This project is funded by the National Science Foundation.

#### The Question

Is it possible to enumerate all feasible designs of opto-mechanical devices from the specification of its desired input-output behavior and knowledge of available primitive components?

#### Approach

We introduce a new, systematic and complete compositional technique for conceptual design of imaging optical systems from behavioral specifications. There are two key ideas: (1) modeling the structure and behavior of optical components using affine transformations, and (2) using a systematic search algorithm integrated with a powerful algebraic constraint solver to compute three-dimensional layouts of optical components in the design. By embedding our tool in a constraint programming environment, we are able to automate the conceptual design of optical imaging systems --- a step performed manually by human designers until now. We demonstrate the power of the method by recreating several designs for imaging systems of copiers.

#### Summary of results

First, our layout synthesis algorithm is grounded in the laws of optics with optical components represented as linear transformations on object and image configurations. Second, our algorithm is complete; if enumerates all feasible design families that implement a given specification. Since our algorithm explores designs in increasing order of complexity as specified by the number of primitive components, we can guarantee minimality for the designs produced. Third, our algorithm computes key constraints on positions and orientations for all components in the design. These can then be instantiated further for input into a detailed design program for further optimization and analysis. Our approach allows for rapid generation of conceptual design sketches that can be refined with further computation. Fourth, even though we work from first principles in optics, we can obtain real-time generation of alternative design topologies. Our synthesis algorithm is fully implemented in Mathematica, and all of the examples presented below are the output of our design system rendered using the Optica application.
• An imaging system that transforms an object at (0,0,0) into a real image of the same size at (1,1,0) rotated by 180 degrees about the z axis with respect to the object. Here is the top view and here is the 3-D arrangement of components.
• An imaging system for the Xerox 914 copier designed by our synthesis program. Here is the top view and here is the 3-D arrangement of components. The system transforms an object at (0,0,0) oriented along the x-axis into a real image of the same size at (4,0,0) oriented along the x-axis.
• An imaging system for the Xerox 4000 copier designed by our synthesis program. Here is the top view and here is the 3-D arrangement of components. The system transforms an object at (0,4,0) oriented along the x axis into a real image of the same size at (2,0,0) also oriented along the x axis.