The parabolic axicon: author's reply to comments

George S. Bakken Biological Station, University of Michigan, Ann Arbor,, Michigan 48104. Received 5 July 1974.

I thank Vargady1 for calling my attention to the little-known Auflicht-Dunkelfeldkondensor nach Prof. Hauser shown in Fig. 79 or 81 (depending on the edition) of Rein-ert.2 Reinert gives few details, but the only other refer­ence3 I was able to find refers to the outer reflecting ele­ment as a Parabelringspiegel, which suggests that it is probably the same optical figure that I described as the parabolic axicon.4

However, Vargady has evidently not read my paper care­fully and is in error in his other comments. In my second paragraph4 and in the figure caption for Fig. 1(A), I clearly defined the parabolic axicon as the football-shaped optical surface generated by revolving a parabola about its latus rectum Lr. As shown in my Fig. 1(A),4 light emitted from the point f on the axis Lr is reflected so as to cross the axis of revolution of the parabolic axicon Lr along a continuous line of points from f to the apex, exactly satisfying McLeod's definitions.5,6 The first two applications for the parabolic axicon, in beam-foil spectroscopy and optical pumping, use the axiconic property directly and have no auxiliary elements. I clearly indicated, in the first sen­tence on p. 1292, that the focusing system that I suggested for use in laser-induced fusion was a system incorporating a parabolic axicon and not the parabolic axicon per se.

It might be argued, as Vargady1 does, that an optical ele­ment loses its identity when incorporated into a system, but I disagree. The essential function of the parabolic axi­con or Parabelringspiegel is to image a line source (specifi­cally, the virtual line source produced by the conical cen­tral mirror) onto a point, just the reverse of the normal mode of operation for an axicon. Since it satisfies the functional definition of an axicon set forth by McLeod, the outer football-shaped mirror, but not the entire system, may be called an axicon.

The question of priority for the parabolic axicon qua axi­con (and, since the parabolic axicon is an axicon, for the ax­icon itself5) hinges on Hauser's recognition of the signifi­cance of the line-to-point imaging. I do not have enough information to resolve this question, but it seems probable that Hauser did recognize this. The priority of the focus­ing system itself, although used for a different purpose, clearly belongs to Hauser.

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The other condensers mentioned by Vargady use spheri­cal, paraboloidal, or cardioid mirrors and thus are irrele­vant to the discussion. However, I do wish to correct some historical errors by Vargady1 and Reinert.2 The condens­ing system assigned the date 1738 is the well-known Lie-berkuhn mirror2 and is an ordinary sphere or paraboloid. It was not first invented by Lieberkuhn in 17387 but rather was described by Descartes8 in his Dioptrique in 1637. Descartes specified that the mirror was to be paraboloidal in shape (un miroir creux parabolique), and the accompa­nying figure shows it as such. This interesting microscope condenser also incorporated an optional condensing lens for viewing the object by transmitted light.

References 1. L. 0. Vargady, Appl. Opt. 14, 000 (1975). 2. G. G. Reinert, Praktische Mikrofotografie (Knapp, Berlin,

1937), pp. 53-55 or p. 59, depending on the edition. 3. G. Stade and H. Staude, Mikrophotographie (Akademische

Verlagsgesellschaft, Leipzig, 1939), p. 94. 4. G. S. Bakken, Appl. Opt. 13, 1291 (1974). 5. J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954). 6. J. H. McLeod, J. Opt. Soc. Am. 50, 166 (1960). 7. S. Bradbury, The Evolution of the Microscope (Pergamon Ox­

ford, 1967), p. 87. 8. R. Descartes, Discours de la Methode, plus la Dioptrique, les

Meteores, et la Geometrie (Jan Maire, Leyde, 1637), p. 132, re­printed in Oeuures de Descartes (J. Vrin, Paris, 1965), p. 207.

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