Abstract

A new geometric scheme translates a diffraction grating along the straight central groove of an exponentially curved pattern. Lit by a stationary incident beam, the two-dimensional pattern scales isotropically, scanning wavelength without change to any angles, macroscopic distances, curvatures or aberrations. This is exemplified by a new class of self-focused grating monochromator, analyzed by rigorous light-path expansion and numerical raytracing. All spectral aberrations in pure meridional powers (including defocus, coma and spherical aberration) cancel for any angular deviation, magnification and translation range. The residual mixed powers yield Δλ/λ = 10−3 ~ 10−5 in the soft x-ray for plane and concave gratings at grazing incidence. Over the visible spectrum, Δλ/λ ~ 10−4 is shown for plane gratings mounted at Littrow and at normal incidence in reflection or transmission.

© 2016 Optical Society of America

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Optical Properties of Plane-Grating Monochromator

Keiei Kudo
J. Opt. Soc. Am. 55(2) 150-161 (1965)

References

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  1. M. C. Hettrick, “A Single-Element Plane Grating Monochromator,” Photonics 3(1), 1–44 (2016).
    [Crossref] [PubMed]
  2. D. E. Aspnes, “High-efficiency concave-grating monochromator with wavelength-independent focusing characteristics,” J. Opt. Soc. Am. 72(8), 1056–1061 (1982).
    [Crossref]
  3. E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
    [Crossref]
  4. M. C. Hettrick, “In-focus monochromator: theory and experiment of a new grazing incidence mounting,” Appl. Opt. 29(31), 4531–4535 (1990).
    [Crossref] [PubMed]
  5. A. April and N. McCarthy, “Control of spectral aberrations in a monochromator using a plane holographic chirped grating,” Appl. Opt. 47(15), 2750–2759 (2008).
    [Crossref] [PubMed]
  6. A. N. Simonov, S. Grabarnik, and G. Vdovin, “Stretchable diffraction gratings for spectrometry,” Opt. Express 15(15), 9784–9792 (2007).
    [Crossref] [PubMed]
  7. T. Glaser, S. Schröter, S. Fehling, R. Pöhlmann, and M. Vlček, “Nanostructuring of organic and chalcogenide resists by direct DUV laser beam writing,” Electron. Lett. 40(3), 176–177 (2004).
    [Crossref]
  8. D. Radtke and U. D. Zeitner, “Laser-lithography on non-planar surfaces,” Opt. Express 15(3), 1167–1174 (2007).
    [Crossref] [PubMed]
  9. E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, “Diffraction gratings,” Rep. Prog. Phys. 38(8), 975–1048 (1975).
    [Crossref]
  10. T. Harada, S. Moriyama, T. Kita, and Y. Kondo, “Development of numerical control ruling engine for stigmatic concave grating,” J. Jpn. Soc. Precision Eng. 42(501), 888–892 (1976).
    [Crossref]
  11. F. Polack, B. Lagarde, M. Idir, A. L. Cloup, and E. Jourdain, ”Variable groove depth gratings and their applications in soft x-ray monochromators,”Am. Inst. Phys., Conf. Proc. 879, 639–643 (2007).
  12. M. C. Hettrick, “Corrections to ‘A Single-Element Plane Grating Monochromator’,” http://www.hettrickscientific.com (2016).
  13. M. Lampton, “BEAM FOUR: Optical ray tracer, v. 194,” http://www.stellarsoftware.com (priv. comm., 2016).
  14. H. A. Rowland, “On concave gratings for optical purposes,” Philos. Mag. 16(99), 197–210 (1883).
    [Crossref]

2016 (1)

M. C. Hettrick, “A Single-Element Plane Grating Monochromator,” Photonics 3(1), 1–44 (2016).
[Crossref] [PubMed]

2008 (1)

2007 (2)

2004 (1)

T. Glaser, S. Schröter, S. Fehling, R. Pöhlmann, and M. Vlček, “Nanostructuring of organic and chalcogenide resists by direct DUV laser beam writing,” Electron. Lett. 40(3), 176–177 (2004).
[Crossref]

1990 (1)

1989 (1)

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

1982 (1)

1976 (1)

T. Harada, S. Moriyama, T. Kita, and Y. Kondo, “Development of numerical control ruling engine for stigmatic concave grating,” J. Jpn. Soc. Precision Eng. 42(501), 888–892 (1976).
[Crossref]

1975 (1)

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, “Diffraction gratings,” Rep. Prog. Phys. 38(8), 975–1048 (1975).
[Crossref]

1883 (1)

H. A. Rowland, “On concave gratings for optical purposes,” Philos. Mag. 16(99), 197–210 (1883).
[Crossref]

April, A.

Aspnes, D. E.

Cloup, A. L.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, and E. Jourdain, ”Variable groove depth gratings and their applications in soft x-ray monochromators,”Am. Inst. Phys., Conf. Proc. 879, 639–643 (2007).

Fehling, S.

T. Glaser, S. Schröter, S. Fehling, R. Pöhlmann, and M. Vlček, “Nanostructuring of organic and chalcogenide resists by direct DUV laser beam writing,” Electron. Lett. 40(3), 176–177 (2004).
[Crossref]

Franks, A.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, “Diffraction gratings,” Rep. Prog. Phys. 38(8), 975–1048 (1975).
[Crossref]

Fukui, K.

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

Gale, B.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, “Diffraction gratings,” Rep. Prog. Phys. 38(8), 975–1048 (1975).
[Crossref]

Glaser, T.

T. Glaser, S. Schröter, S. Fehling, R. Pöhlmann, and M. Vlček, “Nanostructuring of organic and chalcogenide resists by direct DUV laser beam writing,” Electron. Lett. 40(3), 176–177 (2004).
[Crossref]

Grabarnik, S.

Harada, T.

T. Harada, S. Moriyama, T. Kita, and Y. Kondo, “Development of numerical control ruling engine for stigmatic concave grating,” J. Jpn. Soc. Precision Eng. 42(501), 888–892 (1976).
[Crossref]

Hettrick, M. C.

Hutley, M. C.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, “Diffraction gratings,” Rep. Prog. Phys. 38(8), 975–1048 (1975).
[Crossref]

Idir, M.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, and E. Jourdain, ”Variable groove depth gratings and their applications in soft x-ray monochromators,”Am. Inst. Phys., Conf. Proc. 879, 639–643 (2007).

Ishiguro, E.

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

Jourdain, E.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, and E. Jourdain, ”Variable groove depth gratings and their applications in soft x-ray monochromators,”Am. Inst. Phys., Conf. Proc. 879, 639–643 (2007).

Kita, T.

T. Harada, S. Moriyama, T. Kita, and Y. Kondo, “Development of numerical control ruling engine for stigmatic concave grating,” J. Jpn. Soc. Precision Eng. 42(501), 888–892 (1976).
[Crossref]

Kondo, Y.

T. Harada, S. Moriyama, T. Kita, and Y. Kondo, “Development of numerical control ruling engine for stigmatic concave grating,” J. Jpn. Soc. Precision Eng. 42(501), 888–892 (1976).
[Crossref]

Lagarde, B.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, and E. Jourdain, ”Variable groove depth gratings and their applications in soft x-ray monochromators,”Am. Inst. Phys., Conf. Proc. 879, 639–643 (2007).

Matsudo, O.

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

McCarthy, N.

Mizutani, N.

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

Moriyama, S.

T. Harada, S. Moriyama, T. Kita, and Y. Kondo, “Development of numerical control ruling engine for stigmatic concave grating,” J. Jpn. Soc. Precision Eng. 42(501), 888–892 (1976).
[Crossref]

Nakamura, E.

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

Palmer, E. W.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, “Diffraction gratings,” Rep. Prog. Phys. 38(8), 975–1048 (1975).
[Crossref]

Pöhlmann, R.

T. Glaser, S. Schröter, S. Fehling, R. Pöhlmann, and M. Vlček, “Nanostructuring of organic and chalcogenide resists by direct DUV laser beam writing,” Electron. Lett. 40(3), 176–177 (2004).
[Crossref]

Polack, F.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, and E. Jourdain, ”Variable groove depth gratings and their applications in soft x-ray monochromators,”Am. Inst. Phys., Conf. Proc. 879, 639–643 (2007).

Radtke, D.

Rowland, H. A.

H. A. Rowland, “On concave gratings for optical purposes,” Philos. Mag. 16(99), 197–210 (1883).
[Crossref]

Sakai, K.

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

Schröter, S.

T. Glaser, S. Schröter, S. Fehling, R. Pöhlmann, and M. Vlček, “Nanostructuring of organic and chalcogenide resists by direct DUV laser beam writing,” Electron. Lett. 40(3), 176–177 (2004).
[Crossref]

Simonov, A. N.

Suzui, M.

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

Vdovin, G.

Verrill, J. F.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, “Diffraction gratings,” Rep. Prog. Phys. 38(8), 975–1048 (1975).
[Crossref]

Vlcek, M.

T. Glaser, S. Schröter, S. Fehling, R. Pöhlmann, and M. Vlček, “Nanostructuring of organic and chalcogenide resists by direct DUV laser beam writing,” Electron. Lett. 40(3), 176–177 (2004).
[Crossref]

Watanabe, M.

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

Yamazaki, J.

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

Zeitner, U. D.

Appl. Opt. (2)

Electron. Lett. (1)

T. Glaser, S. Schröter, S. Fehling, R. Pöhlmann, and M. Vlček, “Nanostructuring of organic and chalcogenide resists by direct DUV laser beam writing,” Electron. Lett. 40(3), 176–177 (2004).
[Crossref]

J. Jpn. Soc. Precision Eng. (1)

T. Harada, S. Moriyama, T. Kita, and Y. Kondo, “Development of numerical control ruling engine for stigmatic concave grating,” J. Jpn. Soc. Precision Eng. 42(501), 888–892 (1976).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Express (2)

Philos. Mag. (1)

H. A. Rowland, “On concave gratings for optical purposes,” Philos. Mag. 16(99), 197–210 (1883).
[Crossref]

Photonics (1)

M. C. Hettrick, “A Single-Element Plane Grating Monochromator,” Photonics 3(1), 1–44 (2016).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, “Diffraction gratings,” Rep. Prog. Phys. 38(8), 975–1048 (1975).
[Crossref]

Rev. Sci. Instrum. (1)

E. Ishiguro, M. Suzui, J. Yamazaki, E. Nakamura, K. Sakai, O. Matsudo, N. Mizutani, K. Fukui, and M. Watanabe, “Constant deviation monochromator for the range 100 A ≤ λ ≤ 1000 A,” Rev. Sci. Instrum. 60(7), 2105–2108 (1989).
[Crossref]

Other (3)

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, and E. Jourdain, ”Variable groove depth gratings and their applications in soft x-ray monochromators,”Am. Inst. Phys., Conf. Proc. 879, 639–643 (2007).

M. C. Hettrick, “Corrections to ‘A Single-Element Plane Grating Monochromator’,” http://www.hettrickscientific.com (2016).

M. Lampton, “BEAM FOUR: Optical ray tracer, v. 194,” http://www.stellarsoftware.com (priv. comm., 2016).

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Figures (8)

Fig. 1
Fig. 1 A divergent groove grating, comprising an exponentially curved pattern. To scan wavelength, the grating translates by distance s(λ) along its central groove (dashed), causing different regions (white) to be illuminated by a stationary incident beam centered on point P (fixed in space). At two sample positions of the scan, the color templates reveal 2D isotropy in the groove pattern; any geometrical aberration is thus fixed over the scan. A self-focusing version (shown here) also exhibits a variation in spacing with ω. Though not affecting the focusing, the groove depth (exemplified by the black sidewalls of a triangular profile) is also shown scaling with the spacing. This option would extend the isotropy to 3D, providing a relative phase diffraction efficiency which is also independent of the scan wavelength.
Fig. 2
Fig. 2 A single-element divergent groove monochromator. A stationary iris (not shown) parallel to and positioned just above the grating surface provides illumination (white strip) fixed in space. In this self-focusing geometry, varied groove spacing in the meridional direction constrains ∆λi0 = 0 (all i), independent of the scan wavelength. The residual aberrations for plane and concave gratings at a grazing angle are given by Fig. 3, and their raytracings shown in Figs. 4 and 5, respectively. Non-grazing mountings are illustrated in Figs. 6 and 7. Plane gratings may employ a long entrance slit, as raytraced in Fig. 8. Note that the optically-defined “vertical” is in the direction of z and z′ (see Sec. 2 of the text).
Fig. 3
Fig. 3 Geometrical aberrations at the exit plane of a self-focused divergent groove grating at a 3° graze angle. Due to the fixed geometry, all aberrations are independent of the scan wavelength. The object is a point and the detailed parameters are given in the text. Curves are light-path calculations of the dominant terms for a plane grating (Δλ21/λ, solid) and a concave grating (Δλ12/λ, dash and Δλ31/λ, dot-dash). Horizontal magnification is ¼ (green), ½ (blue), 1 (black), 2 (orange) and 4 (red). Precise extractions from numerical raytracings (open circles) match the light-path curves to approximately 13 decimal digits.
Fig. 4
Fig. 4 Numerical raytracings of a self-focused divergent groove plane grating at a 3° Graze angle. The monochromator length is 0.5m, the magnification is unity, ρ = 2, and ϕ = 0.00283. Given a spectral scan of 3 octaves (e.g. λ = 1 – 8 nm), the grating size is 13.5 mm × 40 mm and is illuminated by a 13.5 mm × 0.47 mm ellipse. a) The red and blue images shown at the exit slit plane are for a point object emitting two wavelengths separated by 1 part in 1,000. b) The scan intensity profile shows these lines are marginally resolved (26.5% dip), including a 0.004 mm object width and a 0.004 mm width straight exit slit. At either peak, 3.2 × 10−6 sr of that line radiation from the object is geometrically transmitted. c) Image 1 m downstream of the monochromator, tuned to the dip (upper) or to the blue line peak (lower).
Fig. 5
Fig. 5 Numerical raytrace results on the exit plane of a self-focused divergent groove concave grating at a 3° graze angle. The object is a point. (a) Derivation of optimum exit slit curvature and width. (b) Spot diagram for a monochromator length of 9 m, a grating size of 180 mm × 720 mm (may be composed of 4 segments, each 180 mm square), a scan range of 3 octaves, unit magnification, ρ = 2, ϕm ≃ 0.002 and ϕs ≃ 0.004. The “red” and “blue” soft x-ray wavelengths are separated by 1/40,000. (c) Spectral scan through a curved exit slit of 0.003 mm width; the 48% dip degrades to the marginal value of 20–25% after adding the physical diffraction width due to the 41,130 grooves illuminated at a maximum scan wavelength of 8 nm. (d) Spot diagram for a monochromator length of 25 m, a grating size of 500 mm × 2000 mm (may be composed of 4 segments, each 500 mm square), a scan range of 1 octave, a “sweet magnification” of 4, ρ = 1/2, ϕm ≃ 0.004 and ϕs ≃ 0.002. The “red” and “blue” soft x-ray wavelengths are separated by 1/300,000. (e) Spectral scan through a 0.002 mm wide curved exit slit; the 50% dip degrades to the marginal value of 20–25% after adding the physical diffraction width due to the 457,000 grooves illuminated at the maximum scan wavelength of 2 nm.
Fig. 6
Fig. 6 Normal incidence mount of a self-focused divergent groove grating, with the diffracted beam either transmitted (IT) or reflected (IR). The parameters raytraced are O o P ¯ = 53 mm , PI ¯ = 37 mm , β= 20.1°, M = 2, ϕm = .20, ϕs = .0125, 2 ω r = 10.75 mm , 2 σ r = 0.66 mm and a spectral scan from 400 nm to 700 nm over a translation of Sr = 32 mm. The spot diagram is a numerical raytracing at 2 wavelengths (colored here in red and blue) separated by twice the marginal resolution of 1/6000, with the images being identical in transmission and reflection. An exit slit would be aligned to the z″-axis (image length direction) which makes the fixed angle ψ′ = 46.0642° relative to the z-axis shown in the optical schematic.
Fig. 7
Fig. 7 Littrow mount of a divergent groove grating in a retro-focus configuration. The parameters raytraced are O o P ¯ = PI ¯ = 45 mm , β = 40.3°, M = −1, ϕm = 0.200 and ϕs = 0.0125, 2 ω r = 14.3 mm , 2 σ r = 0.562 mm and a spectral scan from 400 nm to 700 nm over a translation of Sr = 32 mm. The spot diagram is a numerical raytracing at two wavelengths (colored here in red and blue) separated by twice the marginal resolution of 1/14000. An exit slit would be aligned to the z″-axis (image length direction) which makes the fixed angle ψ′ = 42.8432° relative to the z-axis shown in the optical schematic.
Fig. 8
Fig. 8 Raytraced spot diagrams. The plane grating image at the origin is the same as given in Fig. 4(a), while the other 16 are for off-plane object points. These are spaced at 1.25 mm intervals along a 10 mm long entrance slit, which is tilted by +16.84° in accordance with Eq. (59). The upper sequence of images (whose centers lie along a parabolic curve) result from a straight entrance slit; the lower sequence of images (whose centers lie on the z″ axis) result from an optimally-curved entrance slit [e.g. the radius given by Eq. (60)]. The individual aberration terms extracted from the two spot diagrams at the extreme edges (outlined in red) are: Δ x 111 ( tilt ) = 0.0175 mm , in agreement with Eq. (62); Δ x 201 ( defocus ) = 0.0030 mm , in agreement with Eq. (63); Δ x 301 ( coma ) = 0.0029 mm and Δ x 401 ( spherical aberration ) = 0.00026 mm . These combine to yield a net broadening of Δ x net ~ 0.005 mm , added to that of the on-axis image. Note that the magnification from object to image is unity in the spectral direction (x″), but is ≃ η = r′/r = 2 in the slit length direction (z″).

Equations (66)

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d ω ( ω , ) = e c d ω ( ω , 0 ) .
c = ( q / S ) ln 2
N ( ω , σ , s ) = r d ω ( 0 , 0 ) i = 0 n + 1 j = 0 n i + 1 N i j ( s ) ω i σ j
N i j ( s ) = N i e c s ( c ) j / j !
1 d ω ( ω , σ ) = 1 r i = 0 n j = 0 n i   ω V i j ω i σ j 1 d σ ( ω , σ ) = 1 r j = 0 n i = 0 n j   σ V i j ω i σ j
ω ( N , ) [ 1 + 4 N N 2 ( d o / r ) e c 1 ] / ( 2 N 2 ) .
F S L ( 2 π Δ w RMS / d ) 2 .
ε m = [ sin ( m π a / d ) / m π ] 2 .
μ m λ / d ( 0 , 0 ) = cos β cos α 2 γ 2 ( 1 ρ ) / ( 1 + ρ )
  i j x = η { i μ N i j + [ A i j / ( Δ ω ) + B i j / ( Δ ω ) ] ω i 1 σ j c o e f f i c i e n t } / sin β
  i j z = η { j μ N i j + [ A i j / ( Δ σ ) + B i j / ( Δ σ ) ] ω i σ j 1 c o e f f i c i e n t }
A i j / ( Δ ω ) = [ cos α x i j sin α + ( R sin α x i j cos α ) b ] / 1 +   A t i j
A i j / ( Δ σ ) = ( σ z i j ) / 1 +   A t i j
B i j / ( Δ ω ) = [ η cos β + ξ i j sin β + ( R η sin β ξ i j cos β ) b ] / η / 1 +   B t i j
B i j / ( Δ σ ) = ( σ ζ i j ) / η / 1 +   B t i j
  A t i j = 2 ω ( cos α x i j sin α ) 2 z i j σ + x i j 2 + z i j 2 + σ 2 + 2 ( R sin α x i j cos α )
  B t i j η 2 = 2 ω ( ξ i j sin β η cos β ) 2 ζ i j σ + ξ i j   2 + ζ i j   2 + σ 2 + 2 ( R η sin β ξ i j cos β )
ξ i j = ( I , J ) ( i , j ) x I J = ( I , J ) ( i , j )   I J x ω I 1 σ J and ζ i j = ( I , J ) ( i , j ) z I J = ( I , J ) ( i , j )   I J z ω I σ J 1 .
  02 z / η = 1 + 1 / η
  11 T = c
  11 z / η = c μ
  20 T = ( 1 κ + 1 κ ρ 1 ρ 2 1 η ) 1 Q 2 N 20
  12 z / η = ( Γ + 1 / η ) τ + c 2 μ
  21 T = [ 2 N 20 ( 1 + Q τ ) + ( 1 1 η + 1 ρ 2 1 ρ κ ) τ ] c
  21 z / η = ( τ / η N 20 ) c μ
  30 T = 3 N 3 + { [ ( 1 η Γ ρ ) 1 κ ( 1 η 2 Γ ρ 2 ) ] 3 Q + [ ( 2 κ 4 η ) 20 T Q 20 T 2 c 2 μ ] } τ 2
  12 T = [ 1 + ( 1 + Q τ ) c 2 ] / 2
  31 z η = ( τ η N 20 N 30 ) c μ + { ( 1 η κ 1 η 2 ) 1 2 Q [   20 T η + Q 2 20 T 2 ] } c μ 2 c 3 μ 3 2
  40 T = { 4 N 40 + [ ( 1 η 3 + 1 ρ 2 ) 5 4 ( 1 η 3 + 1 ρ 4 ) sin 2 β ] 2 Q } + { [ ( 1 η 2 + 1 ρ ) 3 2 ( 1 η 2 + 1 ρ 3 ) sin 2 β ] 2 κ ( Γ 2 + 1 η ) τ 2 2 κ 2 + ( 1 + 1 ρ ) sin 2 β 2 κ 3 } 1 Q [ ( 3 η 2 3 2 η κ ) 20 T + Q η 20 T 2 + ( 2 η 1 κ + 20 T Q ) 30 T τ ] + [ ( 9 η 6 κ )   20 T η Q + ( 5 η 1 κ ) 20 T 2 + Q 20 T 3 2 N 20 τ c 2 ] μ 2 + (   20 T + 1 / η 1 / κ Q ) c 2 μ 2 2
  22 z η = ( 1 + Γ η ) + ( N 20 τ η ) c 2 μ + [ ( 1 η 2 + 1 ρ κ 1 η κ 3 ρ 2 2 η Γ ) 1 2 Q + ( 3 2 1 η ) c 2 μ ] μ + (   20 T η + Q 2 20 T 2 ) μ 20 T Q c 2 μ 2
  31 T = [ 3 N 30 + N 20 τ Γ + 3 2 η 2 ( η κ 2 ) 2 η 20 T Q 20 T Q τ ] c + 21 T ( 1 κ 1 η 20 T Q ) τ + { τ c 2 + [ 3 η 2 ( 3 2 η κ ) + Γ + 1 κ 1 η ] 1 Q + ( 5 η 1 κ 1 ) 20 T + 3 2 20 T 2 Q } c μ + c 3 μ 2 2 .
Δ x i j = | (   i j x cos ψ i 1 , j + 1 z sin ψ ) | p i j ( 2 ω ) i 1 ( 2 σ ) j
Δ λ i j / λ = ( Δ x i j / cos ψ ) ( sin β ) / η / μ .
Δ z i j = | (   i j z cos ψ + i + 1 , j 1 x sin ψ ) | p i j ( 2 ω ) i ( 2 σ ) j 1 .
tan ψ s = c μ ( 1 + 1 / η ) sin β
tan ψ m = [ 2 N 20 + ( 1 ρ / κ ρ 2 + 1 η / κ η ) 1 Q ] / ( c sin β ) .
2 N 20 = ( 1 ρ / κ ρ 2 + 1 η / κ η ) 1 Q + c 2 μ 1 + 1 / η
3 N 30 = τ 2 { ( 1 η κ Γ ρ κ 1 η 2 + Γ ρ 2 ) 3 Q + 20 T ( 2 κ 4 η 20 T 2 Q + [ 2 η N 20 τ ( η + 1 ) τ 1 ] c 2 μ ) } .
4 N 40 = [ 1 η 2 κ + 1 ρ κ 2 η 3 2 ρ 2 ( 1 η 2 κ + 1 ρ 3 κ 1 η 3 1 ρ 4 1 κ 3 1 ρ κ 3 ) tan 2 β 2 1 / η + Γ 2 2 κ 2 ] τ 2 Q [ ( 6 η 3 κ )   20 T 2 η +   20 T 2 Q η + ( 2 η 1 κ + 20 T Q ) 30 T τ ] + [ N 30 ( 1 + 2 / η ) N 20 τ 1 + 1 / η ( 2 c 2 ) + ( 9 η 6 κ )   20 T η Q + ( 5 η 1 κ ) 20 T 2 + 20 T 3 Q ] μ 2 + [ ( 1 η 1 κ ) ( 1 + 2 η ) 1 Q + ( 1 + 3 η ) 20 T + 20 T 2 Q ] c 2 μ 2 2 ( 1 + 1 / η ) + c 4 μ 3 2 ( 1 + 1 / η ) .
  i j T = (   i 1 , j + 1 z / η ) c / ( 1 + 1 / η ) .
Δ λ 21 λ = | 2 ( 1 + Q τ ) N 20 + ( 1 1 η + 1 ρ / κ ρ 2 ) τ + ( Γ + 1 / η ) τ c 2 μ 1 + 1 / η ) | q S Ω sin α ln 2 2 .
( Δ λ 21 λ ) sep | ρ 2 + 1 / ( M ρ ) ( ρ 1 ) ( 1 + M ρ ) | L S Ω q sin γ ln 2 2 f 21
Δ z 12 = 1 90 [ ( z 10 9 + z 12 11 ) 80 ( z 18 17 + z 20 19 ) + 1024 ( z 22 21 + z 24 23 ) ]
Δ x 21 = 1 90 [ ( x 10 9 + x 12 11 ) 80 ( x 18 17 + x 20 19 ) + 1024 ( x 22 21 + x 24 23 ) ]
Δ z 22 = 1 90 [ ( z 12 9 + z 11 10 2 z 8 6 ) 160 ( z 20 17 + z 19 18 2 z 4 2 ) + 4096 ( z 24 21 + z 23 22 2 z 16 14 ) ]
Δ x 31 = 1 90 [ ( x 12 9 + x 11 10 2 x 8 6 ) 160 ( x 20 17 + x 19 18 2 x 4 2 ) + 4096 ( x 24 21 + x 23 22 2 x 16 14 ) ]
1 κ = ( 1 / ρ 2 + 1 / η ) ( 1 + 1 / η ) + ( 2 / η 2 Γ 1 ) Q τ Q 2 c 2 μ ( 1 + 1 / ρ + Q τ ) ( 1 + 1 / η ) 2 ρ 2 + 1 / η ( ρ + 1 ) 2 ( 1 1 / ρ ) 2 2 ( 1 + 1 / η ) c 2 μ
Δ λ 12 λ = | [ ( 1 + Q τ ) c 2 ] + 1 | ( 2 σ ) 2 8
( Δ λ 12 λ ) sep { [ ( 1 ρ 2 + 1 ) c 2 2 ] + 1 } ϕ s   2 8 f 12 { [ 2 + 2 ρ + 2 / ρ + ρ 2 + 1 / ρ 2 ( 1 + M ρ ) 4 ] ( ln 2 8 ) 2 + ( ρ + 1 ) 2 / ( 1 + M ρ ) 2 32 q 2 ( L / S ) 2 } ( L 2 Ω q g S 2 sin γ ) 2 f 12 .
R ex = ( Δ z 02 / 2 ) 2 2 Δ x 12 = (   02 z cos ψ + 11 x sin ψ ) 2 σ 2 2 (   12 x cos ψ   03 z sin ψ ) σ 2 = η [ ( 1 + 1 / η ) + Q c 2 μ / ( 1 + 1 / η ) ] 2 [ 1 + ( 1 + Q τ ) c 2 ] μ ( 2 sin β c 2 μ ) η ( 1 + 1 / η ) 2 ( 1 + 1 / ρ 2 )
Δ x / Δ x 12 = 2 [ 1 ( 1 Δ z 11 / Δ z 02 ) 2 ] .
Δ λ 31 λ 2 c | 3 N 30 + N 20 2 + ( 3 2 η   21 T c ) ( 1 η κ 2 η 2 ) + [ η ( Q 1 ) N 20 + 1 + Q ( 1 +   21 T c ) ] | c 2 μ η + 1 ω 2 σ
( Δ λ 31 λ ) sep |   31 F ρ , M | g Ω q sin γ ln 2 8 f 31 ; 31 F ρ , M [ ρ 4 + ρ 2 M ρ M + 1 M 2 ρ ] 6 ( ρ 2 1 ) ( ρ + 1 ) + { 7 2 M ρ 5 + ( 3 M + 1 4 ) ρ 4 ( 9 2 M + 1 ) ρ 3 ( 4 M 3 4 ) ρ 2 + ( 1 2 M + 3 ) ρ + ( M 5 4 ) + ( 1 2 M 2 ) / ρ + 1 / ρ 2 / 4 } c 2 μ ( M ρ + 1 ) ( ρ 2 1 ) ( ρ + 1 )
M sweet = 1 / ρ 2
Δ λ 21 λ = | [ η + cos 2 β * η + 1 ] / sin β * + [ 2 η 2 ( η + 1 ) 2 tan 2 ψ ] sin β * | ( ln 2 2 ) ( L S ) ( Ω q η )
Δ λ 12 λ = | [ 1 + ( 1 + tan 2 β * ) q 2 ( L S ) 2 ( ln 2 ) 2 ( η + 1 ) 2 ] | ϕ s   2 8
Δ λ 21 λ = | 1 2 + [ 2 ( η + 1 ) 2 tan 2 ψ ] tan 2 β * | ( ln 2 2 ) ( L S ) ( q η ) ϕ m ϕ s tan β *
Δ λ 12 λ = | [ 1 + ( 1 + 2 tan 2 β * ) q 2 ( L S ) 2 ( ln 2 ) 2 ( η + 1 ) 2 ] | ϕ s   2 8 .
tan ψ = ρ tan ψ = ρ μ c ( 1 + 1 / η ) sin β ( ρ 1 ) ( 1 + 1 / η ) c sin γ
R en = [ ( sin β ) cos ψ ρ μ ] / [ 1 + ( 1 Γ ρ 2 ) ( 1 + 1 / η ) 2 τ Q c 2 ] 2 cos ψ c 2 sin γ [ ( 1 + 1 / η ) / η ρ ( 1 + ρ ) ( 1 1 / ρ ) 3 ] .
Δ λ i j h / λ = p i j |   i j k Δ λ / λ | ( 2 ω ) i 1 ( 2 σ ) j z k .
  111 Δ λ / λ = ( 1 + 1 / η ) ( tan 2 ψ ) τ / μ + 1 ( 1 + 1 / η ) ( tan 2 ψ ) / μ + 1
  201 Δ λ plane / λ = [ ( 1 + 1 / η ) tan 2 ψ + ( Γ 1 ) ] τ ( tan ψ ) ( sin β ) / μ [ ( 1 + 1 / η ) ( tan 2 ψ ) / μ 1 ] ( tan ψ ) sin β .
  201 Δ λ curved 201 Δ λ plane λ = 2 τ Γ ρ + 1 ( ρ + 1 ) 2 ( ρ 2 + 1 η ) tan ψ μ sin β 2 ( ρ 2 + 1 / η ρ + 1 ) tan ψ μ sin β .
( Δ λ / λ ) dispersion = Δ β sin β cos β cos α ( Δ x eff M cos ψ ) D L
( Δ λ / λ ) diffraction = 1 | m | N d sin α m r ϕ m ( λ ϕ m ) D L .

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