Generation of Square Lattice of Focal Points by a Modified White Cell E. O. Schulz-DuBois

IBM Zurich Research Laboratory, 8803 Ruschlikon, Switzerland. Received 12 February 1973. Two schemes for low-loss folded optical delay lines are

described hi the literature. Both are closely related to confocal resonators. The Herriott-Schulte delay line1

typically uses two almost equal mirrors, one of them spherical and the other slightly ellipsoidal, at a spacing much smaller than the focal length. When suitably illu­minated by a laser beam, a set of focal points is produced on either mirror that lie on a Lissajous figure.

This figure is of the same type as that produced by two ac voltages of somewhat different frequency when these are supplied to the vertical and horizontal deflection plates of a cathode ray tube. It consists, approximately, of a set of ellipses that are inscribed inside a rectangle. Crowding of focal points occurs especially near the sides of the rectangle. Although this delay line is well suited to work into a single output, it would be less convenient to use it as a tapped delay line with taps representing equal increments in delay time, since one would like to have the taps located on a linear periodic array.

Note, however, that the ellipses include the diagonal straight line as a limiting case. There the distance be­tween adjacent focal points is small near the corners and larger toward the center of the overall pattern. In unpub­lished work,2 Herriott and his coworkers realized a distri­bution of focal points on a straight line (plus a number of ellipses) by careful adjustment of the delay line. The in­cremental delay is the same for adjacent focal points on the straight line and, in this work, was made to represent around 30 round trips. By measuring the ratio of the light intensity in adjacent foci, this system can be used to determine the reflectance of highly reflecting (around 99% or better) mirrors with considerable accuracy.

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Fig. 1. Perspective view of the White cell—a, b, and c are the centers of curvature of mirrors A, B, and C, respectively.

Fig. 2. Projection of the White cell perpendicular to its axis. Further details are discussed in the text.

Fig. 3. Proposed beam-deflection device consisting of a corner cube prism with a lens-shaped entrance face.

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The other delay line scheme is attributed to White3 and has been used to realize long, folded, absorption paths in gas spectroscopy.4 Roughly speaking, the White cell is the combination of two slightly misaligned, interpenetrat­ing confocal resonators. This is illustrated in the per­spective view of Fig. 1. The centers of curvature, a and b, of mirrors A and B are located on the surface of mirror C; and the center of curvature c is located on the dividing line between A and B. The function of the White cell can be explained by reference to Fig. 2, which shows a projec­tion perpendicular to the device axis. Here mirrors A and B are shown by dashed semicircles. Mirror C is shown by a solid square. It is this mirror on which the lattice of focal points will be produced.

First consider the normal operation of the White cell. Laser light enters the device from the outside and is fo­cused at point 0 next to mirror C. The laser beam is ap­erture-shaped, so at the other end of the device it illumi­nates essentially the full area of mirror A, but not B. After reflection from A, the light is focused on mirror C at point 1. To find this point, one has to double the length of the line O-a. After reflection from C, the light this time illuminates mirror B, since its aperture shape is inverted after every back-and-forth trip. By the same argument as before, the light is then focused at point 2. Continuing in the same way, the light alternates between mirrors A and B and in this fashion produces a double row of focal points. In practice, with an ƒ/10 aperture, the focal points may be spaced as close as 30 λ without having much cross-talk. A linear array of focal points is, of course, well suited for the realization of a tapped optical delay line. In that case mirror C would be made slightly transparent, and one would use a corresponding array of photodetectors that can be manufactured in integrated semiconductor technology.

Next consider the requirements for generating a two-dimensional lattice of focal points. As shown in Fig. 2, a last focal point numbered 2n - 2 is realized in the usual operation of the White cell. Thereafter, the light would be focused through mirror A, at a point marked 2n - 1 outside the area of mirror C. Apparently a first require­ment is that the beam is deflected vertically in Fig. 2 so it seems to reenter the device at point 2n. In essence, this requirement is met by a 90° rooftop prism whose apex line is located in the plane of mirror C, and half a lattice spac­ing below the center line passing through a and b (see dotted line at the left of Fig. 2). A second condition con­cerns the deflection of the beam in the horizontal direc­tion of the figure. It is required that, after deflection, the beam travels as much to the right as before to the left; and furthermore the beam aperture should again cover mirror A from where it just came.

These requirements, taken by themselves, would be sat­isfied by a 90° rooftop prism whose apex forms the vertical line connecting points 2n - 1 and n, again located in the plane of mirror C. The combination of first and second requirements is therefore satisfied by a corner cube prism whose corner is located in the plane of mirror C at the point marked X in Fig. 2. There is a third requirement, however. Since the mirrors are spherical, every reflection in the White cell is accompanied by a focusing effect. The same focusing effect must also occur in the beam-deflecting device. This requirement may be met by shap­ing the front entrance of the corner cube prism as a lens, but with twice the focal length of the mirrors since it is passed twice for one beam deflection. The resulting beam-deflecting device is illustrated in Fig. 3, and its ar­rangement relative to mirror C is indicated by a dashed contour in Fig. 2. From what has been stated it is clear

that the combination of the White cell and beam-deflect­ing device produces a rectangular or square lattice of focal points where the points are passed in the sequence indi­cated. In one interesting application, only one vertical row of focal points is utilized as taps of a tapped delay line. In that case, the time delay between adjacent taps may be very large; yet the taps form a small linear array, which makes it possible to use a corresponding array of integrated photodetectors for readout purposes.

Concerning various aspects of this technique, the au­thor enjoyed useful discussions with his colleagues E. Courtens, D. Pohl, A. E. Siegman, and H. Thomas. He is indebted to D. R. Herriott for communicating unpub­lished experimental results.2

References 1. D. R. Herriott and H. J. Schulte, Appl. Opt. 4, 883 (1965). 2. W. Gonros, D. R. Herriott, R. G. Murray, and W. H. Yocom,

Bell Telephone Laboratories, unpublished report dated July 1967.

3. J. U. White, J. Opt. Soc. Am. 32, 285 (1942). 4. H. J. Bernstein and G. Herzberg, J. Chem. Phys. 16, 30 (1948).

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