Abstract

<p>As an application of the theoretical procedure suggested in the first part of the paper, the design of a general optical system is considered. In order that the formulation may be useful, the general equations deduced in Part I, which merely expressed the classic laws of geometrical optics, should be supplemented by a number of equations expressing the elimination of the principal aberrations: this requires that a literal “ray tracing” be effected. In the tangential plane, this projective ray tracing is merely an application of elementary principles: in the extrameridian case, an artifice is introduced by which points and lines of a tridimensional space can be mapped in a plane through the use of complex coordinates. Though the discussion of such <i>complex mapping</i> requires some supplementary theory to be developed, the extrameridian ray tracing can eventually be performed without greater difficulty than that encountered in the tangential case.</p><p>The conclusions are illustrated by the discussion of the design of a general Schmidt system.</p>

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  1. The shortest way to obtain the result is that of writing [Equation] with tanα = u1, tanβ = u2.
  2. The actual slope of the line P1P2 is [Equation] The last term, however, only introduces a second-order correction in the small term δυ2.
  3. The condition is necessary, but not sufficient: the knowledge of the abscissa alone, in fact, does not give the position of the line, but only the direction.
  4. For instance,E. P. Adams, "Smithsonian Mathematical Formulae" (Washington, 1947), p. 30–33.
  5. In the meridian case, we spoke indifferently of "equinormal" or "equitangent" pairs. In the present case, two concurrent pairs of rays are equinormal but not equitangent, as the tangents can lie in different planes.
  6. It must be remembered that the mere condition of coplanarity of the two rays imposes that (p2 - p1)/(u2 - u1) = (q2 - q1) / (υ2- υ1).
  7. The convention ∂u/∂β=u implies that all the partial derivatives of u with respect to β be u; so that the finite variation Δu corresponding to a finite Δξ is given by MacLaurin's expansion as Δu=u{Δβ+(Δβ)2/2! + (Δβ)3/3!+ … = u {eΔβ-1}, as should be.
  8. Reference is made to Fig. 12 for symbols of points: in the present case, a figure cannot be drawn, as the points are complex.
  9. This means, physically, that the absolute angle between the entrance ray and the normal differs from its meridian projection by a second-order angle.
  10. The reader will remark that the formula differs from formula (20) established in Sec. 13. This is due to the fact that the entrance ray of Fig. 17 is directed to the left while that of Fig. 7 goes from left to right. The lines of the two rays being the same, the normal of Fig. 7 is the tangent of 17, and vice versa.
  11. In other words, the line is the locus of the points (u,υ) that map physical lines passing through x0, y0.
  12. The formula is valid for the mirror by making ν1 = - ν2.
  13. A shorter way of approach is indicated in Sec. 29.
  14. This is a general property of the present step by step procedure.
  15. It is found that d3ξ = aλ + d2ξ, and d2ξ = Bλ + d2ξ, where a and B are rational expressions in terms of the constants u10, u20, etc. ...
  16. The minus sign is taken to conform with Fig. 10 where the displaced point B2 is assumed to be on the right of B0 when facing the first surface.
  17. The "first differentials" are zero as the displaced ray meets the first surface in the design point.
  18. One surface was obliged, so that a single design condition could be imposed: in the actual case, Abbe's condition was accepted.

Adams, E. P.

For instance,E. P. Adams, "Smithsonian Mathematical Formulae" (Washington, 1947), p. 30–33.

Other (18)

The shortest way to obtain the result is that of writing [Equation] with tanα = u1, tanβ = u2.

The actual slope of the line P1P2 is [Equation] The last term, however, only introduces a second-order correction in the small term δυ2.

The condition is necessary, but not sufficient: the knowledge of the abscissa alone, in fact, does not give the position of the line, but only the direction.

For instance,E. P. Adams, "Smithsonian Mathematical Formulae" (Washington, 1947), p. 30–33.

In the meridian case, we spoke indifferently of "equinormal" or "equitangent" pairs. In the present case, two concurrent pairs of rays are equinormal but not equitangent, as the tangents can lie in different planes.

It must be remembered that the mere condition of coplanarity of the two rays imposes that (p2 - p1)/(u2 - u1) = (q2 - q1) / (υ2- υ1).

The convention ∂u/∂β=u implies that all the partial derivatives of u with respect to β be u; so that the finite variation Δu corresponding to a finite Δξ is given by MacLaurin's expansion as Δu=u{Δβ+(Δβ)2/2! + (Δβ)3/3!+ … = u {eΔβ-1}, as should be.

Reference is made to Fig. 12 for symbols of points: in the present case, a figure cannot be drawn, as the points are complex.

This means, physically, that the absolute angle between the entrance ray and the normal differs from its meridian projection by a second-order angle.

The reader will remark that the formula differs from formula (20) established in Sec. 13. This is due to the fact that the entrance ray of Fig. 17 is directed to the left while that of Fig. 7 goes from left to right. The lines of the two rays being the same, the normal of Fig. 7 is the tangent of 17, and vice versa.

In other words, the line is the locus of the points (u,υ) that map physical lines passing through x0, y0.

The formula is valid for the mirror by making ν1 = - ν2.

A shorter way of approach is indicated in Sec. 29.

This is a general property of the present step by step procedure.

It is found that d3ξ = aλ + d2ξ, and d2ξ = Bλ + d2ξ, where a and B are rational expressions in terms of the constants u10, u20, etc. ...

The minus sign is taken to conform with Fig. 10 where the displaced point B2 is assumed to be on the right of B0 when facing the first surface.

The "first differentials" are zero as the displaced ray meets the first surface in the design point.

One surface was obliged, so that a single design condition could be imposed: in the actual case, Abbe's condition was accepted.

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