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Spatial Coherence in Optics

Spatial coherence is a measure of the correlation of the phase of an electromagnetic wave at different points in space transverse to the direction of propagation. It determines the ability of a wavefront to exhibit well-defined interference patterns when sampled at separated locations.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Basic understanding of wave fronts, phase difference, conditions for sustained interference, and Young's double-slit experiment requirements.

PGT extension

Quantitative treatment of transverse coherence length, Van Cittert-Zernike theorem, mutual coherence function, and finite source size effects on fringe visibility.

Prerequisites

Basic Concepts of Diffraction
Double-Slit Diffraction
Fraunhofer Diffraction

Definition and Physical Origin of Spatial Coherence

Spatial coherence evaluates the cross-correlation of optical fields at two distinct points separated transversely at the same instant in time.
It arises fundamentally from the spatial extent of the light source; extended sources produce independent emission centers that wash out interference fringes.
A point source at infinity or an infinitely small pinhole theoretically yields perfect spatial coherence across the entire transverse plane.

Transverse Coherence Width

The transverse coherence width (or length) l_c defines the maximum spatial separation between two points in the wavefront over which the phase remains correlated.
It is inversely proportional to the angular size subtended by the source at the observation plane.
For a quasi-monochromatic source of size s at a distance R, the transverse coherence length is approximately given by l_c ≈ λR / s.

Degree of Spatial Coherence and Visibility

The complex degree of spatial coherence γ₁₂(τ) quantifies the fringe visibility in a double-slit arrangement.
Fringe visibility V is defined as (I_max - I_min) / (I_max + I_min), which directly equals the magnitude of the complex degree of coherence |γ₁₂|.
When |γ₁₂| = 1, the source is completely spatially coherent; when |γ₁₂| = 0, the source is completely spatially incoherent.

Role of Apertures and Spatial Filtering

Placing a narrow pinhole in front of an extended incoherent source spatially filters the light, increasing the spatial coherence of the transmitted beam.
According to the Van Cittert-Zernike theorem, the spatial coherence function of an incoherent source is the Fourier transform of the source intensity distribution.
Laser sources exhibit extremely large spatial coherence due to stimulated emission and resonant cavity transverse mode selection.

Derivations

Transverse Coherence Width from an Extended Source

Conditions: Quasi-monochromatic extended source of width s, observed at distance R through two slits separated by distance d.

Consider two infinitesimal elements of an extended source separated by distance x that emit light independently.
The phase difference introduced at the two observation points (separated by d) due to the displacement x of the source element is given by dφ ≈ (2π / λ) (xd / R).
Interference fringes due to different source elements wash out when the phase difference reaches 2π, which defines the limiting separation d_max = l_c ≈ λR / s.

Result: $l_{c} = \frac{\lambda R}{s}$

Visibility and Complex Degree of Coherence

Conditions: Two-beam interference with electric field amplitudes E₁ and E₂.

The total intensity is expressed as $I = I_1 + I_2 + 2 \sqrt{I_1 I_2} Re{\gamma_{12}(\tau)}$, where γ₁₂ is the complex degree of coherence.
For equal intensity beams (I₁ = I₂ = I₀), the maximum intensity is 4I₀(1 + |γ_12|) and minimum intensity is 4I_0(1 - |γ_12|).
Substitute I_max and I_min into the visibility formula $V = (I_{max} - I_{min}) / (I_{max} + I_{min})$.

Result: $V = |\gamma_{12}|$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Transverse Coherence Length	$$l_{c} = \frac{\lambda R}{s}$$	l_c = coherence length, $\lambda = wavelength$, R = distance from source, s = source size m	Quasi-monochromatic light, small angle approximation
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 Coherence Magnitude	$$\|\gamma_{12}(\tau)\| = \frac{2 \sqrt{I_1 I_2}}{I_1 + I_2} \|V\|$$	γ₁₂ = complex degree of coherence, I₁, I₂ = intensities from individual beams Dimensionless	General partial coherence
Spatial Coherence Radius (Van Cittert-Zernike)	$$r_{c} = 0.61 \frac{\lambda}{\theta}$$	r_c = coherence radius, $\lambda = wavelength$, $\theta = angular source size$ m	Circular uniform incoherent source
Phase Difference Across Wavefront	$$\delta \phi = \frac{2pi d x}{\lambda R}$$	d = slit separation, x = source element position, R = distance rad	Fraunhofer diffraction regime
Maximum Slit Separation for Interference	$$d_{max} le \frac{\lambda R}{s}$$	d_max = maximum allowable slit distance, $\lambda = wavelength$, R = distance, s = source diameter m	For observable interference fringes (V > 0)
Mutual Coherence Function	$$\gamma_{12}(\tau) = \langle E_1(t+\tau) E_2^*(t) \rangle$$	Γ₁₂ = mutual coherence function, E₁, E₂ = electric fields, $\tau = time delay$ W/m² (or V²/m²)	Stationary random optical fields
Normalized Mutual Coherence (Complex Degree)	$$\gamma_{12}(\tau) = \frac{\gamma_{12}(\tau)}{\sqrt{\gamma_{11}(0) \gamma_{22}(0)}}$$	γ₁₂ = complex degree of coherence, Γ₁₁, Γ₂₂ = self-coherence terms Dimensionless	Standardized normalization

Graphs and experiments

Young's Double-Slit Experiment with Extended Source

Draw: A source of width s illuminating two pinholes separated by d, projecting interference fringes on a screen where visibility decreases as d exceeds l_c.

Interpret: Demonstrates that fringe contrast (visibility) drops to zero when the separation between pinholes exceeds the transverse coherence length of the illumination.

Exam use: Tested conceptually and numerically in PGT/TGT regarding fringe disappearance conditions.

Visibility vs. Slit Separation Graph

Draw: A curve plotting fringe visibility V (y-axis from 0 to 1) against slit separation d (x-axis), showing a decaying oscillatory or monotonic decrease depending on source profile.

Interpret: As the distance between the two sampling points increases, the spatial correlation decreases, resulting in washed-out interference fringes.

Exam use: Interpreting graphs of visibility to determine coherence width or source dimensions.

Spatial Filtering Setup (Michelson stellar interferometer principle)

Draw: Light from an extended star passing through two movable mirrors/apertures, producing fringes whose disappearance measures stellar angular diameter.

Interpret: Utilizes spatial coherence measurement to determine astronomical object dimensions.

Exam use: Application-based questions in PGT physics regarding stellar diameter measurement.

Solved numericals

TGT core

Light of wavelength 600 nm from a narrow slit is used in a double-slit setup. If the source width is 0.2 mm and placed at a distance of 1 m from the slits, calculate the maximum slit separation for which interference fringes can be observed.

Identify given values: $\lambda = 600 nm = 6 \times 10^{-7} m$, R = 1 m, $s = 0.2 mm = 2 \times 10^{-4} m$.
Use the formula for maximum transverse coherence length: d_max = λR / s.
Substitute the values: d_max = (6 × 10^-7 × 1) / (2 × 10^-4).
Calculate result: d_max = 3 × 10^-3 m = 3 mm.

Answer: 3 mm

PGT extension

In a Young's double-slit experiment, the intensity maxima and minima in the interference pattern are measured to be 9 units and 1 unit respectively. Calculate the magnitude of the complex degree of spatial coherence (|γ₁₂|) between the two slits, assuming equal beam intensities.

Identify given values: I_max = 9, I_min = 1.
Recall the formula for fringe visibility: $V = (I_{max} - I_{min}) / (I_{max} + I_{min})$.
Substitute values: $V = (9 - 1) / (9 + 1) = 8 / 10 = 0.8$.
Relate visibility to the complex degree of spatial coherence for equal beam intensities: $V = |\gamma_{12}|$.
Therefore, |gamma_{12}| = 0.8.

Answer: 0.8

PGT extension

A circular incoherent source of diameter 2 cm is located at a distance of 5 m from an observation plane. Using light of wavelength 500 nm, find the transverse spatial coherence radius r_c.

Identify given parameters: source size s = 2 cm = 0.02 m, R = 5 m, $\lambda = 500 nm = 5 \times 10^{-7} m$.
Calculate the angular source size θ ≈ s / $R = 0.02 / 5 = 4 \times 10^{-3} rad$.
Use the coherence radius relation for a circular source: r_c = 0.61 λ / θ.
Substitute values: r_c = (0.61 × 5 × 10^-7) / (4 × 10^-3).
Calculate result: r_c = 7.625 × 10^-5 m = 0.076 mm.

Answer: 0.076 mm

Assumptions and limitations

Quasi-monochromatic approximation: Bandwidth Δν is much smaller than the central frequency ν₀.
Small angle approximation is assumed in deriving transverse coherence length from source geometry.
The Van Cittert-Zernike theorem strictly applies in the Fraunhofer (far-field) diffraction regime.

Common misconceptions

Common error: Spatial coherence and temporal coherence are identical concepts.

Correct: Spatial coherence refers to phase correlation across transverse spatial points, whereas temporal coherence refers to phase correlation at different times along the propagation axis.

They depend on different physical properties of the source: spatial coherence relates to source transverse size, while temporal coherence relates to spectral bandwidth.

Common error: An ordinary incandescent bulb has high spatial coherence.

Correct: An incandescent bulb has very low spatial coherence due to its large, independent thermal emission centers.

Spatial coherence is inversely proportional to source size; large extended sources destroy spatial phase correlation.

Quick revision

Spatial coherence measures phase correlation across transverse points on a wavefront.
Transverse coherence length is given by l_c ≈ λR / s.
Fringe visibility V equals the magnitude of the complex degree of spatial coherence |gamma_{12}|.
Spatial filtering using a pinhole increases spatial coherence by reducing the effective source size.

Must remember

Spatial coherence evaluates the cross-correlation of optical fields at two distinct points separated transversely at the same instant in time.
The transverse coherence length is given by $l_c \approx \frac{\lambda R}{s}$, where $\lambda$ is wavelength, $R$ is distance from the source, and $s$ is source size.
Fringe visibility is defined as $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$ and directly equals the magnitude of the complex degree of spatial coherence $|\gamma_{12}|$.
According to the Van Cittert-Zernike theorem, the spatial coherence function of an incoherent source is the Fourier transform of the source intensity distribution.
Lasers exhibit extremely large spatial coherence due to stimulated emission and resonant cavity transverse mode selection.

Glossary

Spatial CoherenceEvaluation of the phase correlation of optical fields at two distinct points separated transversely at the same instant in time.

Transverse Coherence Width ($l_c$)Maximum spatial separation between two points in the wavefront over which the phase remains correlated.

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

Van Cittert-Zernike TheoremTheorem stating that the spatial coherence function of an incoherent source is the Fourier transform of the source intensity distribution.

Spatial Coherence vs. Temporal Coherence

Feature	Spatial Coherence	Temporal Coherence
Definition	Phase correlation at different transverse points at the same time	Phase correlation at the same spatial point at different times
Primary Source Factor	Transverse extent or size of the source ($s$)	Spectral bandwidth or monochromaticity ($\Delta \lambda$ or $\Delta \nu$)
Key Formula	$l_c \approx \frac{\lambda R}{s}$	$l_t \approx \frac{\lambda^2}{\Delta \lambda}$

Practice Quiz

Choose an option to reveal the explanation.

1. What does spatial coherence primarily evaluate in an optical field?

Explanation: Spatial coherence evaluates the cross-correlation of optical fields at two distinct points separated transversely at the same instant in time.

2. Which type of light source theoretically yields perfect spatial coherence across the entire transverse plane?

Explanation: A point source at infinity or an infinitely small pinhole theoretically yields perfect spatial coherence across the entire transverse plane.

3. The transverse coherence width $l_c$ is inversely proportional to which parameter?

Explanation: Transverse coherence length is inversely proportional to the angular size subtended by the source at the observation plane.

4. For a quasi-monochromatic source of size $s$ at a distance $R$, the transverse coherence length $l_c$ is approximately:

Explanation: The transverse coherence length formula is $l_c \approx \frac{\lambda R}{s}$ for a source of size $s$ at distance $R$.

5. How is fringe visibility $V$ defined in a standard two-beam interference experiment?

Explanation: Fringe visibility $V$ is defined as $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$.

6. When the magnitude of the complex degree of coherence $|\gamma_{12}|$ equals 1, what does it indicate about the source?

Explanation: When $|\gamma_{12}| = 1$, the source is completely spatially coherent.

7. What is the physical effect of placing a narrow pinhole in front of an extended incoherent source?

Explanation: Placing a narrow pinhole in front of an extended incoherent source spatially filters the light, increasing the spatial coherence of the transmitted beam.

8. According to the Van Cittert-Zernike theorem, the spatial coherence function of an incoherent source is related to the source intensity distribution by which mathematical operation?

Explanation: According to the Van Cittert-Zernike theorem, the spatial coherence function of an incoherent source is the Fourier transform of the source intensity distribution.

9. Why do laser sources exhibit extremely large spatial coherence?

Explanation: Laser sources exhibit extremely large spatial coherence due to stimulated emission and resonant cavity transverse mode selection.

10. What is the primary physical origin of spatial coherence degradation in thermal sources?

Explanation: Spatial coherence arises fundamentally from the spatial extent of the light source; extended sources produce independent emission centers that wash out interference fringes.

11. What does the condition $|\gamma_{12}| = 0$ signify for an optical source?

Explanation: When $|\gamma_{12}| = 0$, the source is completely spatially incoherent.

12. Which of the following correctly distinguishes spatial coherence from temporal coherence?

Explanation: Spatial coherence refers to phase correlation across transverse spatial points, whereas temporal coherence refers to phase correlation at different times along the propagation axis.

13. Why does an ordinary incandescent bulb possess very low spatial coherence?

Explanation: An incandescent bulb has very low spatial coherence due to its large, independent thermal emission centers.

14. What is the unit of the complex degree of spatial coherence $\gamma_{12}$?

Explanation: The complex degree of spatial coherence $\gamma_{12}$ is a dimensionless quantity.

15. For observable interference fringes ($V > 0$), what is the condition on maximum slit separation $d_{max}$?

Explanation: The maximum allowable slit distance for observable interference fringes is $d_{max} \le \frac{\lambda R}{s}$.

PYQ Patterns

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

1. If a quasi-monochromatic source of diameter $s = 2\text{ mm}$ is at a distance $R = 1\text{ m}$ emitting light of wavelength $\lambda = 500\text{ nm}$, what is the approximate transverse coherence length $l_c$?

Explanation: Using $l_c \approx \frac{\lambda R}{s} = \frac{500 \times 10^{-9} \text{ m} \times 1\text{ m}}{2 \times 10^{-3} \text{ m}} = 2.5 \times 10^{-4}\text{ m} = 0.25\text{ mm}$.

2. In a double-slit experiment, the maximum intensity is measured as $9I_0$ and the minimum intensity as $I_0$. What is the fringe visibility $V$?

Explanation: Visibility $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{9I_0 - I_0}{9I_0 + I_0} = \frac{8I_0}{10I_0} = 0.8$.

3. What is the physical role of a spatial filter consisting of a microscope objective and a pinhole in a laser beam path?

Explanation: A spatial filter cleans up the laser beam profile by eliminating diffraction rings and higher-order transverse modes, ensuring a clean Gaussian profile with high spatial coherence.

4. According to the Van Cittert-Zernike theorem, if the source intensity distribution is uniform and circular, the spatial coherence radius $r_c$ is proportional to:

Explanation: The spatial coherence radius for a circular uniform incoherent source is given by $r_c = 0.61 \frac{\lambda}{\theta}$, which is proportional to $\frac{\lambda}{\theta}$.

5. Why is it difficult to observe interference fringes using sunlight directly without a preliminary pinhole or slit?

Explanation: Sunlight comes from an extended disk (large angular size $s/R$), resulting in a tiny transverse coherence length, washing out interference unless a narrow pinhole is used.

6. If the separation between two slits in a Young's experiment exceeds the transverse coherence length of the source ($d > l_c$), what happens to the interference pattern?

Explanation: When slit separation exceeds the transverse coherence length ($d > l_c$), the phase relationship between the two slits is lost, fringe visibility drops to zero, and interference fringes disappear.

Timed Topic Test

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

1. Which of the following statements about spatial coherence is correct?

Explanation: Spatial coherence evaluates phase correlation across transverse spatial points in an optical wavefront.

2. Calculate the transverse coherence length for light of wavelength $\lambda = 600\text{ nm}$ from a star of angular diameter subtending $10^{-5}\text{ radians}$ at Earth.

Explanation: Using $r_c = 0.61 \frac{\lambda}{\theta} = 0.61 \times \frac{600 \times 10^{-9}}{10^{-5}} = 0.61 \times 6 \times 10^{-2} = 0.0366\text{ m} = 3.66\text{ cm}$.

3. In partial coherence theory, the mutual coherence function $\Gamma_{12}(\tau)$ is defined as:

Explanation: The mutual coherence function is defined as $\Gamma_{12}(\tau) = \langle E_1(t+\tau) E_2^*(t) \rangle$ for stationary random optical fields.

4. What is the relationship between fringe visibility $V$ and the complex degree of coherence magnitude $|\gamma_{12}|$?

Explanation: The magnitude of the complex degree of coherence is related to visibility via $|\gamma_{12}(\tau)| = \frac{I_1 + I_2}{2\sqrt{I_1 I_2}} V$, meaning $V = |\gamma_{12}|$ when beam intensities are equal ($I_1 = I_2$). For general partial coherence, visibility incorporates this factor.

5. If a source size $s$ is doubled while keeping $R$ and $\lambda$ constant, what happens to the transverse coherence length $l_c$?

Explanation: Since $l_c \approx \frac{\lambda R}{s}$, coherence length is inversely proportional to source size $s$. Doubling $s$ halves $l_c$.

6. Which source exhibits the lowest spatial coherence among the following?

Explanation: An unfiltered incandescent tungsten bulb has a large extended thermal emitting area, giving it extremely low spatial coherence.

7. The normalized mutual coherence function $\gamma_{12}(\tau)$ is expressed as:

Explanation: The complex degree of coherence is defined as the normalized mutual coherence function: $\gamma_{12}(\tau) = \frac{\Gamma_{12}(\tau)}{\sqrt{\Gamma_{11}(0) \Gamma_{22}(0)}}$.

8. In Young's double-slit experiment, if $I_{max} = 4\text{ units}$ and $I_{min} = 0\text{ units}$, the fringe visibility is:

Explanation: Visibility $V = \frac{4 - 0}{4 + 0} = \frac{4}{4} = 1.0$, indicating perfect coherence and contrast.

9. What is the primary consequence of the Van Cittert-Zernike theorem in optical imaging and astronomy?

Explanation: The theorem forms the basis of stellar interferometry, where measuring spatial coherence (visibility) at different antenna/telescope spacings determines the source intensity distribution (e.g., star diameter).

10. Under what condition is the small angle approximation used in deriving transverse coherence length valid?

Explanation: The small angle approximation $s \ll R$ is assumed when relating transverse coherence length to angular source size.

Spatial Coherence in Optics

Scope and prerequisites

TGT core

PGT extension

Prerequisites

Definition and Physical Origin of Spatial Coherence

Transverse Coherence Width

Degree of Spatial Coherence and Visibility

Role of Apertures and Spatial Filtering

Derivations

Transverse Coherence Width from an Extended Source

Visibility and Complex Degree of Coherence

Formula sheet

Graphs and experiments

Young's Double-Slit Experiment with Extended Source

Visibility vs. Slit Separation Graph

Spatial Filtering Setup (Michelson stellar interferometer principle)

Solved numericals

Light of wavelength 600 nm from a narrow slit is used in a double-slit setup. If the source width is 0.2 mm and placed at a distance of 1 m from the slits, calculate the maximum slit separation for which interference fringes can be observed.

In a Young's double-slit experiment, the intensity maxima and minima in the interference pattern are measured to be 9 units and 1 unit respectively. Calculate the magnitude of the complex degree of spatial coherence (|γ12|) between the two slits, assuming equal beam intensities.

A circular incoherent source of diameter 2 cm is located at a distance of 5 m from an observation plane. Using light of wavelength 500 nm, find the transverse spatial coherence radius rc.

Assumptions and limitations

Common misconceptions

Quick revision

Must remember

Glossary

Spatial Coherence vs. Temporal Coherence

Practice Quiz

1. What does spatial coherence primarily evaluate in an optical field?

2. Which type of light source theoretically yields perfect spatial coherence across the entire transverse plane?

3. The transverse coherence width $l_c$ is inversely proportional to which parameter?

4. For a quasi-monochromatic source of size $s$ at a distance $R$, the transverse coherence length $l_c$ is approximately:

5. How is fringe visibility $V$ defined in a standard two-beam interference experiment?

6. When the magnitude of the complex degree of coherence $|\gamma_{12}|$ equals 1, what does it indicate about the source?

7. What is the physical effect of placing a narrow pinhole in front of an extended incoherent source?

8. According to the Van Cittert-Zernike theorem, the spatial coherence function of an incoherent source is related to the source intensity distribution by which mathematical operation?

9. Why do laser sources exhibit extremely large spatial coherence?

10. What is the primary physical origin of spatial coherence degradation in thermal sources?

11. What does the condition $|\gamma_{12}| = 0$ signify for an optical source?

12. Which of the following correctly distinguishes spatial coherence from temporal coherence?

13. Why does an ordinary incandescent bulb possess very low spatial coherence?

14. What is the unit of the complex degree of spatial coherence $\gamma_{12}$?

15. For observable interference fringes ($V > 0$), what is the condition on maximum slit separation $d_{max}$?

PYQ Patterns

1. If a quasi-monochromatic source of diameter $s = 2\text{ mm}$ is at a distance $R = 1\text{ m}$ emitting light of wavelength $\lambda = 500\text{ nm}$, what is the approximate transverse coherence length $l_c$?

2. In a double-slit experiment, the maximum intensity is measured as $9I_0$ and the minimum intensity as $I_0$. What is the fringe visibility $V$?

3. What is the physical role of a spatial filter consisting of a microscope objective and a pinhole in a laser beam path?

4. According to the Van Cittert-Zernike theorem, if the source intensity distribution is uniform and circular, the spatial coherence radius $r_c$ is proportional to:

5. Why is it difficult to observe interference fringes using sunlight directly without a preliminary pinhole or slit?

6. If the separation between two slits in a Young's experiment exceeds the transverse coherence length of the source ($d > l_c$), what happens to the interference pattern?

Timed Topic Test

1. Which of the following statements about spatial coherence is correct?

2. Calculate the transverse coherence length for light of wavelength $\lambda = 600\text{ nm}$ from a star of angular diameter subtending $10^{-5}\text{ radians}$ at Earth.

3. In partial coherence theory, the mutual coherence function $\Gamma_{12}(\tau)$ is defined as:

4. What is the relationship between fringe visibility $V$ and the complex degree of coherence magnitude $|\gamma_{12}|$?

5. If a source size $s$ is doubled while keeping $R$ and $\lambda$ constant, what happens to the transverse coherence length $l_c$?

6. Which source exhibits the lowest spatial coherence among the following?

7. The normalized mutual coherence function $\gamma_{12}(\tau)$ is expressed as:

8. In Young's double-slit experiment, if $I_{max} = 4\text{ units}$ and $I_{min} = 0\text{ units}$, the fringe visibility is:

9. What is the primary consequence of the Van Cittert-Zernike theorem in optical imaging and astronomy?

10. Under what condition is the small angle approximation used in deriving transverse coherence length valid?

In a Young's double-slit experiment, the intensity maxima and minima in the interference pattern are measured to be 9 units and 1 unit respectively. Calculate the magnitude of the complex degree of spatial coherence (|γ₁₂|) between the two slits, assuming equal beam intensities.

A circular incoherent source of diameter 2 cm is located at a distance of 5 m from an observation plane. Using light of wavelength 500 nm, find the transverse spatial coherence radius r_c.