Abstract

As an example of a mounting in which the optical components are not on the Rowland cylinder, the Seya-Namioka mounting is treated mainly from the standpoint of physical optics. An ambiguity in the physical meaning of Beutler’s focal conditions is clarified; using the corrected condition a brief summary of the optical conditions in this mounting is given. Astigmatism, spectral line shape, instrumental resolving power, and optimum grating width are discussed in detail and, for convenience of practical application, numerical results are also given for the following conditions: 1-m concave grating with 15 000 lines/in., first order spectrum, with ratios of the radius of curvature of the grating to the distances between the grating center and the entrance and exit slits 1.2223, and 1.2230, respectively, and the angle between the lines connecting the grating center to the entrance and exit slit 70° 15′.

© 1959 Optical Society of America

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References

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  1. Y. Fujioka and R. Ito, Sci. of Light (Tokyo) 1, 1 (1951); Tousey, Johnson, Richardson, and Toran, J. Opt. Soc. Am. 41, 696 (1951);R. A. Sawyer, Experimental Spectroscopy (Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1944), second edition, pp. 161–164.Instead of moving a concave grating the exit slit can be moved along the Rowland circle with fixed entrance slit and grating. However, this is practically not convenient and has lack of compactness. For this kind of mounting, see T. J. M. Sluyters and E. De Hass, Rev. Sci. Instr. 29, 597 (1958).
    [Crossref]
  2. In a commercial vacuum ultraviolet monochromator, a simple link motion is used. See also Bair, Cross, Dawson, Wilson, and Wise, J. Opt. Soc. Am. 43, 681 (1953).
    [Crossref]
  3. M. Seya, Sci. of Light (Tokyo) 2, 8 (1952).
  4. T. Namioka, Sci. of Light (Tokyo) 3, 15 (1954).
  5. H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).
  6. P. D. Johnson, Rev. Sci. Instr. 28, 833 (1957).
    [Crossref]
  7. R. Onaka, Sci. of Light (Tokyo) 7, 23 (1958).
  8. T. Namioka, J. Opt. Soc. Am. 49, 446 (1959).
    [Crossref]
  9. Mack, Stehn, and Edlén, J. Opt. Soc. Am. 22, 245 (1932).
    [Crossref]
  10. H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945). For the corrected theory, see reference 8.
    [Crossref]
  11. For details, see references 3 and 5.
  12. For details, refer to Sec. II of reference 8.
  13. Image points and points at which diffraction takes place are, in the sense of geometrical optics, in “one to one” correspondence for a given point light source. In geometrical optics, an image is defined in this case as a figure produced by intersection of the diffracted rays and a given screen surface regardless of whether the quality of the image is good or bad. Equations (3) and (4) give only directions in space, β0 and z0′/r0′, for the central ray and thus z0′ and r0′ cannot be determined independently but are uniquely determined when the equation of the screen surface is known.
  14. For the reason why we considered (5) but not (6), refer to the discussion following Eq. (4″) of reference 8.
  15. Refer to pp. 316 and 319 of reference 10.
  16. Hurzeler, Inghram, and Morrison, J. Chem. Phys. 27, 313 (1957);J. Chem. Phys. 28, 76 (1958).
    [Crossref]
  17. In Sec. V and VI we are dealing with the resolving power of the grating itself and not including any effects due to the finite slit width, Doppler effect, sensitivity of the detector, etc. Therefore, in practical cases such effects must be taken into consideration.

1959 (1)

1958 (1)

R. Onaka, Sci. of Light (Tokyo) 7, 23 (1958).

1957 (3)

Hurzeler, Inghram, and Morrison, J. Chem. Phys. 27, 313 (1957);J. Chem. Phys. 28, 76 (1958).
[Crossref]

H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).

P. D. Johnson, Rev. Sci. Instr. 28, 833 (1957).
[Crossref]

1954 (1)

T. Namioka, Sci. of Light (Tokyo) 3, 15 (1954).

1953 (1)

1952 (1)

M. Seya, Sci. of Light (Tokyo) 2, 8 (1952).

1951 (1)

Y. Fujioka and R. Ito, Sci. of Light (Tokyo) 1, 1 (1951); Tousey, Johnson, Richardson, and Toran, J. Opt. Soc. Am. 41, 696 (1951);R. A. Sawyer, Experimental Spectroscopy (Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1944), second edition, pp. 161–164.Instead of moving a concave grating the exit slit can be moved along the Rowland circle with fixed entrance slit and grating. However, this is practically not convenient and has lack of compactness. For this kind of mounting, see T. J. M. Sluyters and E. De Hass, Rev. Sci. Instr. 29, 597 (1958).
[Crossref]

1945 (1)

1932 (1)

Bair,

Beutler, H. G.

Cross,

Dawson,

Edlén,

Fujioka, Y.

Y. Fujioka and R. Ito, Sci. of Light (Tokyo) 1, 1 (1951); Tousey, Johnson, Richardson, and Toran, J. Opt. Soc. Am. 41, 696 (1951);R. A. Sawyer, Experimental Spectroscopy (Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1944), second edition, pp. 161–164.Instead of moving a concave grating the exit slit can be moved along the Rowland circle with fixed entrance slit and grating. However, this is practically not convenient and has lack of compactness. For this kind of mounting, see T. J. M. Sluyters and E. De Hass, Rev. Sci. Instr. 29, 597 (1958).
[Crossref]

Greiner, H.

H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).

Hurzeler,

Hurzeler, Inghram, and Morrison, J. Chem. Phys. 27, 313 (1957);J. Chem. Phys. 28, 76 (1958).
[Crossref]

Inghram,

Hurzeler, Inghram, and Morrison, J. Chem. Phys. 27, 313 (1957);J. Chem. Phys. 28, 76 (1958).
[Crossref]

Ito, R.

Y. Fujioka and R. Ito, Sci. of Light (Tokyo) 1, 1 (1951); Tousey, Johnson, Richardson, and Toran, J. Opt. Soc. Am. 41, 696 (1951);R. A. Sawyer, Experimental Spectroscopy (Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1944), second edition, pp. 161–164.Instead of moving a concave grating the exit slit can be moved along the Rowland circle with fixed entrance slit and grating. However, this is practically not convenient and has lack of compactness. For this kind of mounting, see T. J. M. Sluyters and E. De Hass, Rev. Sci. Instr. 29, 597 (1958).
[Crossref]

Johnson, P. D.

P. D. Johnson, Rev. Sci. Instr. 28, 833 (1957).
[Crossref]

Mack,

Morrison,

Hurzeler, Inghram, and Morrison, J. Chem. Phys. 27, 313 (1957);J. Chem. Phys. 28, 76 (1958).
[Crossref]

Namioka, T.

T. Namioka, J. Opt. Soc. Am. 49, 446 (1959).
[Crossref]

T. Namioka, Sci. of Light (Tokyo) 3, 15 (1954).

Onaka, R.

R. Onaka, Sci. of Light (Tokyo) 7, 23 (1958).

Schäffer, E.

H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).

Seya, M.

M. Seya, Sci. of Light (Tokyo) 2, 8 (1952).

Stehn,

Wilson,

Wise,

J. Chem. Phys. (1)

Hurzeler, Inghram, and Morrison, J. Chem. Phys. 27, 313 (1957);J. Chem. Phys. 28, 76 (1958).
[Crossref]

J. Opt. Soc. Am. (4)

Optik (1)

H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).

Rev. Sci. Instr. (1)

P. D. Johnson, Rev. Sci. Instr. 28, 833 (1957).
[Crossref]

Sci. of Light (Tokyo) (4)

R. Onaka, Sci. of Light (Tokyo) 7, 23 (1958).

M. Seya, Sci. of Light (Tokyo) 2, 8 (1952).

T. Namioka, Sci. of Light (Tokyo) 3, 15 (1954).

Y. Fujioka and R. Ito, Sci. of Light (Tokyo) 1, 1 (1951); Tousey, Johnson, Richardson, and Toran, J. Opt. Soc. Am. 41, 696 (1951);R. A. Sawyer, Experimental Spectroscopy (Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1944), second edition, pp. 161–164.Instead of moving a concave grating the exit slit can be moved along the Rowland circle with fixed entrance slit and grating. However, this is practically not convenient and has lack of compactness. For this kind of mounting, see T. J. M. Sluyters and E. De Hass, Rev. Sci. Instr. 29, 597 (1958).
[Crossref]

Other (6)

In Sec. V and VI we are dealing with the resolving power of the grating itself and not including any effects due to the finite slit width, Doppler effect, sensitivity of the detector, etc. Therefore, in practical cases such effects must be taken into consideration.

For details, see references 3 and 5.

For details, refer to Sec. II of reference 8.

Image points and points at which diffraction takes place are, in the sense of geometrical optics, in “one to one” correspondence for a given point light source. In geometrical optics, an image is defined in this case as a figure produced by intersection of the diffracted rays and a given screen surface regardless of whether the quality of the image is good or bad. Equations (3) and (4) give only directions in space, β0 and z0′/r0′, for the central ray and thus z0′ and r0′ cannot be determined independently but are uniquely determined when the equation of the screen surface is known.

For the reason why we considered (5) but not (6), refer to the discussion following Eq. (4″) of reference 8.

Refer to pp. 316 and 319 of reference 10.

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Figures (7)

F. 1
F. 1

Schematic diagram of the optical system. A(x,y,z) is an illuminated point on the entrance slit; B(x′,y′,z′) is an image point; and P(u,w,l) is a point on the grating rulings. The origin, O, of the Cartesian coordinate system is the center of the grating rulings 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. The signs of α and β are opposite if A and B lie on different sides of the xz 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.

F. 2
F. 2

The length of the astigmatic image for a point light source per unit length of the grooves. [z′]ast is the length of the astigmatic image for a point light source, L is the total length of the grating rulings and α the angle of incidence. The solid curve is for the conditions; ρ = R/r = 1.2223, p′ = R/r′ = 1.2230, and C = 70° 15′. R is the radius of curvature of the grating, r and r′ are the distances of the entrance and exit slits from the grating center, which is the tangent point to the Rowland circle, respectively, and C the angle between the lines connecting the grating center to the entrance and exit slits. The dashed curve shows the values of [z′]ast/L for the case that all optical components are mounted on a Rowland circle but keeping C = 70° 15′. Wavelengths corresponding to the values of α are also shown at the upper coordinate for both positive and negative spectrum.

F. 3
F. 3

The length of illuminated grooves, Lb, necessary to give spectral lines with approximately uniform intensity over the exit slit of the length Zb′. α is the angle of incidence and a0 the half length of the illuminated part of the entrance slit. The apparatus constants used here are ρ = R/r = 1.2223, ρ′ = R/r′ = 1.2230, and C = 70° 15′ (for the definitions of the constants, refer to the caption of Fig. 2).

F. 4
F. 4

Spectral line shapes at 1510 A for several values of the grating width, W: (a) W = 2.0 cm, (b) W = 2.4cm, (c) W = 2.8cm, (d) W = 3.2 cm, (e) W = 4.0 cm, (f) W = 5.2 cm, (g) W = 6.0cm, (h) W = 7.2cm, and (i) W = 8.0 cm. is a parameter proportional to an amount of displacement along the run of the spectrum from the position where the central maximum would be expected from the grating equation. One unit of corresponds to 0.0426 A in wavelength units. The scale corresponding to 0.1 A is shown in Fig. 4(a). The solid curves are the intensity-curves for a point light source and the dashed curves for the line light source of 16.3 μ width and zero height. (Δ)r and (Δ)R shown by arrows in the figures are parameters related to the minimum resolvable wavelength differences in the case of a point light source and of a line light source, respectively. The intensity distributions are calculated under the conditions; 1-m concave grating with 15 000 lines/ in., negative first order, ρ = R/r = 1.2223, ρ′ = R/r′ = 1.2230, and C = 70° 15′ (for the definitions of the constants, refer to the caption of Fig. 2).

F. 5
F. 5

Spectral line shapes at 5818 A for several values of the grating width, W, and for a point light source. The numbers drawn in the figures are the grating widths in cm units. In this figure only the principal maxima are shown and the secondary maxima which are very similar to those in Fig. 4 are omitted here. is a parameter proportional to an amount of displacement along the run of the spectrum from the position where the central maximum would be expected from the grating equation. One unit of corresponds to 0.1642 A in wavelength units. The intensity curves are obtained under the conditions; 1-m concave grating with 15 000 lines/in, negative first order, ρ = R/r = 1.2223, ρ′ = R/r′ = 1.2230, and C = 70° 15′ (for the definitions of the constants, refer to the caption of Fig. 2).

F. 6
F. 6

The resolving power of the grating at 1510 A. The solid curve shows the resolving power of the grating for which the secondary maxima are not intense enough to be troublesome. The dashed curve is the resolving power obtained by applying a conventional definition of resolving power to the intensity curves having very strong secondary maxima. Weff at the right-hand coordinate shows the width of a plane grating which has resolving power equivalent to that of the concave grating. Both curves are obtained from the intensity curves shown in Fig. 4. Note that R is the resolving power of the grating itself and does not include any factors due to the finite slit width, Doppler effect, sensitivity of the detector, and etc. Therefore, in practical cases these effects must be taken into consideration, namely, the values of resolving power shown here will be reduced further.

F. 7
F. 7

The total intensity of the image at 1510 A. This curve is obtained by integrating over the whole area under the intensity curves (solid ones) in Fig. 4. W is the grating width.

Equations (62)

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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 and F / l = 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 + w ( cos 2 α r cos α R + cos 2 β 0 r cos β 0 R ) + O ( w 2 R 2 ) = 0 ,
Δ ( z r ) + l ( 1 r cos α R + 1 r cos β 0 R ) + O ( w 2 R 2 ) = 0
cos 2 α r cos α R + cos 2 β 0 r cos β 0 R = 0 .
( F 2 + F 2 ) / w = 0 ,
w ( cos 2 α r cos α R + cos 2 β r cos β R ) + 3 2 w 2 [ sin α r ( cos 2 α r cos α R ) + sin β r ( cos 2 β r cos β R ) ] + = 0 ,
r = R cos α and r = R cos β 0 ,
r = and r = R cos 2 β 0 / ( cos α + cos β 0 ) .
r = cos 2 β 0 ( cos α + cos β 0 R cos 2 α r )
f = cos 2 α r cos α R + cos 2 β 0 r cos β 0 R ,
I / ρ = 0 ; I / ρ = 0 ; I / C = 0 ,
I = Ω 1 Ω 2 f 2 d k , ρ = R / r , ρ = R / r ,
α = 1 2 C + k , β 0 = ( 1 2 C k ) , Ω 1 k Ω 2 ,
Ω 1 Ω 2 f f ρ d k = 0 ,
Ω 1 Ω 2 f f ρ d k = 0 ,
Ω 1 Ω 2 f f C d k = 0 .
f = f ( α 0 ) + [ f α ] α = α 0 Δ α + 1 2 [ 2 f α 2 ] α = α 0 ( Δ α ) 2 + ,
cos 2 α 0 r + cos 2 ( α 0 C ) r cos α 0 + cos ( α 0 C ) R = 0 ,
sin 2 α 0 r sin 2 ( α 0 C ) r + sin α 0 + sin ( α 0 C ) R = 0 ,
2 cos 2 α 0 r 2 cos 2 ( α 0 C ) r + cos α 0 + cos ( α 0 C ) R = 0 ,
| cos 2 α 0 cos 2 ( α 0 C ) [ cos α 0 + cos ( α 0 C ) ] sin 2 α 0 sin 2 ( α 0 C ) sin α 0 + sin ( α 0 C ) 2 cos 2 α 0 2 cos 2 ( α 0 C ) cos α 0 + cos ( α 0 C ) | = 0 .
C 1 = α 0 + tan 1 ( 3 sec α 0 2 tan α 0 ) ,
C 2 = α 0 tan 1 ( 3 sec α 0 + 2 tan α 0 ) ,
C 3 = 0
d C d α 0 = 0 and [ r α 0 ] C , r = const . ,
d C 1 d α 0 = sec 2 α 0 ( 2 3 sin α 0 ) ( 1 3 sin α 0 ) 1 + ( 3 sec 2 α 0 2 tan α 0 ) 2 = 0
ρ = 1 4 sin 2 α 0 sec ( α 0 C ) cosec C + 1 2 sin α 0 cosec C + 1 2 sec ( α 0 C )
ρ = 2 sec α 0 ρ , α 0 1 2 π
Δ r ( Δ α ) 2 R ( 2 R 1 r ) R 2 ( Δ α ) 2 ,
C = 70 ° 15 and r r = 81.8 cm for R = 1 m .
[ z r ( 1 + z 2 r 2 ) 1 2 + z r ( 1 + z 2 r 2 ) 1 2 ] + l ( 1 r + 1 r cos α + cos β 0 R ) + O ( w 2 R 2 ) = 0 ;
z = z r r + l [ 1 + r r r R ( cos α + cos β 0 ) ]
= z ρ ρ + l [ 1 + ρ ρ 1 ρ ( cos α + cos β 0 ) ]
[ z ] ast = [ z ] l = 1 2 L [ z ] l = 1 2 L = L [ 1 + ρ ρ 1 ρ ( cos α + cos β 0 ) ] ,
l j = ± [ ( j 1 2 ) λ R ρ + ρ ( cos α + cos β 0 ) ] 1 2 , j = 1 , 2 , 3 , ,
L 1 = 2 | l 1 | = [ 2 λ R ρ + ρ ( cos α + cos β 0 ) ] 1 2
Z t = 2 a 0 ρ ρ + L [ 1 + ρ ρ 1 ρ ( cos α + cos β 0 ) ]
Z b = | 2 a 0 ρ ρ L [ 1 + ρ ρ 1 ρ ( cos α + cos β 0 ) ] | .
L a = 2 a 0 ρ ρ Z b 1 + ρ ρ 1 ρ ( cos α + cos β 0 ) for 2 a 0 ρ ρ > [ z ] ast
L b = 2 a 0 ρ ρ + Z b 1 + ρ ρ 1 ρ ( cos α + cos β 0 ) for 2 a 0 ρ ρ < [ z ] ast .
| n = 1 2 N 1 2 N σ 1 2 L 1 2 L δ n exp [ 2 π i F / λ ] d l | 2
| n = 1 2 N 1 2 N σ δ n exp [ 2 π i F / λ ] | 2 .
| 1 2 W 1 2 W δ n exp [ 2 π i F / λ ] d w | 2 .
2 π i F λ = 2 π i ( r + r ) λ + i π 3 ( w + A w 2 + B w 3 + D w 4 ) ,
= 6 m Δ λ σ λ .
A = 3 λ R [ ρ cos 2 α + ρ cos 2 β 0 ( cos α + cos β 0 ) ] ,
B = 3 λ R 2 [ ρ sin α cos α ( ρ cos α 1 ) + ρ sin β 0 cos β 0 ( ρ cos β 0 1 ) ] ,
D = 3 2 λ R 3 [ 4 ρ 2 sin 2 α cos α ( ρ cos α 1 ) + 4 ρ 2 sin 2 β 0 cos β 0 ( ρ cos β 0 1 ) + ρ + ρ ( cos α + cos β 0 ) ρ cos 2 α ( ρ cos α 1 ) 2 ρ 2 cos 2 β 0 ( ρ cos β 0 1 ) 2 ] .
I = ( δ n ) 2 | 1 2 W 1 2 W exp [ { 2 π i ( r + r ) / λ } + { i π 3 ( w + A w 2 + B w 3 + D w 4 ) } ] d w | 2 = ( δ n ) 2 | 1 2 W 1 2 W exp [ i π 3 ( w + A w 2 + B w 3 + D w 4 ) ] d w | 2 = 4 ( δ n ) 2 [ I 1 2 + I 2 2 ] ,
I 1 = 0 1 2 W cos [ π 3 ( A w 2 + D w 4 ) ] cos [ π 3 ( w + B w 3 ) ] d w ,
I 2 = 0 1 2 W sin [ π 3 ( A w 2 + D w 4 ) ] cos [ π 3 ( w + B w 3 ) ] d w .
R = 1 m , σ = 2.54 / 15 000 cm , m = 1 , ρ = 1.2223 , ρ = 1.2230 , C = 70 ° 15 , }
λ = 1510.025 A ( α = 32 ° , β 0 = 38 ° 1 5 ) ,
λ = 5818.14 A ( α = 23 ° , β 0 = 47 ° 1 5 ) .
( Δ ) r = 6 m ( Δ λ ) r / σ λ .
R λ ( Δ λ ) r = 6 m σ ( Δ ) r = m N [ 6 / ( Δ ) r W ] .
( Δ λ ) R = [ ( Δ λ ) r 2 + ( Δ λ ) s 2 ] 1 2 .
R λ ( Δ λ ) r = λ [ ( Δ λ ) R 2 ( Δ λ ) s 2 ] 1 2 = m λ σ [ { λ ( Δ ) R 6 } 2 ( s ρ cos β 0 R ) 2 ] 1 2
( Δ ) R = 6 m ( Δ λ ) R / σ λ .