A design method using transformation matrix to obtain the trace of the light through the optical components controlled by high-precision actuators

We introduce a method using the matrix for mechanical control to obtain the trace of the light determined by driving the high-precision actuators, which can control the angle and position of the optical components. To design a comprehensive optical system using several mirrors like the interferometer, it's crucial to trace the position of the light as it reflects or passes through at each component. By being able to calculate actuator control with ray tracing, it becomes possible to design the size and operation of systems such as interferometers. That is the design method we introduce.

 In the matrix method in optics, the position of the light and its angle are used to trace the position of the light. However, in most cases, optical components are fixed and analyzed under such static conditions [1]. When dealing with moving optical components using actuators, we focus on two specific points. Using the transformation matrix, commonly employed in robotics, we can transform the coordinate system to the point where the light is irradiated [2]. Because the transformation matrix includes the actuator parameters, this ray-tracing serves as the design method.

 Lastly, we show the simple equations for the complicated system using n-th optical components, following as

{G_n} = {f (G_n-1, G_n-2)}.         (1)

Here, {G_n} is the position where the light is irradiated at n-th component and f is a function.

[1] A. Gerrard, and J. M. Burch, Introduction to matrix methods in optics, Courier Corporation (1994).

[2] J.C. John, Introduction to Robotics: Mechanics and Control, Prentice Hall (2004).