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1. The Optical Field
2. Maxwell's Equations
3. The Wave Equation
4. Solutions to the Wave Equation

Optical fields lie on the electromagnetic spectrum in the frequency range from 10¹⁴ to 10¹⁶ Hertz. The hard ultraviolet and x-ray regimes lie at frequencies above the optical region and the mid and far infrared and microwave regimes lies at frequencies below. The most distinguishing aspect of optical fields is that they are visible to the human eye. As there is little fundamental difference between physical phenomena observed for the visual spectrum and phenomena observed in the near infrared and ultraviolet, however, these regions are often included in the optical regime.

Two characteristics distinguish optics from other electromagnetic technologies. First, optical wavelengths (0.1 to 10 microns) are small compared to typical aperture sizes and measurement distances, meaning that optical fields are usually considered in far-field and weakly diffracting limits. Second, optical fields are generated and detected by atomic-scale quantum interactions rather than by antennas or resonators. The first characteristic leads one to analyze optical propagation under approximations to the Maxwell equations appropriate to the short wavelength regime. In many cases, one can neglect diffraction and consider optical excitations to consist of straight "beams." In such cases, one uses geometrical analysis based on ray tracing to understand the general behavior of optical systems. The second characteristic means that optical field generation and detection is not amenable to direct artificial control and that optical fields are susceptible to different and more profound sources of noise and uncertainty than other regions of the spectrum.

Optical fields are central to three technologies: sensing, communication and illumination. Sensing is the oldest of these, since cavemen had eyes, but fire for illumination came not much later and fire for smoke signal communication soon followed. (The earliest long distance communication might have been "Urg's hut is on fire.") In the modern world, optical sensing spans a range from microscopy to astronomical telescopy and includes still, cinematic and video photography, lidar, reflectometry, and a number of other applications. Optical sensing is increasingly dominated by electronic sensors and digital processing. Despite the long history of smoke signals, lighthouses, lanterns and signal lights, optical communications has bloomed as an industry only in the past 20 years. The key to this development has been the explosion in demand for data transmission, the development of astonishingly low loss optical fiber and the development of semiconductor laser sources and detectors. Illumination has a longer history as a major industry than either sensing or communications. Illumination today includes the range from flashlights to laser tools for surgery. In view of the greater potential for research and engineering at the higher tech end of the scale, modern engineering courses tend to emphasize lasers over lightbulbs. There are other optical technologies which are difficult to classify in these three areas, for example, optical data storage mixes sensing, communications and illumination. This course covers sensing applications of optical fields. In the UIUC ECE department ECE 371 GP covers communications applications and ECE 355 covers lasers.

Just as there are three major optical applications, there are three main questions to be resolved in analyzing optical systems: how is the field generated?, how does it propagate? and how is it detected? These questions play a role in all three technologies, the significance and focus varies. Communications system development focuses primarily on modulation and detection technologies. Illumination system development focuses primarily on source issues. Sensor systems are most concerned with how the field propagates from the object or system being sensed to the detection system. While the nature of the illuminating field (e.g. its polarization, spectral and coherence properties) and the fidelity of the detection system are important, the enabling issue for sensing is how to detect information which can be back propagated to identify the object.

The primary reason for the emphasis on field propagation in sensing systems is that the most sensors detect parallel spatial information. In contrast, the spatial pattern generated by most illumination systems contain little information and communication systems limit the field to a single spatial mode or a very small number of modes in a guiding fiber. Since the spatial pattern of the field is information rich in sensing applications, it is important to understand how this information is transformed by the spatial distortions of propagation. The propagation of the optical field is described by Maxwell's equations. Since, as will be discussed later in the course, the optical field itself is not directly detectable, statistical correlations and intensity measures must supplement Maxwell's equations to fully understand propagation.

Maxwell's equations in differential form are

is the electric field, is the electric displacement, B is the magnetic induction, H is the magnetic field, J is the current density and is the charge density. The first equation is Faraday's Law and expresses the experimental fact that a time varying magnetic field induces an electromotive force. The second equation is Ampere's Circuital Law and expresses the experimental fact that a current or time varying electric field induces a magnetic field. The last two equations are Gauss's Law and express the experimental facts of Coloumb's law and charge conservation.

The field variables are related by the material equations

where P is the polarization of material and M is the magnetization. In most optical materials, M=0 and P is a function of E. The simplest and most common case is

Then

where

and

As a result of the high frequencies and small wavelengths of optical fields, charge interactions with the optical field involve quantum mechanical effects which cannot accurately be analyzed by continuous models. In optical diffraction problems, is always neglected. In principle, J is also neglected, although the current density can be used formally to add a loss component to the pemitivitty, . We also note that the vectors E and D need not be colinear, meaning that may in general be tensor valued. A medium in which is (not) tensor valued is called (isotropic) anisotropic.

Using the material relations, we substitute in the Maxwell equations to find the wave equations in isotropic media

The equations are reduced to a simpler form by the vector identity

From the Gauss's Law we know that

,

where we have assumed for the moment that is scalar valued. Thus,

and

A medium in which (the permittivity is spatially constant) is homogeneous. Most common materials are approximately homogeneous. Media in which this is not the case include optical fiber, graded index lenses, and volume holograms.

In homogeneous isotropic media, the wave equations are

Fields satisfying the wave equations are 4D disturbances in space-time. Since a wave disturbance propagates without dissipation, there are no spatially bounded solutions to the wave equation in source and absorber free space. Pulse packets in localized in space-time are potential solutions, but such solutions diffractively dissipate on propagation. There are no simple exact solutions of finite energy. This section reviews the basic nature of solutions and builds intuition for wave propagation. Lecture 3 considers solutions in more detail using Fourier methods.

Plane waves are the simplest solutions to the wave equation. Plane waves are unbounded and constant in two dimensions and propagate along a third dimension. Assuming that the constant dimensions are x and y, the derivative of the fields vanish along these dimension and the wave equation for the electric field reduces to

,

()1

where we have chosen to write the equation only for one polarization component of the electric field. Note that for the plane wave to satisfy Gauss's law the z component of the field must be zero.

Eqn. has many possible solutions, the simplest being

where and is any twice differentiable function. This solution is a wave propagating in the negative (plus) z direction if the plus (negative) sign is selected.

We consider some simple animations to provide some feeling for solutions in the optical domain. The first example is an pulse with a center wavelength of 0.5 microns, in the green/blue region of the optical spectrum. The pulse oscillates at a center frequency of at approximately Hz. This particular pulse has a Gaussian envelop. The pulse width of the Gaussian is 10 femtoseconds. The field amplitude of the pulse can be written . The animation below shows the field amplitude as a function space along the axis of propagation. Each frame in the animation represents a time shift of 20 femtoseconds. The pulse propagates to the right at a velocity of 0.3 microns/femtosecond.

The solution need not be spatially confined. For Fourier analysis of fields, unbounded harmonic solutions such as are popular. For such fields, one perceives wave propagation in the phase of the excitation. An animation of this field propagating in the plus z direction is shown below:

The next lecture discusses the advantages of using harmonic solutions, such as the one sketched above, to analyze fields.

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David J. Brady, Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, dbrady@uiuc.edu