Abstract

The subject of this study is the theoretical background of astigmatism at spheres and its compensation. As a result of this study, it is shown that it is possible to construct White cells free of astigmatism for one light pass or more by separating the two rows of foci on the field mirror by a distance h. This row distance h depends on the number of passes of the light beam, the common radius of curvature R, the distance between entrance and exit focus, and the distance between the two central points on the objective mirrors. The theoretically derived mathematical equations describing this relationship are compared with results of computer simulations, and excellent agreement is observed.

© 1992 Optical Society of America

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References

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  1. J. U. White, “Long optical paths of large aperture,” J. Opt. Soc. Am. 32, 285–288 (1942).
    [CrossRef]
  2. T. H. Edwards, “Multiple-traverse absorption cell design,” J. Opt. Soc. Am. 51, 98–102 (1961).
    [CrossRef]
  3. T. R. Reesor, “Astigmatism of a multiple path absorption cell,” J. Opt. Soc. Am. 41, 1059–1060 (1951).
    [CrossRef]
  4. W. J. Riedel, “An anastigmatic White cell for IR diode laser spectroscopy,” in Monitoring of Gaseous Pollutants by Tunable Diode Lasers, R. Grisar, ed. (Kluwer, Dordrecht, The Netherlands, 1989).
    [CrossRef]
  5. G. S. Monk, Light: Principles and Experiments (McGraw-Hill, New York, 1937).

1961 (1)

1951 (1)

1942 (1)

Edwards, T. H.

Monk, G. S.

G. S. Monk, Light: Principles and Experiments (McGraw-Hill, New York, 1937).

Reesor, T. R.

Riedel, W. J.

W. J. Riedel, “An anastigmatic White cell for IR diode laser spectroscopy,” in Monitoring of Gaseous Pollutants by Tunable Diode Lasers, R. Grisar, ed. (Kluwer, Dordrecht, The Netherlands, 1989).
[CrossRef]

White, J. U.

J. Opt. Soc. Am. (3)

Other (2)

W. J. Riedel, “An anastigmatic White cell for IR diode laser spectroscopy,” in Monitoring of Gaseous Pollutants by Tunable Diode Lasers, R. Grisar, ed. (Kluwer, Dordrecht, The Netherlands, 1989).
[CrossRef]

G. S. Monk, Light: Principles and Experiments (McGraw-Hill, New York, 1937).

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Figures (13)

Fig. 1
Fig. 1

Back view of a White cell showing the ray trace for two light passes. From the entrance focus P0 the light beam hits the first objective mirror O1 and is then focused onto the field mirror F at point H1. From there the beam impinges on the second objective mirror O2 and is reflected back to F at point P1. After this the beam hits O1 again and is focused at H2. After the second reflection on O2 the light leaves the White cell at point P2.

Fig. 2
Fig. 2

Point-to-point imaging process by the arc of curvature in the second apex of an ellipse. The ellipse with its two focal points P1 and P2 allows an exact imaging process with separation between object and image.

Fig. 3
Fig. 3

By rotating the arrangement around the g axis point P2 becomes the meridional image M. All the reflected rays intersect the g axis, giving rise to the sagittal image S. The distance between D and M is the meridional distance m; the distance between D and S is the sagittal distance s. The astigmatic difference d is the difference between s and m.

Fig. 4
Fig. 4

Compensation of astigmatism by complementary imaging processes. Two identical imaging processes at spherical mirrors, lying in planes perpendicular to each other, permit a total imaging process from P to P′ without astigmatism. Z and Z′ are the centers of curvature of the objective mirrors at D and D′.

Fig. 5
Fig. 5

Top view of a White cell without astigmatism for one light pass. The field mirror F at H folds the optical arrangement shown in Fig. 4 and reflects all the rays from the objective mirror O onto O′. Z and Z′ are the centers of curvature and D and D′ lie on the principal axis.

Fig. 6
Fig. 6

Ray trace of an anastigmatic White cell projected onto the plane that contains the entrance and exit focuses P and P′ and point H on the field mirror. This projection is called the first-order imaging process.

Fig. 7
Fig. 7

Astigmatic difference in relation to h, the distance between the two rows of foci, for R = 625 mm and for p = 20 and 40 mm. The results of computer simulations are indicated by pluses; the outlined curves are the graphs that correspond to Eq. (12).

Fig. 8
Fig. 8

(a) Second-order imaging process. The central ray from the entrance focus P0 hits the field mirror at point H1, P1, and H2 and leaves the White cell at focus P2. (b) The second-order imaging process may be split into two first-order processes. Complementary rays from (a) are combined into symmetrical pairs. The total astigmatic difference computes as the sum of the differences for the corresponding first-order processes.

Fig. 9
Fig. 9

Fourth-order imaging process. According to Fig. 8(a) it may be split into four processes of first order.

Fig. 10
Fig. 10

Value of V, the ratio of row distance h and focus distance p, as a function of n, the number of passes. The results of computer simulations Vs and theoretical considerations Vth are marked with triangles and pluses, respectively.

Fig. 11
Fig. 11

Construction of a ray trace resulting from a displacement Δ of the central point from the second apex D to A.

Fig. 12
Fig. 12

Values of Vth(0) (pluses) and Vs(Δ) (triangles) as a function of n. Vs(Δ) shows an almost linear decrease for four passes and more.

Fig. 13
Fig. 13

Values of Vth(0) (pluses), Vth(Δ) (boxes), and Vs(Δ) (triangles) as a function of n. The theoretical values Vth(Δ) are in good agreement with Vs(Δ); the values were calculated by computer simulations.

Equations (58)

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c 2 = b ( R - b ) .
c 2 = a 2 - b 2 = b ( a 2 / b - b ) .
d = P 2 S = s - a ,
d = a ( 1 / cos 2 α - 1 ) = 2 a sin 2 α / ( 1 - 2 sin 2 α ) ,
a = R cos α ,             c = a sin α ,
d = 2 c 2 / [ R cos α ( 1 - 2 sin 2 α ) ] .
1 / a + 1 / m = 2 / R cos α ,
1 / a + 1 / s = 2 cos α / R .
s = a / ( 2 cos 2 α - 1 ) = a / cos 2 α ,
d = s - a = a ( 1 / cos 2 α - 1 ) .
d = 2 c 2 / R .
d 0 = 4 c 2 / R p 2 / R .
d = d 0 [ 1 - ( h / p ) 2 ] = ( p 2 - h 2 ) / R ,
R * d tot = p 1 2 - h 2 + p 2 2 - h 2 = 9 δ 2 - h 2 + δ 2 - h 2 = 10 δ 2 - 2 h 2 = 5 p 2 / 2 - 2 h 2 .
p k = 2 p - ( 2 k - 1 ) δ .
R * d tot = k = 1 n ( p k 2 - h 2 ) = k = 1 n [ 2 n - ( 2 k - 1 ) ] 2 δ 2 - n h 2 = l = 1 n ( 2 l - 1 ) 2 δ 2 - n h 2             ( l : = n - k + 1 ) = n ( 4 n 2 - 1 ) δ 2 / 3 - n h 2 ,
R * d tot = n [ 4 p 2 ( 1 - 1 / 4 n 2 ) / 3 - h 2 ] .
V = h / p = ( 4 / 3 ) 0.5 [ 1 - 1 / ( 2 n ) 2 ] 0.5 ;
P 0 T = p 1 cos β ,
H 1 T = p 1 sin β ,
H 1 T = p 1 sin β * tan α ,
H 1 H 1 = p 1 sin β / cos α ,
sin β tan β = Δ / R ,             tan α p 1 / 2 R ,             cos α 1 ,
p 1 = P 0 H 1 = p 1 ( cos β - sin β * tan α )
p 1 2 = p 1 2 cos 2 β ( 1 - tan β * tan α ) 2 ,
p 1 2 [ 1 - ( Δ / R ) 2 ] ( 1 - Δ * p 1 / R 2 ) ,
p 1 2 [ 1 - ( Δ / R ) 2 - Δ * p 1 / R 2 ] ,
w 1 = H 1 H 1 Δ * p 1 / R .
Δ = p - δ / 2 = p - p / 2 n .
Δ = ( 2 n r - 1 ) * p / 2 n .
p k = ( 2 n + 1 - 2 k ) p / n ,
p k 2 = ( 2 n + 1 - 2 k ) 2 * [ 1 - ( Δ / R ) 2 - Δ ( 2 n + 1 - 2 k ) p / ( n R 2 ) ] ( p / n ) 2 ,
= p / n R ,             Ω = ½ ( 2 n r - 1 ) ,
Δ = Ω R ,             p k = ( 2 n + 1 - 2 k ) R ,
p k 2 = ( 2 n + 1 - 2 k ) 2 × { 1 - Ω [ Ω + ( 2 n + 1 - 2 k ) ] 2 } R 2 2 .
a = a + σ ,             m = m + σ m ,             s = s + σ s .
1 / a + 1 / m = 1 / a + 1 / m = 1 / ( a + σ ) + 1 / ( m + σ m ) ( 1 / a + 1 / m ) × 1 + ( σ + σ m ) / ( a + m ) 1 + ( σ m + σ m a ) / a m ,             σ m σ 0.
1 + ( σ + σ m ) / ( a + m ) = 1 + ( σ m + σ m a ) / a m ,
σ m = - σ ( m / a ) 2 .
σ s = - σ ( s / a ) 2 .
d = s - m = s - m + σ s - σ m ,
= d - ( s 2 - m 2 ) σ / a 2 ,
= d [ 1 - ( d + 2 m ) σ / a 2 ] .
d = d ( 1 - 2 σ / R ) .
R * d k = R * d k , 1 + R * d k , 2 ,
= 1 2 [ p k 2 ( 1 - 2 * m = 1 k - 1 w m / R ) - h 2 ] + 1 2 [ p k 2 ( 1 - 2 * m = 1 n w m / R - 2 * × m = k + 1 n w m / R ) - h 2 ] ,
= p k 2 ( 1 - 2 * m = 1 n w m / R + w k / R ) - h 2 .
w k = Δ p k / R = ( 2 n + 1 - 2 k ) Ω R 2 ,
m = 1 n w m = m = 1 n ( 2 n + 1 - 2 m ) Ω R 2 = n 2 Ω R 2 ,
R * d k = p k 2 { 1 - [ 2 n 2 - ( 2 n + 1 - 2 k ) ] Ω 2 } - h 2 .
R * d tot + n h 2 = k = 1 n p k 2 { 1 - [ 2 n 2 - ( 2 n + 1 - 2 k ) ] × Ω 2 } ,             ( l : = n + 1 - k ) ,
= R 2 2 l = 1 n ( 2 l - 1 ) 2 * { 1 - Ω [ Ω + ( 2 l - 1 ) ] 2 } × { 1 - [ 2 n 2 - ( 2 l - 1 ) ] Ω 2 } ,
= R 2 2 { l = 1 n ( 2 l - 1 ) 2 [ 1 - ( 2 n 2 + Ω ) Ω 2 + 2 n 2 Ω 3 4 ] + l = 1 n ( 2 l - 1 ) 3 ( 2 n 2 - Ω ) Ω 2 4 - l = 1 n ( 2 l - 1 ) 4 Ω 2 4 } .
R * d tot + n h 2 = R 2 2 n 3 ( 4 / 3 ) ( 1 - 1 / 4 n 2 ) × [ 1 - ( 2 n 2 + Ω ) Ω 2 ] ,
= n p 2 ( 4 / 3 ) ( 1 - 1 / 4 n 2 ) × [ 1 - ( 2 n 2 + Ω ) Ω p 2 / n 2 R 2 ] .
V = ( 4 / 3 ) 0.5 ( 1 - 1 / 4 n 2 ) 0.5 [ 1 - ( 2 n 2 + Ω ) Ω p 2 n 2 R 2 ] 0.5 ,
Ω = ½ ( 2 n p - p ) / p = n r - ½ .
V = ( 4 / 3 ) 0.5 ( 1 - n r p 2 / R 2 ) .

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