When two or more waves of the same frequency are present in the same location, the sum of their intensities may not equal the intensity of the total wavefunction. The interference is understood as the difference between the intensity of the total wavefunction and the sum of the individual wavefunction intensities.
The interference equation is used to talculate the intensity of the total wavefunction. The third term is the interference between the two waves, where φ is equal to the sum of the phases of the two waves.
When adding wavefunctions of different phases, these wavefunctions can be drawn as a superposition of vectors, where the intensity of the wavefunction in the magnitude and the phase is the angle of the wavefunction vector.
Consider the case in which two waves, represented by two vectors are equal in magnitude, but 180 degrees out of phase of each other. In this case, the intensity of the total wavefunction is zero. If there is no phase difference between the two wavefunction vectors, then the interference of the two waves is zero and the maximum intensity of the system is reached.
It has been mentioned that Wave Optics and Geometrical Optics are insufficient to take measurements of the intensity of rays and waves. However, by determining the level to which waves interfere with each other, a relative intensity can be measured. The interferometer is an instrument that detects the intensity of the a superposition of waves of a varied phase difference. A wave is split using a beamsplitter and each split wave is reflected after different (or possibly the same) distances and recombined. After recombination of the optical waves, the interference is measured by amount of loss in the system and subsequently the distances of the mirrors. Applications include metrology, measurements of refractive index and spectrometry.
Three prominent examples of interferometers are the Mich-Zehnder interferometer, the Michelson interferometer and the Sagnac interferometer.
B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.