A new optical illusion has been added to our geometric optics repertoire: A combination of four converging lenses creates a 3D cloaking effect.
What it shows
The demonstration is a replica of an experiment described by J.S. Choi and J.C. Howell (Univ of Rochester, NY) in their paper entitled Paraxial Ray Optics Cloaking.1 Using standard lenses, the optical devise produces an image of the background with unity magnification. Objects, such as your hand, appear invisible when inserted into the devise. This is basically an optical illusion and happens because the light rays that make up the imaged background pass around the object as if it isn't there.
How it works:
Four lenses are used to produce a non-inverted image of a grid background with unity magnification. If one looks through the lenses and changes the viewing angle slightly, the image of the grid shifts accordingly and gives the illusion that one is simply looking through the lenses at the background. The lenses are configured so that the light rays that create the backgound image pass through a point on the optical axis. Thus, if an object is placed inside the lens configuration, it will not be visible as long as the object is located in an annulus-shaped space around the optical axis outside the bundle of rays. The cloaking space is depicted in this idealized illustration (not to scale) in which the background is very far away.
Setting it up
The two outside lenses are 20.0 cm focal length achromat doublets. The center two lenses are 7.5 cm focal length achromat doublets. The prescription given by Choi and Howell is that the separation of the first two and last two lenses be equal to the sum of their focal lengths: t1 = f1+f2 = 27.5 cm. The two inside lenses should be separated by an amount equal to t2 = 2f2(f1+f2)/(f1-f2) = 33 cm. This assumes thin lenses and that the background grid is far away, conditions not met in this set-up. The background grid is approximately 2 meters away. The 20.0 cm lens is 1.05 cm thick and the 7.5 cm lens is 2.3 cm thick. Thus, adjustments to t1and t2 must be made to secure an image having unity magnification. For a background distance of 2 meters, t1 = 25.6 cm and t2 = 33.6 cm work well. These distances are between the closest surfaces of the lenses.
To avoid pincushion distortion, it is important to orient the lenses properly by following these general guidelines: when collimating a point source, the first air-to-glass interface should have the greater radius of curvature (e.g. the flatter side faces the point source). Conversely, when focusing a collimated beam, the air-to-glass interface with the smaller radius of curvature should face the incoming light.
When looking into the last lens, we wish the grid background to fill the lens, so one must not be too close to keep the exit pupil small. The other issue is depth-of-field which is influenced not only by the camera's lens focal length and aperture, but also by the distance from camera to object. Again, a simple rule: The closer you are to the object you are focusing on, the fuzzier everything else will be — or, put another way, the less your depth-of-field will be. Since we need to be far away to satisfy the exit pupil constraint, but want the image to be large, we need to use a telephoto lens. We gain depth-of-field by going farther away and lose depth-of-field by using a telephoto lens. A satisfactory compromise is the following.
To show the cloaking effect to the audience, set up a video camera approximately 5 meters away from the lenses — this ensures that the grid fills the lens. Use the 150 mm Cannon zoom lens and adjust it for a small aperture (large f-stop) to gain depth-of-field. Rely on the camera's auto-gain. Take care to align the camera along the optical axis of the cloaking lenses. Finally, to achieve the best overall focus, focus the camera lens a little beyond the hand you intend to image and take advantage of yet another rule-of-thumb: 1/3 of the depth-of-field is in front of the object you're focusing on and the other 2/3 is behind. So by focusing a little beyond the object of interest, the object itself will remain in focus and you have a better chance of bringing the background into focus.
Before demonstrating the cloaking, it might be worthwhile to walk in front of the grid in the background to convince the audience that the image in the lens is truly an image of the background.
The illustrtation above depicts a cloaking region outside a 2 cm diameter bundle of light rays between the two center lenses. The last photo shows a test of that prediction; a 2 cm diameter tube is not visible when looking into the device (all the light rays pass through the inside of the tube). Cloaking aside, this demonstration can be used as an interesting example of ray tracing (a little more interesting than telescope ray tracing).
J.S. Choi and J.C. Howell, "Paraxial Ray Optics Cloaking," Optics Express 22(24), pp 29465-29478 (2014).