Aberration compensation in two-dimensional reﬂective optical systems

. We present a novel approach to minimize aberrations in imaging systems. The energy distributions at the source and target of an optical system play a crucial role in designing freeform surfaces through illumination optics methodologies. We quantify the on-axis and o ﬀ -axis aberrations using a merit function that depends on the energy distributions. The minimization of the merit function yields optimal energy distributions, which subsequently enable us to design freeform reﬂector surfaces that cause the least aberrations. We validate our method by testing it for two conﬁgurations, a single-reﬂector system with a parallel source to a near-ﬁeld target, and a double-reﬂector system with a parallel source to a point target.


Introduction
In optical imaging systems, aberrations affect the quality of an image adversely.Freeform surfaces are very suitable for minimizing aberrations as they are designed with a higher degree of flexibility than simple conic shapes.In illumination optics, inverse methods are used to compute freeform surfaces.These methods are based on converting a given source distribution to a desired target distribution.We use the techniques of freeform reflector design from illumination optics to optimize imaging systems.

Mathematical Model
When a set of off-axis rays passes through an optical system, the resulting image is an aberrated spot.This spot differs from an ideal, in-focus spot produced by on-axis rays.Aberrations are quantified by the root-mean-square (RMS) spot size.Our goal is to minimize aberrations when various sets of off-axis parallel light rays pass through an optical system.Consider a 2-D optical system with the z-axis as the optical axis.The source S and target T lie on the x-axis and the y-axis, respectively.Let α be the angle that a set of parallel rays forms with the optical axis.This angle is measured counterclockwise from the z-axis to the incoming light rays.The case α = 0 • is called the base case.This case refers to the on-axis light rays.
We compute the reflectors for the base case by means of illumination optics techniques.For 2-D systems, the shape of the reflector is the solution of an ordinary differential equation (ODE) subject to some initial conditions.The ODE can be derived by combining the equation for energy conservation with the optical map.
As an example, we briefly present the method to compute the reflector z = u(x) for a single-reflector system with a parallel to near-field target (see Fig. 1).The illuminance at the source S and the target T is given by f (x) and g(y), respectively.We assume that there exists an optical map y = m(x) such that the total energy at the source and the target is conserved.The conservation of energy together with the optical map gives us We solve Eq. 1 (either + or − sign) subject to the transport boundary condition [1] to obtain a mapping y = m(x).This mapping should be equivalent to the optical map y = M(x, u, u ) for the base case, which is derived using the law of reflection.Thus, we obtain another ODE where H is a known function.The reflector z = u(x) is obtained by numerically solving Eq. 2.
We aim to minimize aberrations and to ensure least fluctuations in spot sizes across all sets of parallel rays.We use a ray tracer to find target coordinates for multiple sets of off-axis uniformly generated parallel light rays passing through the base case optical system.The RMS spot size is calculated using the ray tracer.It depends on the angle of incoming rays and the energy distributions used to compute the reflectors.We subsequently take the RMS of the already calculated spot sizes.It is called the merit function and the value of the merit function is unique for specific energy distributions used to determine reflectors for the base case.This measures the aberration deviations for off-axis light rays passing the optical system.
We validate and compare our method for two configurations (see Fig. 1), a single-reflector system (parallel to near-field target), and a double-reflector system (parallel to point target).

Algorithm for Aberration Compensation
The algorithm we propose is described below:

Compute Reflectors for the Base Case
The reflectors are computed using initial energy distributions utilizing the methods described in [1].

RMS Spot Size
An incoming parallel beam under an angle α is directed towards a reflector that has been computed using the energy distribution f .The RMS spot size (W α ) f , corresponding to this beam is estimated by the standard deviation of target coordinates y, i.e., (3)

Merit Function
The deviation in aberrations of on-axis and off-axis parallel light rays is measured by a merit function

Optimization Method
In order to obtain minimum value of the merit function, we solve a constrained optimization problem The derivative-free Nelder-Mead simplex method [2] is employed to solve this optimization problem.
The optimal energy distributions which design the freeform reflectors that ensure the least changes in spot sizes among all sets of off-axis rays are found using the minimization procedure.

Numerical Examples
We consider the source distribution f as an element of the vector space generated by a span of orthogonal basis functions.The Nelder-Mead method is used for unconstrained optimization problems.For ensuring that the constraint f ≥ 0 is satisfied, we choose a positive function F , such that the source distribution f is given by We select the even-degree Legendre polynomials P 2i as the orthogonal basis functions.This guarantees a symmetrical energy distribution and consequently a rotationally symmetric reflector.We generate off-axis parallel rays from a source of unit length under angles α ∈ {−5 • , −4 • , . . ., 5 • }.The minimization procedure is tested for the following:  For both optical system configurations, the minimum value of the merit function (in Table 1) ensures that the deviation in aberrations corresponding to various sets of parallel rays is the minimum.The results obtained from the minimization procedure are consistent (shown in Fig. 2) for both cases 1 and 2. The optimized double-reflector system is better than the optimized single-reflector system for minimizing aberrations of off-axis rays.

Figure 2 :
Figure 2: Spot sizes for various angles α in optimal reflector systems.

Table 1 :
. Value of the merit function W( f ) for different distributions f used in the minimization procedure.