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## Thin Lens Calculations

One of the primary diagnostics for shock tube experiments is imaging. This is accomplished via one of two mechanisms, the first being through the field imaging (e.g. Schlieren, shadowgraphy), and the second being planar imaging using a laser sheet. Thus, it is necessary to either change the diameter of a collimated light source, or to convert a collimated laser beam into a wide, but very thin, laser sheet.
To begin design, it may be helpful to use the following graphical applet. The arbitrary cooridinates in the applet may be scaled appropriately to the desired geometry of the imaging application. The design of a two lens combination may then be finalized with the javascript calculations below the applet. These tools will enable the designer to choose the appropriate lenses and provide guidance on the approximate (due to the thin lens approximation) spacing between the lenses.

## Using two lenses to produce a single focal length

There are occasions where a desired focal length cannot be achieved due to a lack of availability. For example, a very long focal length lens is desired, such as f=2.5 m, but a small lens is not available in this focal length. An equivalent focal length (EFL) may be produced with two lenses having focal lengths f1 and f2 separated by a distance d.

Input
mm, f1, focal length of first lens
mm, f2, focal length of second lens
mm, d, distance between lenses
Output
mm, EFL, equivalent focal length of two lens system

## Changing the diameter of a collimated beam

The figure to the right shows a beam of initial diameter x1 being shrunk to a final diameter of x2 and the distance between the two lenses is d. If it were desired to expand the beam, the plano-concave (negative focal length) lens would be placed first and have focal length f1, and the plano-convex lens would be placed second and have focal length f2.

Input
mm, x1, diameter of incoming beam
mm, f1, focal length of first lens
mm, f2, focal length of second lens
Output
mm, d, distance between lenses
mm, x2, diameter of outgoing beam

To use a script for EES

## Creating a laser sheet from a collimated beam

Figure A to the right shows a beam of initial diameter x1 being focused to a point, which is a distance f1 (the focal length of the lens) away from lens 1. Figure B adds a negative cylindrical lens that expands the laser beam in one dimension to form a light sheet. The lenses are separated by a distance d and the width of the laser sheet at the distance f1 from the first lens is x2.

Input
mm, x1, diameter of incoming beam
mm, f1, focal length of first lens
mm, f2, focal length of second lens
mm, d, distance between lenses
Output
mm, x2, width of laser sheet
radians, angle of diverging light sheet

To use a script for EES

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 File last updated: Aug. 7, 2008 Feedback, questions or accessibility issues: oakley@engr.wisc.edu Copyright © 2004 The Board of Regents of the University of Wisconsin System.