Light waves have a secret superpower: temporal coherence. This measures how well a wave's phases match up over time as it travels. The more coherent the light, the longer it can interfere with itself, creating stunning patterns.

Temporal coherence is like a light wave's attention span. Lasers, with their single color, have the longest coherence times. White light, with its rainbow of colors, has the shortest. This property affects how light behaves in interferometers and other optical setups.

Temporal Coherence and Coherence Time

Temporal coherence and spectral bandwidth

Temporal coherence measures the correlation between phases of a light wave at different points along propagation direction
- Quantifies the ability of a light source to interfere with a delayed version of itself (self-interference)
Temporal coherence is inversely related to the spectral bandwidth of a light source
- Narrower spectral bandwidth results in higher temporal coherence (lasers)
- Broader spectral bandwidth leads to lower temporal coherence (white light)
Monochromatic light sources exhibit the highest temporal coherence due to their narrow spectral bandwidth (single wavelength)
White light sources have low temporal coherence because of their broad spectral bandwidth (wide range of wavelengths)

Coherence time calculations

Coherence time ( $\tau_c$ $τ_{c}$ ) is the time over which the phase of a light wave remains predictable
- Related to the spectral bandwidth ( $\Delta \nu$ ) by: $\tau_c \approx \frac{1}{\Delta \nu}$
Longer coherence time implies light source maintains phase coherence over a longer duration
- Allows observation of interference effects over larger path length differences
Shorter coherence time means phase coherence is lost more quickly
- Interference effects only observable for smaller path length differences
Coherence time determines the maximum path length difference ( $\Delta L_{max}$ $Δ L_{ma x}$ ) for observable interference:
- $\Delta L_{max} \approx c \tau_c$ , where $c$ is the speed of light

Temporal coherence in interferometers

In interferometers (Michelson, Mach-Zehnder), visibility of interference fringes depends on temporal coherence of light source
High temporal coherence (long coherence time) leads to high-contrast, clearly visible interference fringes
- Fringes remain visible for larger path length differences between interferometer arms
Low temporal coherence (short coherence time) results in low-contrast or no visible interference fringes
- Fringes only visible for small path length differences, quickly disappear as path length difference increases
Fringe visibility ( $V$ ) is related to the degree of coherence ( $\gamma$ ) by: $V = |\gamma|$
Degree of coherence decreases with increasing path length difference and decreasing temporal coherence

Coherence properties of light sources

Monochromatic light sources (lasers) have the highest temporal coherence
- Emit light with a single wavelength, very narrow spectral bandwidth
- Produce high-contrast interference fringes over large path length differences
Quasi-monochromatic light sources (LEDs, filtered lamps) have moderate temporal coherence
- Emit light with a narrow range of wavelengths, relatively narrow spectral bandwidth
- Produce visible interference fringes, but fringe contrast decreases more quickly with increasing path length difference compared to monochromatic sources
White light sources (incandescent bulbs, sun) have low temporal coherence
- Emit light with a wide range of wavelengths, broad spectral bandwidth
- Produce interference fringes with very low contrast, visible only for extremely small path length differences