Abstract

Analytic calculations are presented which describe aberrations of the in-plane and off-plane varied line-space grating designs we recently proposed [ Appl. Opt. 22, 3921 ( 1983)]. Ray traces confirming these results to within typical accuracies of 10% are illustrated for several examples. Spectral field aberrations are calculated for convenient focal surfaces, and optimal field curvatures are calculated and ray traced. An improvement of the off-plane fan grating is proposed, where the angular spacings of the grooves are varied to achieve a large decrease in grating aberrations. However it is shown that, in conical diffraction, the net resolution can also be dominated by a diminished dispersive power compared to in-plane grating mounts. Curved groove in-plane grating designs are ray traced, revealing no substantial degradation in imaging performance by restricting such curves to concentric circles. However, it is also shown that the general case of hyperbolic grooves can be fabricated by use of visible or UV holography, with small residual aberrations. We designate this new class of holographic gratings as Type V. Misalignment aberrations of high resolution in-plane gratings, for the in situ cases of off-axis illumination, grating and detector displacements, and grating rotational misalignment, are calculated and found to be generally small.

© 1984 Optical Society of America

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References

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  1. M. C. Hettrick, S. Bowyer, “Variable Line-Space Gratings: New Designs for Use in Grazing Incidence Spectrometers,” Appl. Opt. 22, 3921 (1983).
    [CrossRef] [PubMed]
  2. M. V. R. K. Murty, “Use of Convergent and Divergent Illumination with Plane Gratings,” J. Opt. Soc. Am. 52, 768 (1962).
    [CrossRef]
  3. M. V. R. K. Murty, N. C. Das, “Narrow-Band Filter Consisting of Two Aplanatic Gratings,” J. Opt. Soc. Am. 72, 1714 (1982).
    [CrossRef]
  4. J. T. Hall, “Focal Properties of a Plane Grating in a Convergent Beam,” Appl. Opt. 5, 1051 (1966).
    [CrossRef] [PubMed]
  5. J. D. Baumgardner, “Theory and Design of Unusual Concave Gratings,” Thesis, U. Rochester (1969).
  6. W. C. Cash, “X-Ray Spectrographs Using Radial Groove Gratings,” Appl. Opt. 22, 3971 (1983).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 206 and 207.
  8. E. G. Loewen, L. Bartle, “Triangular and Sinusoidal Grooves in Holographic Gratings—Manufacture and Test Results,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 27 (1980).
  9. A. J. Caruso, G. H. Mount, B. E. Woodgate, “Absolute S- and P-Plane Polarization Efficiencies for High Frequency Holographic Gratings in the VUV,” Appl. Opt. 20, 1764 (1981).
    [CrossRef] [PubMed]
  10. G. H. Mount, W. G. Fastie, “Comprehensive Analysis of Gratings for Ultraviolet Space Instrumentation,” Appl. Opt. 17, 3108 (1978).
    [CrossRef] [PubMed]
  11. J. Kielkopf, “Echelle and Holographic Gratings Compared for Scattering and Spectral Resolution,” Appl. Opt. 20, 3327 (1981).
    [CrossRef] [PubMed]
  12. C. A. Wallace, G. D. Ludbrook, M. Stedman, “Manufacture and Measurement of Ion-Etched X-Ray Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 315, 165 (1981).
  13. J. Lerner, “Aberration Corrected Holographically Recorded Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 72 (1980).
  14. Y. Sakayanagi, “Ruling of a Curved Grating,” Sci. Light 3, 79 (1955).
  15. M. C. Hettrick, “Extreme Ultraviolet Explorer Spectrometer Option Study,” MCH/EUVE/321/82, U. California, Berkeley (1982).
  16. M. Neviere, D. Maystre, W. R. Hunter, “Use of Classical and Conical Diffraction Mountings for XUV Gratings,” J. Opt. Soc. Am. 68, 1106 (1978).
    [CrossRef]
  17. R. A. M. Keski-Kuha, “Layered-Synthetic Microstructure Technology Considerations for the Extreme Ultraviolet,” submitted ??
  18. T. B. Andersen, “Automatic Computation of Optical Aberration Coefficients,” Appl. Opt. 19, 3800 (1980).
    [CrossRef] [PubMed]

1983 (2)

1982 (1)

1981 (3)

1980 (3)

J. Lerner, “Aberration Corrected Holographically Recorded Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 72 (1980).

T. B. Andersen, “Automatic Computation of Optical Aberration Coefficients,” Appl. Opt. 19, 3800 (1980).
[CrossRef] [PubMed]

E. G. Loewen, L. Bartle, “Triangular and Sinusoidal Grooves in Holographic Gratings—Manufacture and Test Results,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 27 (1980).

1978 (2)

1966 (1)

1962 (1)

1955 (1)

Y. Sakayanagi, “Ruling of a Curved Grating,” Sci. Light 3, 79 (1955).

Andersen, T. B.

Bartle, L.

E. G. Loewen, L. Bartle, “Triangular and Sinusoidal Grooves in Holographic Gratings—Manufacture and Test Results,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 27 (1980).

Baumgardner, J. D.

J. D. Baumgardner, “Theory and Design of Unusual Concave Gratings,” Thesis, U. Rochester (1969).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 206 and 207.

Bowyer, S.

Caruso, A. J.

Cash, W. C.

Das, N. C.

Fastie, W. G.

Hall, J. T.

Hettrick, M. C.

M. C. Hettrick, S. Bowyer, “Variable Line-Space Gratings: New Designs for Use in Grazing Incidence Spectrometers,” Appl. Opt. 22, 3921 (1983).
[CrossRef] [PubMed]

M. C. Hettrick, “Extreme Ultraviolet Explorer Spectrometer Option Study,” MCH/EUVE/321/82, U. California, Berkeley (1982).

Hunter, W. R.

Keski-Kuha, R. A. M.

R. A. M. Keski-Kuha, “Layered-Synthetic Microstructure Technology Considerations for the Extreme Ultraviolet,” submitted ??

Kielkopf, J.

Lerner, J.

J. Lerner, “Aberration Corrected Holographically Recorded Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 72 (1980).

Loewen, E. G.

E. G. Loewen, L. Bartle, “Triangular and Sinusoidal Grooves in Holographic Gratings—Manufacture and Test Results,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 27 (1980).

Ludbrook, G. D.

C. A. Wallace, G. D. Ludbrook, M. Stedman, “Manufacture and Measurement of Ion-Etched X-Ray Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 315, 165 (1981).

Maystre, D.

Mount, G. H.

Murty, M. V. R. K.

Neviere, M.

Sakayanagi, Y.

Y. Sakayanagi, “Ruling of a Curved Grating,” Sci. Light 3, 79 (1955).

Stedman, M.

C. A. Wallace, G. D. Ludbrook, M. Stedman, “Manufacture and Measurement of Ion-Etched X-Ray Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 315, 165 (1981).

Wallace, C. A.

C. A. Wallace, G. D. Ludbrook, M. Stedman, “Manufacture and Measurement of Ion-Etched X-Ray Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 315, 165 (1981).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 206 and 207.

Woodgate, B. E.

Appl. Opt. (7)

J. Opt. Soc. Am. (3)

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

C. A. Wallace, G. D. Ludbrook, M. Stedman, “Manufacture and Measurement of Ion-Etched X-Ray Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 315, 165 (1981).

J. Lerner, “Aberration Corrected Holographically Recorded Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 72 (1980).

E. G. Loewen, L. Bartle, “Triangular and Sinusoidal Grooves in Holographic Gratings—Manufacture and Test Results,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 27 (1980).

Sci. Light (1)

Y. Sakayanagi, “Ruling of a Curved Grating,” Sci. Light 3, 79 (1955).

Other (4)

M. C. Hettrick, “Extreme Ultraviolet Explorer Spectrometer Option Study,” MCH/EUVE/321/82, U. California, Berkeley (1982).

J. D. Baumgardner, “Theory and Design of Unusual Concave Gratings,” Thesis, U. Rochester (1969).

R. A. M. Keski-Kuha, “Layered-Synthetic Microstructure Technology Considerations for the Extreme Ultraviolet,” submitted ??

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 206 and 207.

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Figures (16)

Fig. 1
Fig. 1

In-plane orthogonal coordinate system. Projections are displayed on the (a) central dispersion plane y = 0, (b) grating plane z = 0, and (c) a plane x = t0. The 3-D distances are indicated rather than the projected distances.

Fig. 2
Fig. 2

Distance parameters which specify the in-plane grating geometry.

Fig. 3
Fig. 3

Ray trace spot diagrams for the straight groove in-plane grating specified in the text but with λ* = 150 Å: (a)–(d) along the sagittal circle of radius L0; (e)–(g) along the tangential circle of radius 0.56 L0; and (h)–(k) along a plane passing through λ*. Wavelengths within each of the triplets are separated by Δλ/λ = 0.01. Linear scales are indicated at the top.

Fig. 4
Fig. 4

Curves of extremum aberrations in wavelength and image height as functions of wavelength for the straight groove grating of Fig. 3. Results obtained through 1-D binning of numerical ray tracing. Bumps in the data signify only smoothed statistical noise.

Fig. 5
Fig. 5

Field curvature for minimization of wavelength aberrations for in-plane grating geometries.

Fig. 6
Fig. 6

Ray trace spot diagrams for (a) straight grooves, (b) stigmatic hyperbolas, (c) quasi-stigmatic hyperbolas generated by visible holography, and (d) concentric grooves. In all cases, the tangential focal surfaces have been employed, and each wavelength in the triplets is separated by Δλ/λ = 1/400.

Fig. 7
Fig. 7

Recording geometry for the Type V holographic grating. The outer circle includes the object point F, the zero-order image m, and the correction wavelength λ*. The inner circle passes through the two recording sources and the virtual recording source on the opposite side of the grating.

Fig. 8
Fig. 8

Aberrations of the Type V grating as a function of distance along a 1-D section of the grating pupil (y = 0). A recording wavelength of λR = 3637 Å was simulated and the ray traces were performed at λ* = 304 Å. The fractional errors in the line spacing are indicated for both a single recording and a multipartite recording. The attainable resolution λ/Δλ equals dd.

Fig. 9
Fig. 9

Three-dimensional perspective of a concentric groove grating.

Fig. 10
Fig. 10

Curves of extremum aberrations in wavelength and image height as functions of wavelength for the concentric groove grating specified in the text. Stigmatism is enforced at λ* = 20 Å. For the sagittal focal line, the image height is identically zero for all wavelengths.

Fig. 11
Fig. 11

Off-plane coordinate system. Projections are shown for the (a) plane containing the central groove, (b) grating plane, and (c) focal plane. Within the focal plane projection is also shown a groove profile geometry which maximizes efficiency at λ*.

Fig. 12
Fig. 12

Tandem gratings, each having conventional uniform line spacing: (a) 3-D perspective, (b) an example ray trace for extreme UV spectroscopy revealing a spectral resolution in the λ/Δλ = 100–200 range.

Fig. 13
Fig. 13

Ray traces for oriental fan grating showing (a) the conical wavelength map and the spot diagrams for various cases: (b)–(f) no displacement of the ruling focus behind the mirror focus ΔRF = 0; (g)–(k) an optimal displacement Δ R F L 0 sin γ 0 2; and (l)–(p) an optimal displacement plus an optimized angular variation of the groove spacings. In all cases the sagittal detector plane has been employed.

Fig. 14
Fig. 14

Curves of extremum aberrations in wavelength and image height as functions of wavelength for the off-plane grating specified in the text. The correction wavelength is 150 Å. The light curves summarize the results of Fig. 13 for the sagittal focal plane, while the dark curves indicate the ray trace results for the optimally curved cylinders.

Fig. 15
Fig. 15

Off-axis in-plane grating illumination, revealing the equivalent error in incident angle Δα0, produced by a displacement of the mirror focus from F to F′.

Fig. 16
Fig. 16

Practically attainable spectral resolution for a grazing incidence grating in a converging beam as a function of the mean graze angle γ0. L0 is the distance from grating center to focus. Other parameters assumed for the grating mounting are indicated in the text. The solid lines indicate the ability to dispersively separate images from collecting mirrors (focal length F) of various image qualities (ɛ). The horizontal dot–dash lines indicate the grating aberrations for an optimized fan groove pattern having varied angular spacings. The wavelengths for which single-coating reflectance is 50% are indicated at the bottom.

Tables (1)

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Table I Aberration Coefficients for Off-Plane Solutions

Equations (82)

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m N λ * = ( t 1 - x N ) 2 + h 1 2 - ( t 0 - x N ) 2 + h 0 2 - P 0 ,
P 0 = t 1 2 + h 1 2 - t 0 2 + h 0 2 .
Δ * = L * - L + m N λ * - [ L * ( 0 , 0 , 0 ) - L ( 0 , 0 , 0 ) ] .
L = L 0 2 sin 2 α 0 + ( L 0 cos α 0 - x ) 2 + y 2 ; L * = L 0 2 sin 2 β 0 + ( L 0 cos β 0 - x ) 2 + y 2 ;
m N λ * = L 0 2 sin 2 α 0 + ( L 0 cos α 0 - x ) 2 - L 0 2 sin 2 β 0 + ( L 0 cos β 0 - x ) 2 .
L = 1 - 2 x cos α 0 + x 2 + y 2 ; L * = 1 - 2 x cos β 0 + x 2 + y 2 ,
m N λ * = 1 - 2 x cos α 0 + x 2 - 1 - 2 x cos β 0 + x 2 .
Δ * = ( cos β 0 - cos α 0 ) [ 1 / 2 x y 2 + 3 / 4 x 2 y 2 ( cos β 0 + cos α 0 ) - 3 / 8 x y 4 + ] = ( m λ * / d ) [ 1 / 2 x y 2 + 3 / 4 x 2 y 2 ( cos β 0 + cos α 0 ) - 3 / 8 x y 4 + ] .
λ / Δ λ = ( m λ / d 0 ) / ( δ Δ / δ x ) ,
H = δ Δ / δ y ,
δ Δ * / δ x = ( m λ * / d 0 ) [ 1 / 2 y 2 + 3 / 2 x y 2 ( cos β 0 + cos α 0 ) + ] ,
δ Δ * / δ y = ( m λ * / d 0 ) ( x y + 3 / 2 x 2 y - 3 / 2 x y 3 + ) .
λ * / Δ λ * = 8 f y - 2 + ,
H * / L 0 = ( m λ * / d 0 ) / ( 2 α max f x f y ) ,
L 0 = 485.2 mm ; t 0 = 481.5 mm ; h 0 = 59.9 mm ; P 0 = 0.26 mm ; t 1 = 463.9 mm ; h 1 = 143.1 mm ; f y = 6.24 ; d 0 = 4350 A ˚ ; λ * = 160 A ˚ ; m = - 1.
Δ = Δ * + ( L - L + m N λ ) - ( L * - L + m N λ * ) = L - L * + m N ( λ - λ * ) ,
L = 1 - 2 x cos β + x 2 y 2 ; L * = 1 - 2 x cos β 0 + x 2 + y 2 ;
m N ( λ - λ * ) = ( λ / λ * - 1 ) ( 1 - 2 x cos α 0 + x 2 - 1 - 2 x cos β 0 + x 2 ) .
cos β 0 - cos α 0 = ( m λ * / d 0 ) ,
cos β - cos α 0 = m λ / d 0 ,
cos β 0 - cos β = ( m / d 0 ) ( λ * - λ ) ,
Δ - Δ * = ( m / d 0 ) [ - 1 / 2 x y 2 ( λ * - λ ) + 1 / 2 ( m λ / d 0 ) ( λ * - λ ) x 2 + ] .
δ Δ / δ x = 1 / 2 ( m λ / d 0 ) [ y 2 + 2 x m ( λ * - λ ) / d 0 ] ,
δ Δ / δ y = ( m λ / d 0 ) x y .
λ / Δ λ = 8 / [ f y - 2 + 8 m ( λ * - λ ) / d 0 / ( f x α 0 ) ] ,
H / L 0 = ( m λ / d 0 ) / ( 2 α max f x f y ) .
L R / L 0 = 1 - ¼ [ ( β / α ) 2 - 1 ] α 0 2 ( λ R / λ * - 1 ) .
λ * / Δ λ * = f x 2 / ( λ R / λ * - 1 ) ,
L = ( L 0 cos α 0 - x ) 2 + L 0 2 cos 2 α 0 tan 2 β ,
L * = ( L 0 cos α 0 - x ) 2 + L 0 2 cos 2 α 0 tan 2 β 0 ,
m N λ * = ( L 0 cos α 0 - x ) 2 + L 0 sin 2 α 0 - ( L 0 cos α 0 - x ) 2 + L 0 2 cos 2 α 0 tan 2 β 0 - P 0 ,
Δ = 3 / 2 ( m λ / d 0 ) [ m ( λ - λ * ) / d 0 ] x 2 ,
λ / Δ λ = 1 / 3 α 0 f x / m ( λ - λ * ) / d 0 .
L = ( L 0 cos γ 0 - x ) 2 + [ y + y ( λ * ) ] 2 + [ z ( λ * ) ] 2 ,
L * = ( L 0 cos γ 0 - x ) 2 + [ y - y ( λ * ) ] 2 + [ z ( λ * ) ] 2 .
y ( λ * ) = L 0 m λ * / ( 2 d 0 ) ;
z ( λ * ) = L 0 sin γ 0 1 - 1 / 4     ( m λ * / d 0 / sin γ 0 ) 2 .
L = 1 - 2 x cos γ 0 + x 2 + ( m λ * / d 0 ) y + y 2 ,
L * = 1 - 2 x cos γ 0 + x 2 - ( m λ * / d 0 ) y + y 2 .
L * - L = ( m λ * / d 0 ) [ - y - x y cos γ 0 + 1 / 2 ( 1 - 3 cos 2 γ 0 ) x 2 y + 1 / 8 ( 4 - m 2 λ * 2 / d 0 2 ) y 3 - x 3 y + 3 / 2 x y 3 - x 4 y + 3 x 2 y 3 - 3 / 8 y 5 + ] .
L * - L = ( m λ * / d 0 ) a i j x i y j ,
λ Δ / λ = sin γ 0 / ( sin γ max - sin γ min ) ,
N = [ ( L 0 cos γ 0 + Δ R F ) / d 0 ] arctan [ y / ( L 0 cos γ 0 - x + Δ R F ) ] .
b 01 = + 1 , b 11 = ( 1 + 1 / 2 γ 0 2 + 5 / 24 γ 0 4 + ) - Δ R F ( 1 - 1 / 2 γ 0 2 + ) + Δ R F 2 + , b 21 = ( 1 + γ 0 2 + 2 / 3 γ 0 4 + ) - Δ R F ( 2 + 3 γ 0 2 + ) + 3 Δ R F 2 + , b 03 = ( - 1 / 3 - 1 / 3 γ 0 2 - 2 / 9 γ 0 4 + ) + Δ R F ( 2 / 3 + γ 0 2 + ) - Δ R F 2 + , b 13 = - 1 - 3 / 2 γ 0 2 + 3 Δ R F - 6 Δ R F 2 + , b 31 = + 1 + 3 / 2 γ 0 2 - 3 Δ R F + 6 Δ R F 2 + , b 23 = - 2 + 8 Δ R F + , b 41 = + 1 - 4 Δ R F + .
Δ = ( m λ * / d 0 ) [ γ 0 2 x y + 5 / 2 γ 0 2 x 2 y + 1 / 6 y 3 + ] .
λ / Δ λ = ( m λ * / d 0 ) / ( δ Δ / δ y ) ,
H / L 0 = ( 1 / γ 0 ) ( δ Δ / δ x ) ,
λ / Δ λ [ γ 0 f x - 1 + 5 / 8 f x - 2 + 1 / 8 f y - 2 ] - 1 .
H / L 0 ( m λ * / d 0 ) ( γ 0 + 5 / 2 f x - 1 ) / f y ,
γ 0 = 7.089 ° ; L 0 = 485.47 mm ; d 0 = 2125 A ˚ ; γ min = 6.02 ° ; γ max = 8.62 ° ; f y = 6.2 ; λ * = 150 A ˚ .
Δ = ( m λ * / d 0 ) [ 1 / 2 γ 0 2 x 2 y + 1 / 6 y 3 + 1 / 2 x y 3 + ] .
λ / Δ λ [ 1 / 8 f x - 2 + 1 / 8 f x - 2 ] - 1 8 f 2 , where f 1 / f x - 2 + f y - 2 ,
H / L 0 ( 1 / 2 ) ( m λ * / d 0 ) / ( f x f y ) .
( L 0 cos γ 0 - x ) 2 + L 0 2 sin 2 γ 0 = ( L 0 cos γ 0 - x ) × ( L 0 cos γ 0 - x + Δ R F ) ,
N = ( L 0 cos γ 0 + Δ R F ) / d 0 0 θ d θ / ( 1 + η θ 2 ) .
N = ( L 0 cos γ 0 + Δ R F ) / d 0 × ( θ - η θ 3 / 3 + η 2 θ 5 / 5 + ) N ( η = 0 ) - ( η / 3 ) L 0 / d 0 × arctan 2 [ y / ( L 0 cos γ 0 - x + Δ R F ) ] .
L = 1 - 2 x cos γ 0 + x 2 - ( m λ * / d 0 ) y + y 2 + 2 [ m ( λ * - λ ) / d 0 ] y ,
L * = 1 - 2 x cos γ 0 + x 2 - ( m λ * / d 0 ) y + y 2 ,
N = ( 1 / d 0 ) [ y + ( 1 - γ 0 2 / 2 ) x y + ( 1 - γ 0 2 ) x 2 y - ( 1 / 3 - γ 0 2 ) y 3 + ] ,
Δ - Δ * = [ 1 / 2 cos γ 0 m ( λ - λ * ) / d 0 m λ / d 0 ] x + [ 1 / 2 m ( λ - λ * ) / d 0 sin 2 γ 0 ] x 2 y - [ 1 / 2 m ( λ - λ * ) / d 0 m λ / d 0 ] y 2 + [ 1 / 6 m ( λ - λ * ) / d 0 ] y 3 .
λ / Δ λ 8 / [ f x - 2 + f y - 2 + 8 m ( λ - λ * ) / d 0 / f y ) ] .
Δ = Δ + ( 1 / 2 ) ( Δ x D / L 0 ) y 2 .
Δ x D = R ( cos ϕ - cos ϕ * ) ,
Δ x D / L 0 - ( 1 / 2 ) ( m / d 0 ) 2 λ ( λ * - λ ) L 0 / R .
Δ α 0 ( F / L 0 ) ɛ .
Δ λ = ( d 0 / m ) × ( Δ β 0 sin β 0 - Δ α 0 sin α 0 ) .
Δ λ / λ = 2 ( F / L 0 ) ɛ / ( β 0 2 / α 0 2 - 1 α 0 ) .
Δ λ / λ = ( F / L 0 ) ɛ / sin γ 0 / β 0 / α 0 - 1 .
ruled width [ ( L 0 / f x ) / sin γ 0 ] ( β 0 / α 0 + 1 ) / 2.
Δ λ / λ = ( F / L 0 ) ɛ / tan δ / sin γ 0 / 2 ,
groove length ( L 0 f x ) / sin γ 0 + ( L 0 / f y ) tan δ / sin γ 0 .
1 / d ( lines / mm ) = 50 / Δ λ ( A ˚ ) × ɛ ( sec of arc ) × F / L 0 / m × cos δ .
Δ M = L * - L + m N λ * ( 1 + D λ / λ * ) - Δ * ,
L = ( cos α 0 - x - Δ x ) 2 + ( y + Δ y ) 2 + ( sin α 0 + Δ z ) 2 ,
L * = ( cos β 0 - x - Δ x ) 2 + ( y + Δ y ) 2 + ( sin β 0 + Δ z ) 2
L * = ( cos α 0 - x - Δ x ) 2 + ( y + Δ y ) 2 + ( cos α 0 tan β 0 - Δ z ) 2 ,
D λ / λ 2 ( Δ z F / L 0 ) / α 0 / ( β 0 2 / α 0 2 - 1 ) .
Δ λ / λ * ( 3 / f x ) ( Δ z F / L 0 ) [ β 0 4 / α 0 4 + 2 / 3 β 0 2 / α 0 2 - 5 / 3 ) / ( β 0 2 / α 0 2 - 1 ) 2 ] .
Δ λ / λ * 2 ( Δ x F / L 0 ) / α 0 / f x / ( β 0 2 / α 0 2 - 1 ) .
Δ λ / λ * 2 ( β 0 / α 0 ) 2 ( Δ x D / L 0 ) / α 0 / f x / ( β 0 2 / α 0 2 - 1 ) .
Δ λ / λ * 2 τ y / α 0 / ( β 0 α 0 - 1 ) ,
Δ λ / λ * 4 τ y ( β 0 / α 0 - 1 ) / f x .

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