The two-wave interference equation is typically written as I = I₁ + I₂ + 2 √I₁I₂ cos(φ₂ - φ₁) where I₁ and I₂ are the irradiance values for two interfering waves, while φ₁ and φ₂ are the respective phases with units of radians. Variable I is the total irradiance for the two interfering waves.

The calculator below simulates two-wave interference when the phase of each wave is described as the sum of three Zernike polynomials. Press and hold the Shift key to remove color and view greyscale luminance. To visualize the combined wavefront without interference, press and hold the Ctrl key on your keyboard.

Conversion between wavefront (units nm) and phase (units radians) is possible because a phase delay of 2π radians corresponds to a distance of one wavelength. Irradiance typically has units of Watts/m², but in this example, only the relative irradiance is significant.

Broad spectrum sources, such as sunlight, have many wavelengths and generally create a multi-color interference pattern. Each wavelength has a different phase delay where the cosine term will be a maximum. This is shown in the figure below.

The top of the image shows two overlapping waves traveling to the right, while the middle of the image shows their sum. The sum is the interference pattern and changing the delay between the two waves affects the combined irradiance. This process occurs for each pixel in the wavefront interference calculator.

The bottom of the image shows the amplitude of the combined wave for colors across the visible spectrum. For a non-zero delay, changing the wavelength also affects the irradiance of the interference pattern.

Derivation

The interference equation is derived in many textbooks[1], and for completeness is reproduced here.

The electric field (E) for each monochromatic wave has amplitude (A) and phase (φ), both of which may be spatially varying. Following the decreasing phase convention, the electric field for two light waves propagating in the z-direction with the same polarization state is written as E₁ = A₁ cos(kz - ωt - φ₁) E₂ = A₂ cos(kz - ωt - φ₂) where k = 2 π / λ and ω = 2 π ν are related to basic properties of wavelength (λ) and frequency (ν) of the light.

When the light waves overlap, the electric fields coherently add together. E = E₁ + E₂

People can not directly observe the electric field, and instead see the time-averaged square of the field, which is called irradiance (I). I₁ = ½ c ε E₁² I₂ = ½ c ε E₂² In this equation, 'c' is the speed of light in a vacuum while 'ε' is known as dielectric permitivity. The dielectric constant characterizes the material properties the light is traveling through.

For the sum of two waves, then I = ½ c ε (E₁ + E₂)² I = ½ c ε (E₁² + E₂² + 2 E₁ E₂) I = I₁ + I₂ + c ε E₁ E₂

Using the equation of the electric field in terms of amplitude and phase, the expression contains a product of cosines I = I₁ + I₂ + c ε A₁A₂ cos(kz - ωt - φ₁) cos(kz - ωt - φ₂) which has a trigonometric identity allowing for simplification. I = I₁ + I₂ + ½ c ε A₁A₂ [cos(φ₂ - φ₁) + cos(2kz - 2ωt - φ₁ - φ₂)]

Visible light oscillates approximately ω = 10¹⁵ cycles per second and is too fast to directly observe. Therefore the term oscillating at twice the frequency (2ω) will average to zero over any realistic measurement time.

The interference equation therefore simplifies to I = I₁ + I₂ + ½ c ε A₁A₂ cos(φ₂ - φ₁)

Next, the irradiance for each wave can be written in terms of the electric field amplitude. I₁ = ¼ c ε A₁² I₂ = ¼ c ε A₂² The extra factor of ½ is because amplitude describes peak value, but irradiance is the time average. For a cosine wave, the conversion from peak to average is a factor of ½.

With this last step, the interference equation can be simplified to the typical notation I = I₁ + I₂ + 2 √I₁I₂ cos(φ₂ - φ₁)

Final remarks

For simplicity, several details about color have been ignored in this discussion. To learn more about colors associated with two-wave interference, see the page about the soap bubble colors.

To aid with quantitative analysis of the simulation, each Zernike polynomial is normalized to have maximum value = 1. Ordering of the terms follows the convention specified by Noll[2].

References

1. M.Born and E.Wolf, "Principles of Optics", 7th ed., section 7.2, "Interference of two monochromatic waves", Cambridge University Press, 2019. (ISBN: 978-1-108-47743-7) https://doi.org/10.1017/9781108769914.
2. R. Noll, "Zernike polynomials and atmospheric turbulence", Journal of the Optical Society of America, vol. 66, 1976, pp.207-211. https://doi.org/10.1364/JOSA.66.000207