 4

A comparison of spot diagrams from instruments utilizing high-blaze gratings, hence a steep grating angle; (a) is an Ebert spectrometer; (b) a Czerny–Turner spectrometer where coma is absolutely corrected through the center of the pattern by tilting the second mirror, R1 = R2 = 200 cm; (c) is the same instrument as (b) except that the tilt of the second mirror has been reduced, bringing coma to the center of the pattern, but partially balancing the diagram; (d) is a Czerny–Turner spectrometer where R1 = 200 cm and R2 = 130 cm. The off-axis angle of the second mirror is approximately that of the collimating mirror. Note the change in vertical scale for (b) and (c). In (b) and (c) the curvature of the astigmatic pattern is more pronounced than in the low-blaze case (Fig. 3). The spot diagram (d) indicates that simply tilting the second mirror without a reduction in the radius of curvature will not be a satisfactory correction. All traces originate from the slit center. The numerical scale is in microns.

F. 5
F. 5

In (a), (1) is an approximate prolate ellipse, (2) a circle, and (3) an approximate oblate ellipse. They have a common center at point P. These curves correspond to the directions of the normal to the blaze step at blaze wavelength shown in (b), A, B, and C. The optical path of S M1G is for a central ray striking the grating positioned at A, B, or C in drawing (b). The central ray is shown parallel to the OX axis as in Fig. 2. But this is not necessary since slit curvature is independent of the trajectory of the central ray from M1 to G.

F. 6
F. 6

These spot diagrams all originate from 2.75 cm above the center of a circular slit. This would be a slit length of 5.5 cm. They are for low-blaze grating instruments; (a) is an Ebert spectrometer; (b) is a Czerny–Turner where R1 = R2 = 200 cm, and (c) is a Czerny–Turner spectrometer where R1 = 200 cm and R2 = 180 cm. The comatic shift towards the right, as compared to Fig. 3 is quite pronounced and, as explained in the text and Fig. 7, limits the slit length. The numerical scale is in microns.

F. 7
F. 7

This schematically illustrates a general curve for the exit slit. The arrows show the direction of coma, length is indicative of the amount and zeros show the points of coma correction. Curve (a) is for a high-blaze symmetrical arrangement or a low-blaze symmetrical arrangement with large angle (χ). It shows the large amount of coma near the central region and decrease towards the slit ends. Curve (b) is for a high- or low-blaze unsymmetrical arrangement in which there is absolute coma correction near the slit center, e.g., aligning the instrument by imaging a point source. The coma increases towards the slit ends. Curve (c) represents a symmetric arrangement of low-blaze angle and small angle (χ), an unsymmetric, low blaze arrangement with large angle (χ), or a high-blaze, unsymmetric arrangement. Coma is introduced near the slit center and is balanced towards the ends.

Equations (21)

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

Φ = k τ B λ δ λ ( l / F ) ( 2 W H sin φ cos χ / λ ) ,
Φ = k τ B λ ( δ λ ) 2 ( l / F ) ( 2 W H sin φ cos χ / λ )
δ λ h / δ λ l sin φ l cos φ h / sin φ h cos φ l ,
δ λ h / δ λ l ( sin φ l cos φ h / sin φ h cos φ l ) 1 2 .
n λ / d = cos ψ ( sin α g + sin β g ) ,
Δ β g = 2 ( sin α g 0 / cos β g 0 ) tan ψ Δ ψ ( cos α g 0 / cos β g 0 ) Δ α g ,
F = r + r w ( sin α + sin β ) + w 2 2 [ ( cos 2 α r cos α R ) + ( cos 2 β r cos β R ) ] + 1 2 w 3 [ sin α r ( cos 2 α r cos α R ) + sin β r ( cos 2 β r cos β R ) ] + 1 2 w 4 [ sin 2 α r 2 ( cos 2 α r cos α R ) + sin 2 β r 2 ( cos 2 β r cos β R ) ] + w 4 8 R 2 [ ( 1 r cos α R ) + ( 1 r cos β R ) ] + .
F w = w ( cos 2 α r 2 cos α R ) + 3 2 w 2 ( sin α cos 2 α r 2 sin α cos α r R ) + 2 w 3 ( sin 2 α cos 2 α r 3 sin 2 α cos α r 2 R ) + w 3 2 R 2 ( 1 r 2 cos α R ) + = 0 .
r = r 0 = ( R / 2 ) cos α .
Δ F i = 0 w i F w d w ,
0 w i F w d w = w i 0 F w d w ,
Δ F i = ± ( w i 3 / R 2 ) sin α .
Δ F m = ± ( w m 3 / R 1 2 ) sin α m )
Δ F m = w m 3 R 2 2 sin α m .
w m 3 W m 3 8 cos 3 β g cos 3 α g cos 3 α m cos 3 α m .
Δ F m + Δ F m = 0 .
sin α m = R 2 2 / R 1 2 ( cos α g cos α m / cos β g cos α m ) 3 sin α ,
R 2 = [ R 1 2 sin α m sin α m / ( cos α g cos β g cos α m cos α m ) 3 ] 1 2 .
sin α m = ( R 2 2 / R 1 2 ) ( cos 3 α g / cos 3 β g ) sin α m ,
R 2 = R 1 ( sin α m cos 3 β g / sin α m cos 3 α g ) 1 2 ,
W 1 = { 64 λ N / [ 1 R 1 3 + 1 R 2 3 ( cos β g 0 cos α g 0 cos α m cos α m ) 4 ] } 1 2 ,