Introduction

Linear systems theory applies to systems in which the effect of multiple inputs is the sum of the effects due to each input individually. The linearity of electromagnetic phenomena is commonly described by ``the principle of linear superposition,'' which states that the total field excited by multiple sources is the sum of the fields due to the individual sources. Linear superposition means that at least the passive components of electro-optic systems are best studied using linear systems theory. This link overviews the basic concepts of 2-D linear systems theory.

Often, one analyzes optical systems without specific reference to the driving sources or the detector. Without a source, no fields are generated in the interior of the system. Assuming that waves enter the system and propagate from one side to the other, the inputs are the fields on the entrance boundary and the outputs are the fields on the exit boundary. Examples of systems which can be modeled in this way include imaging systems, where the entrance boundary is the object plane and the exit boundary is the image plane and optical correlators and pattern recognition systems, where the entrance boundary is again an object plane but the exit boundary is a correlation plane, and general scattering and diffraction systems. Each of these systems can be modeled using linear systems theory. In addition, concepts from linear systems theory are very useful in understanding the linear aspects of nonlinear systems, such as laser systems and nonlinear optical devices.

A linear systems analysis of an optical system is done in three steps:

1. Find the modes which propagate in the system. Once one understands what modes to use and how propagation from the input to the output effects the modes, a linear system is fully characterized. In optical systems, modes typically are plane waves, spherical waves, resonator modes, or waveguide modes.
2. Decompose the input, e.g. the field on the entrance boundary in terms of the modes and transform the modes to the exit boundary.
3. Sum the transformed modal decomposition to find the transformation from the actual input field and to the output field.

The most common modal decomposition is into plane waves, which for monochromatic fields are phasors of the form

These fields are solutions to the wave equation

where is the magnetic permeability, is the electric permittivity, and is the radial frequency. is the wavevector of the plane wave and is the wavenumber. While is, in general, a three dimensional vector , the range of values it can assume is constrained to a two dimensional surface, the sphere . This constraint means that we can perform plane wave decompositions of systems using only two dimensions in . On a fundamental level our ability to reduce modal decompositions of fields to two dimensions is a result of the fact that fields in a volume containing no sources are completely determined by the boundary conditions on the volume.

Plane waves represent a complete set of modes for physically realizable solutions to the wave equations in unbounded space. The representation of fields in terms of the plane wave modes is Fourier analysis. Since, at a single frequency, the domain of allowed spatial frequencies is two dimensional, 2-D Fourier analysis is central to optical systems. The sections below briefly overview the 2-D Fourier transformation and its properties.

The Fourier transform

In two dimensions, the Fourier transform of a function is

The inverse Fourier transform is

A proof that

relys on the identity

where is the Dirac delta function, which has the properties and

for all

Properties of Fourier transforms

1. The linearity theorem: , where and are constants.
2. The shift theorem:
3. The similarity theorem:
4. The convolution theorem: where .

Linear Transformations

A transformation is a mapping from an input function, for example to an output function . The transformation is expressed

is a linear transformation if

A linear transformation can be expressed in integral form using the Dirac delta function. An arbitrary function can be represented using the delta functio as

Substituting this representation in the transformation, we find

where is the impulse response of the transformation.

A linear transformation is shift invariant if the impulse response takes the form

In this case,

which is a convolution. If we take the Fourier transform of this equation and apply the convolution theorem, we find where is the transfer function of the transformation.

Note that the Fourier transform is a linear transformation but is not shift invariant. The ideal imaging system, which exactly replicated the field from one plane in another plane, is linear and shift invariant. Diffraction from one plane to another in free space is another example of a linear and shift invariant transformation.