Abstract

The problem of mounting a twenty-one foot concave grating and slit at oblique incidence, so that the focus will lie on a previously established track having less than one inch adjustment range, is discussed. A simple technique, employing only a good transit, is described. The development of this method depends on a precise knowledge of the nature of the general focal curve. This information is found by a brief examination of grating theory. A table of permissible departures of the slit from the Rowland circle is given, such that the error in path length to the sharpest image is less than λ/4. Application of classical grating theory to the case of oblique incidence and diffraction results in an expression for maximum useful grating width. This expression is shown to differ only by a factor of 1.06 from the expression derived in a recent article by Mack, Stehn and Edlén.

© 1933 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Mack, Stehn, and Edlén, “On the Concave Grating Spectrograph,” J. Opt. Soc. Am. 22, 260 (1932).
  2. A communication from Dr. Mack advises that the paragraph quoted was not intended to be a discussion of the general focal curve. According to that author, the quoted paragraph should begin somewhat as follows—“By the Upsala method of adjustment, the photographic plate and the slit are placed quite accurately on a circle of radius ρ/2, tangent to the grating circle.” The remainder of the paragraph then discusses correctly the errors in phase introduced by the special misadjustment described there.
  3. This result is mentioned with others in the discussion of the Rowland mounting in Kayser, Handbuch der Spectroscopie, Vol.  1, 443, p. 468. The discussion of various slit adjustments in 448, p. 476 is quite complete, and applicable to most mounts.
  4. This comparison is also embodied in reference 8 of Mack, Stehn and Edlén’s paper. The limiting useful values of H (directly proportional to grating width) as calculated by Rayleigh’s criterion and by the new generalization of that criterion are shown to be 1.120 and 1.177 which bear the ratio of 1 : 1.053.

1932 (1)

Mack, Stehn, and Edlén, “On the Concave Grating Spectrograph,” J. Opt. Soc. Am. 22, 260 (1932).

Edlén,

Mack, Stehn, and Edlén, “On the Concave Grating Spectrograph,” J. Opt. Soc. Am. 22, 260 (1932).

Kayser,

This result is mentioned with others in the discussion of the Rowland mounting in Kayser, Handbuch der Spectroscopie, Vol.  1, 443, p. 468. The discussion of various slit adjustments in 448, p. 476 is quite complete, and applicable to most mounts.

Mack,

Mack, Stehn, and Edlén, “On the Concave Grating Spectrograph,” J. Opt. Soc. Am. 22, 260 (1932).

Stehn,

Mack, Stehn, and Edlén, “On the Concave Grating Spectrograph,” J. Opt. Soc. Am. 22, 260 (1932).

Handbuch der Spectroscopie (1)

This result is mentioned with others in the discussion of the Rowland mounting in Kayser, Handbuch der Spectroscopie, Vol.  1, 443, p. 468. The discussion of various slit adjustments in 448, p. 476 is quite complete, and applicable to most mounts.

J. Opt. Soc. Am. (1)

Mack, Stehn, and Edlén, “On the Concave Grating Spectrograph,” J. Opt. Soc. Am. 22, 260 (1932).

Other (2)

A communication from Dr. Mack advises that the paragraph quoted was not intended to be a discussion of the general focal curve. According to that author, the quoted paragraph should begin somewhat as follows—“By the Upsala method of adjustment, the photographic plate and the slit are placed quite accurately on a circle of radius ρ/2, tangent to the grating circle.” The remainder of the paragraph then discusses correctly the errors in phase introduced by the special misadjustment described there.

This comparison is also embodied in reference 8 of Mack, Stehn and Edlén’s paper. The limiting useful values of H (directly proportional to grating width) as calculated by Rayleigh’s criterion and by the new generalization of that criterion are shown to be 1.120 and 1.177 which bear the ratio of 1 : 1.053.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Section of grating spectrograph, perpendicular to direction of rulings. P and P′ are successive rulings on the surface of the grating, 0 is the point of tangency with the grating of the axis along which the constant grating interval, σ, is measured. A, the slit, and A′, the point at which the diffracted light is observed, are in general at arbitrary positions in this plane of the section shown. For final focus, A and A′ are made to lie on the Rowland circle, R, which is tangent to the grating at 0 and has the radius of curvature, ρ, of the grating as its diameter.

Fig. 2
Fig. 2

Exaggerated diagram illustrating method of placing focal curve coincident with a previously established circle of diameter ρ. S1, G1, K1 are preliminary positions of the slit, grating, and grating center-of-curvature. Images of slit by λ5461 for this arrangement are indicated by A1′ and B1′. C1 is the center of the circle through O1, A1′, and B1′. 2Δr is the departure of S1 from this circle. S2, G2, K2 are the corrected positions of the slit, grating, and grating center-of-curvature. C2, the center of the circle computed through O2, A2′, and B2′, coincides with C, the center of the established circle. This focal circle passes through the slit, S2. Therefore the Rowland circle of G2 is the focal curve, and coincides with the established circle (drawn heavily in the figure).

Tables (1)

Tables Icon

Table I Greatest permissible departures, Δr, of slit from Rowland circle.

Equations (16)

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

a 2 + b 2 = r 2 ,             2 ρ x = x 2 + y 2 ,
u 2 = ( x - a ) 2 + ( y - b ) 2 = r 2 - 2 b y + y 2 + x 2 - 2 a x = ( r - b y / r ) 2 + ( a / r 2 - 1 / ρ ) a y 2 + ( 1 - a / ρ ) x 2 .
u = ( r - b y / r ) + 1 2 ( a / r 2 - 1 / ρ ) a y 2 ( r - b y / r ) + = ( r - b y / r ) + 1 2 r ( a / r 2 - 1 / ρ ) a y 2 ( 1 + b y / r 2 + ) + .
v = ( r - b y / r ) + 1 2 r ( a / r 2 - 1 / ρ ) a y 2 ( 1 + b y / r 2 + ) + ;
u + v = ( r + r ) - ( b r + b r ) y + [ a r ( a r 2 - 1 ρ ) + a r ( a r 2 - 1 ρ ) ] y 2 2 + [ a b r 3 ( a r 2 - 1 ρ ) + a b r 3 ( a r 2 - 1 ρ ) ] y 3 2 + .
sin θ + sin θ = ± m λ / σ .
( a / r ) ( a / r 2 - 1 / ρ ) + ( a / r ) ( a / r 2 - 1 / ρ ) = 0.
cos 2 θ / r + cos 2 θ / r = ( cos θ + cos θ ) / ρ .
( cos 2 θ / r 2 ) Δ r + ( cos 2 θ / r ) Δ r 2 = 0.
cos 2 θ / r 2 = cos 2 θ / r 2 = 1 / ρ 2 .
a / r 2 - 1 / ρ = ( ρ cos θ - r ) / r ρ = Δ r / r ρ , a / r 2 - 1 / ρ = Δ r / r ρ = - Δ r / r ρ ,
Δ 3 ( u + v ) = [ a b r 3 ( a / r 2 - 1 / ρ ) + a b r 3 ( a / r 2 - 1 / ρ ) ] y 3 2 = ( a b r 4 ρ - a b r 4 ρ ) Δ r y 3 2 = ( sin θ cos θ r 2 ρ - sin θ cos θ r 2 ρ ) Δ r y 3 2 = ( tan θ - tan θ ) Δ r 2 ( y ρ ) 3 .
u = r - b y / r + ( x 2 / 2 r ) ( 1 - a / ρ ) , u + v = r + r - ( sin θ + sin θ ) y + ( x 2 / 2 ) [ ( 1 - cos 2 θ ) / r + ( 1 - cos 2 θ ) r ] .
Δ 4 ( u + v ) = y 4 8 ρ 3 [ sin 2 θ cos θ + sin 2 θ cos θ ] = y 4 8 ρ 3 [ tan θ sin θ + tan θ sin θ ] .
2 Y m = 2 [ 2 λ ρ 3 tan θ sin θ + tan θ sin θ ] 1 4 .
2 Y opt . = 2.36 β = 2.36 [ 2 λ ρ 3 ( 2 / π ) tan θ sin θ + tan θ sin θ ] 1 4 = 1.06 ( 2 Y m ) .