Coherence and Optical Systems

As indicated in the syllabus, much of this course focuses on representations of the statistical states of optical fields. In view of the considerable detail we will accord this topic in reviewing MW, we will not go into the formalisms of coherence theory in this note. By wave of introduction, however, it is worth considering why coherence theory assumes such a large role in optical systems analysis.

Optical fields lie at frequencies of 10¹⁴ to 10¹⁵ hertz on the electromagnetic spectrum. At higher frequencies in the x-ray and gamma ray ranges, the particle nature of quantized fields becomes more and more prevalent. These high energy photons are emitted by nuclear processes and high energy particle accelerators. Counting statistics, rather than wave coherence, are usually of interest at high energies. At lower frequencies in the radio range, fields are generated and detected by continuous charge fluctuations in antennas. The number of photons in radio fields is generally so large that particle-like phenomena and counting statistics are not relevant. In the microwave, infrared and optical ranges, radiating fields are generated and detected by discrete phenomena such as spontaneous atomic emission or black body radiation rather than continuous current distributions, but the number of photons in the field is often large enough that wave analysis of the field is more useful than quantized analysis.

By coherence theory we mean the use of statistical measures to characterize optical fields. Coherence is a measure of the statistical correlation between fields drawn from different points in space and time. Coherence theory is crucial in the analysis of optical fields for two reasons:

1. The frequency of optical fields is so large compared to the spectral resolution of the filters and detectors used to analyze it that it is not possible to use truly monochromatic fields in analyzing optical systems. Even if it were possible to filter to a truly monochromatic field, the energy in the filtered field would be so small that quantum fluctuations would likely assume a large role in field analysis.
2. Even if the field is completely coherent and subject to the Maxwell equations, field detection always occurs via discrete and statistical quantum mechanical processes.

In view of point 1, one often analyzes optical systems using quasi-monochromatic fields. A quasi-monochromatic field is a field in which the carrier frequency greatly exceeds the bandwidth. Using diffractive filters, an optical field can be filtered to a resolution of 0.1 to 1 nm. On a 1000 nm center wavelength, this field has a carrier frequency which is 10³ to 10⁴ times greater than its bandwidth. Using lasers, one can greatly exceed this resolution. A typical gas laser might have a bandwidth of 1-100 MHz. An actively stabilized laser might have a bandwidth as small as a few hertz, meaning that it the ratio of the center wavelength to the bandwidth has an astounding value of 10¹⁴.

Coherence theory plays a particularly important role when the bandwidth of the source field is larger than the bandwidth of the detector. For Laser fields with bandwidths less than the bandwidth of available electrical detectors, quantum detection noise is still a determining factor in the statistics of measurements, but the field itself may be regarded as coherent. When the bandwidth of the field is greater than the bandwidth of the available detector, the field is likely to be incoherent even if it is quasi-monochromatic.