Abstract

Although the use of prisms is often avoided because of their aberrations, they are suitable for image-tilt correction. In contradiction to current recommendations, zero astigmatism together with zero coma cannot be achieved at minimum deviation. This paper describes two configurations that meet these conditions. A correction of earlier published third-order formulas is presented and compared to ray-tracing results. The results of the third-order theory are used to develop a model for image-tilt correction, while introducing minimal coma and maintaining a flat image field.

© 1988 Optical Society of America

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References

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  1. C. J. Barth, “A High Transmission, 20-Channel Polychromator for the Thomson-Scattering Diagnostic of TORTUR III,” FOM-Instituut voor Plasmafysica, Rijnhuizen, Nieuwegein, Rijnhuizen Report 84-156 (1984).
  2. C. J. Barth, “A High Transmission 20–Channel Polychromator for Observing Non-Maxwellian Electron Velocity Distributions in Plasmas by Means of Thomson Scattering,” Appl. Opt. 27, 2981 (1988).
    [CrossRef] [PubMed]
  3. J. W. Howard, “Formulas for the Coma and Astigmatism of Wedge Prisms Used in Converging Light,” Appl. Opt. 24, 4265 (1985).
    [CrossRef] [PubMed]
  4. F. A. Jenkins, H. E. White, Eds., Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 23.
  5. W. G. Driscoll, W. Vaughan, Eds., Handbook of Optics (McGraw-Hill, New York, 1978), p. 2–41.
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), Sec. 5.5.2.

1988 (1)

1985 (1)

Barth, C. J.

C. J. Barth, “A High Transmission 20–Channel Polychromator for Observing Non-Maxwellian Electron Velocity Distributions in Plasmas by Means of Thomson Scattering,” Appl. Opt. 27, 2981 (1988).
[CrossRef] [PubMed]

C. J. Barth, “A High Transmission, 20-Channel Polychromator for the Thomson-Scattering Diagnostic of TORTUR III,” FOM-Instituut voor Plasmafysica, Rijnhuizen, Nieuwegein, Rijnhuizen Report 84-156 (1984).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), Sec. 5.5.2.

Howard, J. W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), Sec. 5.5.2.

Appl. Opt. (2)

Other (4)

F. A. Jenkins, H. E. White, Eds., Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 23.

W. G. Driscoll, W. Vaughan, Eds., Handbook of Optics (McGraw-Hill, New York, 1978), p. 2–41.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), Sec. 5.5.2.

C. J. Barth, “A High Transmission, 20-Channel Polychromator for the Thomson-Scattering Diagnostic of TORTUR III,” FOM-Instituut voor Plasmafysica, Rijnhuizen, Nieuwegein, Rijnhuizen Report 84-156 (1984).

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Figures (10)

Fig. 1
Fig. 1

Definition of parameters. The x axis is perpendicular to the plane of drawing (see text for description).

Fig. 2
Fig. 2

Two configurations to obtain stigmatic and coma-free imaging.

Fig. 3
Fig. 3

Definition of parameters which are important in analyzing prism aberrations.

Fig. 4
Fig. 4

Spot size Δx (sagittal focus) and Δy (tangential focus) as a function of the prism location Δzp with respect to the object point for a prism with α = −30°, t = 10 mm, and u = 10°: (a) minimum deviation setup i1 = −23°; (b) close to the best configuration i1 = −43° (iopt = −49°).

Fig. 5
Fig. 5

Prism and image point location (Δzp, open circles; Δzs, dots) for different incident angles i1 at which Δxmin is reached. The lines are fitted to the ray-tracing results (within the interval from iopt to zero).

Fig. 6
Fig. 6

Astigmatism Δz(ts) as a function of incident angle achieved by ray-tracing (solid line) and third-order theory (dotted line) if the zero-coma condition is fulfilled. I and II refer to the configurations presented in the Figs. 2(a) and (b), respectively.

Fig. 7
Fig. 7

Sagittal spot size (Δx, dotted line) and circle of least confusion (Δx,y, solid line) for different prism orientations.

Fig. 8
Fig. 8

Spot diagram in the x-y plane at different positions in the z-direction: (a) i1 = ioptm and Δzp = 0, steps in the z-direction of 0.01 mm; (b) minimum deviation setup while zero coma is reached; step, 0.15 mm; (c) i1 = 0 and Δzs = 0; step, 0.01 mm.

Fig. 9
Fig. 9

Definition of parameters to describe a model to correct the tilt γ of an image with height h = |AB|.

Fig. 10
Fig. 10

Field curvature of the tangential prism image as a function of the incident angle of the chief ray. A flat field is achieved at a value of i1, at which the other aberrations are also very small.

Tables (1)

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Table I Results of Image Tilt Correction

Equations (35)

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sin i opt = n · sin α .
t 1 * = t 1 · cos i 1             i 1 i 1 n , t 2 * = t 2 · cos i 2 t 2 = t - t 1             i 2 i 2 1 n , y 1 t 1 · u · cos i 1 , u u n
t 1 * y 1 n u ,
t 2 * ( t - y 1 n u ) .
A = l s - l t t i 2 ( n 2 - 1 ) n 3 ,
C s t u 2 i ( n 2 - 1 ) 2 n 3 ,
S = L - l t u 2 ( n 2 - 1 ) 2 n 3 .
( i 2 - i 1 ) = n n - 1 · δ ,
A = ( i 2 ) 2 t ( n 2 - 1 ) n 3 - y 1 u ( n + 1 n ) ( i 2 + i 1 ) δ ,
C s = i 2 t u 2 ( n 2 - 1 ) 2 n 3 - y 1 u ( n + 1 ) δ 2 n ,
S = t u 2 ( n 2 - 1 ) 2 n 3 + y 1 u ( n + 1 ) ( i 2 + i 1 ) δ 4 n 3 .
S 1 = ( n u ) 3 y { ( 1 n ) 2 - ( 1 n ) 2 } ,
C = n + 1 n u 2 y 1 δ + n 2 - 1 n 3 t u 3 i 2 ,
A = n + 1 n u y 1 δ ( i 1 + i 2 ) + n 2 - 1 n 3 t u 2 ( i 2 ) 2 .
T A y = L A · u = - l n OPD y ,
i 2 - i 1 = - n α ( = n n - 1 · δ )
A = ( n 2 - 1 n 3 ) [ t ( i 1 - α n ) 2 + Δ z p α n 2 ( α n - 2 i 1 ) ] .
C s = ( n 2 - 1 2 n 3 ) [ t ( i 1 - α n ) u 2 - Δ z p α n 2 u 2 ] .
TS = n 2 - 1 2 n 3 · t u 3 .
Δ z ( n - 1 n ) t ,
Δ z s = - ( Δ z p + t n ) .
Δ z p = t n ( i 1 n α - 1 )             for C = 0 ;
Δ z p = t n [ i 1 2 n α ( 2 i 1 - α n ) - 1 ]     for A = 0.
A = - ( n 2 - 1 ) t i 1 2 n 3 + ( n 2 - 1 ) t i 1 α n 2 ,
δ = γ + α ( 1 - 1 n ) .
α = - n n 2 - 1 γ .
Δ z A = t A n ( i 1 + β n α - 1 ) ,             Δ z B = t B n ( i 1 - β n α - 1 ) .
ϕ Δ z A - Δ z B h = t A - t B n h ( i 1 n α - 1 ) + t A + t B n 2 h α β .
ϕ i 1 n 2 - α n - β n 2 h p h .
ϕ = - γ - i 1 ,
i 1 = n 2 n 2 + 1 ( α n - γ + β n 2 · h p h ) .
i 1 = - γ n 4 n 4 - 1 + β n 2 + 1 · h p h .
γ = 13.36 ° . h = 44.3 mm , β = - 9.84 ° , h p = 70 mm , u = 5.8 ° ,             n = 1.513.
i 1 = - 21.23 ° and Δ z p = Δ z A + Δ z B 2 = - 2.33 mm .
α * = γ γ - γ s · α = - 28.15 ° .

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