Home › Physics › Optics › Temporal Coherence in Optics

Optics

Temporal Coherence in Optics

Temporal coherence characterizes the phase correlation of a light wave at a single point in space at two different instants of time. It is a fundamental measure of the monochromaticity and frequency stability of an optical source.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Qualitative understanding of coherence, interference conditions, distinction between temporal and spatial coherence, and path difference limits in classical interferometers like Michelson.

PGT extension

Rigorous mathematical formulation using mutual coherence functions, power spectral density via the Wiener-Khinchin theorem, coherence time, coherence length, and wave packet analysis.

Prerequisites

Superposition principle of waves
Fourier transforms and frequency spectrum analysis
Wave optics and path-to-phase difference conversion

Definition and Physical Origin

Temporal coherence measures the correlation between electric field vectors E(t) and E(t + τ) at the same spatial point.
It arises due to the finite bandwidth (Δν) or frequency spread of real optical sources.
Perfect monochromatic light possesses infinite temporal coherence, whereas actual sources emit wave trains of finite duration.

Coherence Time and Coherence Length

Coherence time (τ_c) is the maximum time interval over which a predictable phase relationship is maintained, given by τ_c ≈ 1 / Δν.
Coherence length (L_c) is the spatial extent of the wave train, expressed as L_c = cτ_c = c / Δν.
For light with a mean wavelength λ₀ and wavelength spread Δλ, L_c ≈ λ₀² / Δλ.

Degree of Temporal Coherence

Quantified by the complex degree of temporal coherence γ₁₂(τ), defined via the normalized auto-correlation function of the electric field.
Fringe visibility V in a two-beam interferometer is directly proportional to the magnitude of the degree of temporal coherence: $V = 2\sqrt{I_{1}I_{2}} / (I_{1} + I_{2}) |\gamma_{12}(\tau)|$.
Interference fringes vanish when the optical path difference exceeds the coherence length of the source.

Relation with Frequency Spectrum (Wiener-Khinchin Theorem)

The temporal coherence function and the power spectral density of a light source form a Fourier transform pair.
A broader spectral width (Δν) results in a narrower coherence function and shorter coherence length.
Laser sources have extremely narrow linewidths, yielding coherence lengths of meters to kilometers.

Derivations

Derivation of Coherence Length from Wavelength Spread

Conditions: Quasi-monochromatic source with mean wavelength λ₀ and spectral width Δλ

Express frequency ν in terms of wavelength: $\nu = c / \lambda$.
Differentiate to find the frequency spread corresponding to wavelength spread: d$\nu = (c / \lambda^{2}) d\lambda$, hence Δ$\nu = (c / \lambda_{0}^{2}) \Delta\lambda$.
Use the definition of coherence time τ_c = 1 / Δν to substitute Δν.
Multiply coherence time by the speed of light c to obtain coherence length L_c = c τ_c.

Result: $L_c = \frac{\lambda_0^2}{\Delta\lambda}$

Relation between Fringe Visibility and Path Difference

Conditions: Two-beam interference with unequal path lengths producing an optical path difference (OPD) Δx

Define the time delay corresponding to the OPD as $\tau = \Deltax / c$.
Express the intensity in the interference pattern as I(τ) = I₀ [1 + |γ(τ)| cos(k c τ)], where γ(τ) is the degree of temporal coherence.
Identify maximum and minimum intensities: I_max = I₀ (1 + |γ(τ)|) and I_min = I₀ (1 - |γ(τ)|).
Calculate visibility $V = (I_{max} - I_{min}) / (I_{max} + I_{min})$.

Result: V = |\gamma(\tau)|

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Coherence Time	$$\tau_c = \frac{1}{\Delta\nu}$$	τ_c = Coherence time, Δ$\nu = Spectral bandwidth$ s (seconds)	Valid for stationary random optical fields
Coherence Length	$$L_c = c \tau_c = \frac{c}{\Delta\nu}$$	L_c = Coherence length, c = Speed of light in vacuum m (meters)	Vacuum or isotropic medium
Coherence Length in terms of Wavelength Spread	$$L_c = \frac{\lambda_0^2}{\Delta\lambda}$$	λ₀ = Mean wavelength, Δ$\lambda = Wavelength spread$ m (meters)	Small spectral width approximation (Δλ \ll \lambda_0)
Fringe Visibility	$$V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$$	I_max = Maximum intensity, I_min = Minimum intensity Dimensionless	Standard two-beam interference
Complex Degree of Temporal Coherence Magnitude	$$\|\gamma_{12}(\tau)\| = \frac{2 \sqrt{I_1 I_2}}{I_1 + I_2} V$$	I₁, I₂ = Beam intensities, V = Visibility Dimensionless	Equal polarization states
Path Difference Limit for Interference	$$\Delta x \le L_c$$	Δx = Optical path difference between interfering beams m (meters)	Observable interference fringes condition
Bandwidth-Coherence Product	$$\Delta\nu \cdot \tau_c \approx 1$$	Δ$\nu = Frequency spread$, τ_c = Coherence time Dimensionless	Fourier transform limit
Time Delay from Path Difference	$$\tau = \frac{\Delta x}{c}$$	$\tau = Time delay$, Δx = Path difference, c = Speed of light s (seconds)	Uniform refractive index medium

Graphs and experiments

Michelson Interferometer fringe visibility vs Path Difference

Draw: A damped cosine oscillation curve where fringe visibility V decreases monotonically from 1 to 0 as path difference Δx increases.

Interpret: Demonstrates that interference fringes wash out when the path difference exceeds the coherence length of the source.

Exam use: Tested conceptually to determine coherence length of sodium lamps or lasers.

Power Spectral Density and Coherence Function

Draw: A Gaussian or Lorentzian spectral profile Δν matched against the corresponding Fourier-transformed temporal envelope τ_c.

Interpret: Shows inverse relationship between spectral width in frequency domain and temporal extent in time domain.

Exam use: Used in PGT numericals relating linewidth to coherence time.

Wave Packet Train representation

Draw: Series of finite sinusoidal wave packets of length L_c separated by random phase jumps.

Interpret: Illustrates that thermal sources emit independent wave bursts with random initial phases.

Exam use: Helps clarify why incoherent sources cannot produce stable interference over long path differences.

Solved numericals

TGT core

Calculate the coherence length of a sodium light source emitting at mean wavelength 589 nm with a spectral wavelength spread of 0.6 nm.

Identify given values: λ₀ = 589 n$m = 589 \times 10^{-9} m$, Δ$\lambda = 0.6 nm = 0.6 \times 10^{-9} m$.
Apply the formula for coherence length: L_c = λ₀² / Δλ.
Substitute values: L_c = (589 \times 10^-9)² / (0.6 \times 10^-9).
Calculate result: L_c = (3.4692 \times 10^-13) / (0.6 \times 10^-9) = 5.78 \times 10^-4 m.

Answer: 5.78 \times 10^-4 m (or 0.578 mm)

PGT extension

A laser source has a frequency bandwidth of 2 MHz. Find its coherence time and coherence length.

Identify given frequency bandwidth: Δ$\nu = 2 MHz = 2 \times 10^{6} Hz$.
Calculate coherence time using τ_c = 1 / Δν.
Substitute values: τ_c = 1 / (2 \times 10⁶) = 5 \times 10^-7 s.
Calculate coherence length using L_c = c τ_c taking $c = 3 \times 10^{8} m/s$.
Multiply: L_c = (3 \times 10⁸) \times (5 \times 10^-7) = 150 m.

Answer: Coherence tim$e = 5 \times 10^-7 s$, Coherence length = 150 m

PGT extension

In a Michelson interferometer experiment using a spectral line, fringes disappear when the movable mirror is translated by 0.15 mm. Estimate the spectral width (Δλ) of the source if its mean wavelength is 600 nm.

Note that fringe disappearance occurs when path difference Δ$x = 2 \times (mirror translation d) = 2 \times 0.15 mm = 0.30 mm = 3 \times 10^{-4} m$.
Equate coherence length L_c to path difference: L_c = Δ$x = 3 \times 10^{-4} m$.
Use formula L_c = λ₀² / Δλ to solve for Δλ.
Rearrange: Δ$\lambda = \lambda_{0}^{2} / L_{c} = (600 \times 10^{-9})^{2} / (3 \times 10^{-4})$.
Calculate: Δ$\lambda = (3.6 \times 10^{-13}) / (3 \times 10^{-4}) = 1.2 \times 10^{-9} m = 1.2 nm$.

Answer: 1.2 nm

Assumptions and limitations

Assumes quasi-monochromatic light where spectral width is much smaller than mean frequency (Δν \ll \nu₀).
Coherence length formulas represent order-of-magnitude estimates based on Fourier transform limits rather than exact lineshape profiles unless specified as Gaussian or Lorentzian.

Common misconceptions

Common error: Temporal coherence depends on the physical size and spatial separation of the light source pinholes.

Correct: Temporal coherence depends strictly on the frequency spectrum (monochromaticity) of the source, whereas spatial coherence depends on source size.

Spatial and temporal coherence are orthogonal properties; temporal relates to time delays at a single point, spatial relates to phase correlations at different points.

Common error: A thermal source like an incandescent bulb has a long coherence length because it appears bright.

Correct: Thermal sources have broad emission bandwidths, resulting in very short coherence lengths (typically micrometers).

Brightness is related to intensity, not frequency purity or coherence.

Quick revision

Temporal coherence is the correlation of a wave with its time-delayed self ($\tau = \Deltax / c)$.
Coherence time τ_c = 1 / Δν and coherence length L_c = λ₀² / Δλ.
Fringe visibility vanishes when optical path difference exceeds coherence length.

Must remember

Temporal coherence measures the phase correlation of electric field vectors $E(t)$ and $E(t + \tau)$ at the exact same spatial point.
Coherence time $\tau_c$ is inversely proportional to the spectral bandwidth $\Delta\nu$ ($\tau_c \approx 1 / \Delta\nu$).
Coherence length $L_c$ is given by $c\tau_c$ or $\lambda_0^2 / \Delta\lambda$, representing the spatial length of the wave train.
Interference fringes completely vanish when the optical path difference $\Delta x$ exceeds the coherence length $L_c$ of the optical source.
According to the Wiener-Khinchin theorem, the temporal coherence function and the power spectral density form a Fourier transform pair.

Glossary

Temporal CoherenceThe degree of phase correlation of a light wave with itself at a later time at the same spatial point.

Coherence Time ($\tau_c$)The maximum time interval over which a predictable phase relationship is maintained by an optical source.

Coherence Length ($L_c$)The spatial length of a wave train over which temporal coherence is preserved, equal to $c\tau_c$.

Fringe Visibility ($V$)A quantitative measure of the contrast of interference fringes defined by $(I_{max} - I_{min}) / (I_{max} + I_{min})$.

Wiener-Khinchin TheoremStates that the power spectral density and the temporal coherence function of a stationary random process form a Fourier transform pair.

Temporal vs Spatial Coherence

Feature	Temporal Coherence	Spatial Coherence
Definition	Correlation between fields at different times at the same point	Correlation between fields at different points at the same time
Governing Factor	Spectral bandwidth ($\Delta\nu$)	Source size and spatial extension
Relevant Parameter	Coherence length ($L_c$)	Coherence area

Practice Quiz

Choose an option to reveal the explanation.

1. What does temporal coherence primarily measure in optics?

Explanation: Temporal coherence measures the phase correlation between $E(t)$ and $E(t + \tau)$ at the same spatial point over a time delay $\tau$.

2. What is the physical origin of finite temporal coherence in real optical sources?

Explanation: Real optical sources emit light with a finite frequency spread or bandwidth ($\Delta\nu$), leading to wave trains of finite duration.

3. Which type of light source possesses infinite temporal coherence in theory?

Explanation: Perfect monochromatic light has a single frequency (zero bandwidth), which implies infinite temporal coherence and infinite coherence length.

4. How is coherence time ($\tau_c$) related to the spectral bandwidth ($\Delta\nu$)?

Explanation: Coherence time is inversely proportional to the spectral bandwidth, given by $\tau_c \approx 1 / \Delta\nu$.

5. What is the formula for coherence length ($L_c$) in terms of coherence time ($\tau_c$) and speed of light ($c$)?

Explanation: Coherence length is the spatial extent of the wave train, expressed as $L_c = c \tau_c = c / \Delta\nu$.

6. For a light source with mean wavelength $\lambda_0$ and wavelength spread $\Delta\lambda$, what is the expression for coherence length ($L_c$)?

Explanation: Using $L_c = c / \Delta\nu$ and $\nu = c / \lambda$, differentiation yields $L_c \approx \lambda_0^2 / \Delta\lambda$.

7. What theorem states that the temporal coherence function and the power spectral density form a Fourier transform pair?

Explanation: The Wiener-Khinchin theorem establishes that the power spectral density and the temporal coherence function form a Fourier transform pair.

8. What happens to interference fringes in a two-beam interferometer when the optical path difference exceeds the coherence length ($L_c$)?

Explanation: When the optical path difference exceeds the coherence length of the source, the phase relationship is lost and interference fringes vanish.

9. How is fringe visibility ($V$) defined in terms of maximum and minimum intensities?

Explanation: Standard fringe visibility is defined as $V = (I_{max} - I_{min}) / (I_{max} + I_{min})$.y

10. Why do thermal sources like incandescent bulbs have very short coherence lengths?

Explanation: Thermal sources emit across a broad frequency spectrum (large $\Delta\nu$), which yields very short coherence times and lengths (typically micrometers).

11. What is the approximate coherence length of a typical laser source due to its extremely narrow linewidth?

Explanation: Lasers have extremely narrow linewidths (very small $\Delta\nu$), which yields coherence lengths ranging from meters to kilometers.

12. Which incorrect misconception is commonly associated with temporal coherence?

Explanation: Temporal coherence depends strictly on the frequency spectrum (monochromaticity), whereas spatial coherence depends on source size.

13. What condition must the optical path difference ($\Delta x$) satisfy for observable interference fringes?

Explanation: Observable interference fringes require the optical path difference to be less than or equal to the coherence length ($|\Delta x| \le L_c$).

14. If a spectral bandwidth $\Delta\nu$ increases, how does it affect the coherence length $L_c$?

Explanation: Since $L_c = c / \Delta\nu$, an increase in spectral bandwidth ($\Delta\nu$) results in a shorter coherence length.

15. What is the dimensionless product of bandwidth and coherence time ($\Delta\nu \cdot \tau_c$) approximately equal to?

Explanation: The bandwidth-coherence product is governed by Fourier transform limits, yielding $\Delta\nu \cdot \tau_c \approx 1$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. An optical source has a mean wavelength of $589 \text{ nm}$ and a wavelength spread of $0.1 \text{ nm}$. What is its approximate coherence length?

Explanation: Using $L_c \approx \lambda_0^2 / \Delta\lambda = (589 \times 10^{-9})^2 / (0.1 \times 10^{-9}) \approx 3.47 \times 10^{-3} \text{ m} = 3.47 \text{ mm}$.

2. If the spectral bandwidth of a light source is doubled, what happens to its coherence time?

Explanation: Coherence time is inversely proportional to spectral bandwidth ($\tau_c \approx 1 / \Delta\nu$). Doubling $\nu$ halves $\tau_c$.

3. In a Michelson interferometer, white light fringes disappear after a very small path difference because:

Explanation: White light contains a continuous broad spectrum of wavelengths, resulting in a very short coherence length and rapid disappearance of fringes.

4. The complex degree of temporal coherence magnitude $|\gamma_{12}(\tau)|$ is directly connected to which parameter in a two-beam interferometer?

Explanation: Fringe visibility $V$ in a two-beam interferometer is proportional to the magnitude of the degree of temporal coherence: $V = \frac{2\sqrt{I_1 I_2}}{I_1 + I_2} |\gamma_{12}(\tau)|$.

5. What is the coherence time of a light source having a frequency bandwidth of $10^8 \text{ Hz}$?

Explanation: Coherence time $\tau_c \approx 1 / \Delta\nu = 1 / 10^8 \text{ Hz} = 10^{-8} \text{ s}$.

6. Which of the following correctly pairs the Fourier transform relationship according to the Wiener-Khinchin theorem?

Explanation: The Wiener-Khinchin theorem establishes that the temporal coherence function and the power spectral density of a light source form a Fourier transform pair.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Temporal coherence is primarily a measure of correlation between electric field vectors at:

Explanation: Temporal coherence specifically evaluates the correlation between $E(t)$ and $E(t + \tau)$ at the same spatial point.

2. Calculate the coherence length of a laser source with a spectral linewidth ($\Delta\nu$) of $1 \text{ MHz}$ ($10^6 \text{ Hz}$).

Explanation: $L_c = c / \Delta\nu = (3 \times 10^8 \text{ m/s}) / (10^6 \text{ s}^{-1}) = 300 \text{ m}$.

3. Which statement regarding spatial and temporal coherence is correct?

Explanation: Spatial and temporal coherence are orthogonal: temporal coherence relates to frequency bandwidth, while spatial coherence relates to source size.

4. If the fringe visibility $V$ in an interferometer is $0.8$ and the beam intensities are equal ($I_1 = I_2$), what is the magnitude of the complex degree of temporal coherence $|\gamma_{12}(\tau)|$?

Explanation: When $I_1 = I_2$, the formula $|\gamma_{12}(\tau)| = \frac{2\sqrt{I_1 I_2}}{I_1 + I_2} V$ simplifies to $|\gamma_{12}(\tau)| = V = 0.8$.

5. What is the time delay $\tau$ corresponding to an optical path difference of $1.5 \text{ m}$ in air (taking $c = 3 \times 10^8 \text{ m/s}$)?

Explanation: Time delay $\tau = \Delta x / c = 1.5 \text{ m} / (3 \times 10^8 \text{ m/s}) = 5 \times 10^{-9} \text{ s}$? Wait, calculation: $1.5 / 3 = 0.5$, so $0.5 \times 10^{-8} = 5 \times 10^{-9} \text{ s}$. Let's verify option values: $5 \times 10^{-9} \text{ s}$ is correct.

6. Why do laser sources achieve coherence lengths of kilometers?

Explanation: Extremely narrow spectral linewidths (small $\Delta\nu$) result in very large coherence times and lengths.

7. What happens to the magnitude of the degree of temporal coherence $|\gamma_{12}(\tau)|$ as the time delay $\tau$ increases well beyond the coherence time $\tau_c$?

Explanation: As the time delay exceeds the coherence time, phase correlation is lost and the degree of temporal coherence approaches zero.

8. Which of the following is a necessary condition for observing clear interference fringes in a Michelson interferometer using a thermal source?

Explanation: Interference fringes are observable only if the optical path difference does not exceed the coherence length ($|\Delta x| \le L_c$).

9. If $I_{max} = 9 \text{ units}$ and $I_{min} = 1 \text{ unit}$ in an interference pattern, what is the fringe visibility $V$?

Explanation: $V = (9 - 1) / (9 + 1) = 8 / 10 = 0.8$.

10. Which misconception is explicitly refuted by optical coherence theory?

Explanation: Brightness is an intensity property, whereas coherence length is determined by frequency purity (bandwidth), making thermal sources short-coherence despite high brightness.