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🔬Modern Optics

🔬modern optics review

5.3 Partial coherence and its effects on interference

4 min readLast Updated on July 22, 2024

Light waves aren't always in perfect sync. Partial coherence measures how well they line up in space and time. This affects how light behaves in interferometers and other optical systems.

Understanding partial coherence is key for many applications. It impacts fringe visibility in interferometers, resolution in imaging systems, and even how we measure distant stars. Knowing these effects helps us design better optical devices.

Partial Coherence and Interference

Partial coherence and light properties

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  • Partial coherence quantifies correlation between phases of light waves at different points in space and time
    • Light sources exhibit varying partial coherence from completely incoherent to fully coherent
  • Temporal coherence relates to spectral purity and monochromaticity of light source
    • Longer coherence time implies more monochromatic source and higher temporal coherence (lasers)
  • Spatial coherence depends on light source size and distance between considered points
    • Smaller source size and larger distances lead to higher spatial coherence (stars, distant objects)
  • Degree of partial coherence affects light waves' ability to interfere constructively or destructively (interference patterns, fringe visibility)

Effects on interference fringe visibility

  • Partial coherence reduces visibility and contrast of interference fringes in interferometers
    • Visibility defined as V=(ImaxImin)/(Imax+Imin)V = (I_{max} - I_{min}) / (I_{max} + I_{min}), where ImaxI_{max} and IminI_{min} are maximum and minimum fringe intensities
    • Lower coherence results in lower visibility and less distinct fringes (blurred or washed out patterns)
  • In Michelson interferometers, partial coherence affects fringe sharpness and contrast
    • Reduced temporal coherence decreases fringe contrast as path difference between interferometer arms increases (white light interferometry)
  • Fabry-Perot interferometers sensitive to both temporal and spatial coherence
    • Partial coherence reduces finesse and sharpness of transmission peaks (broadened peaks, reduced resolution)
  • Partial coherence also affects fringe formation in other interferometric systems (Mach-Zehnder, Sagnac interferometers)

Analysis using coherence functions

  • Mutual coherence function Γ(r1,r2,τ)\Gamma(r_1, r_2, \tau) describes correlation between electric fields at two points r1r_1 and r2r_2 with time delay τ\tau
    • Defined as Γ(r1,r2,τ)=E(r1,t)E(r2,t+τ)\Gamma(r_1, r_2, \tau) = \langle E^*(r_1, t) E(r_2, t+\tau) \rangle, where E(r,t)E(r, t) is electric field at position rr and time tt
  • Complex degree of coherence γ(r1,r2,τ)\gamma(r_1, r_2, \tau) is normalized mutual coherence function
    • Given by γ(r1,r2,τ)=Γ(r1,r2,τ)/Γ(r1,r1,0)Γ(r2,r2,0)\gamma(r_1, r_2, \tau) = \Gamma(r_1, r_2, \tau) / \sqrt{\Gamma(r_1, r_1, 0) \Gamma(r_2, r_2, 0)}
    • γ(r1,r2,τ)|\gamma(r_1, r_2, \tau)| ranges from 0 (incoherent) to 1 (fully coherent)
  • Complex degree of coherence quantifies temporal and spatial coherence properties of light sources
    • Temporal coherence characterized by complex degree of self-coherence γ(τ)=γ(r,r,τ)\gamma(\tau) = \gamma(r, r, \tau) (autocorrelation)
    • Spatial coherence characterized by complex degree of mutual coherence γ(r1,r2,0)\gamma(r_1, r_2, 0) (cross-correlation)

Applications in interferometric systems

  • In interferometric system design, light source coherence properties must be considered for desired performance
    • High-precision measurements often use highly coherent sources (lasers)
    • Applications requiring reduced coherent noise or speckle may prefer partially coherent sources (LEDs, superluminescent diodes)
  • Path difference between interferometer arms should be within source coherence length to maintain high fringe visibility
    • Coherence length related to source spectral bandwidth by Lc=c/ΔfL_c = c / \Delta f, where cc is speed of light and Δf\Delta f is spectral bandwidth
  • White-light interferometry exploits short coherence length of broadband sources for high axial resolution
    • Sample scanned through zero path difference position to obtain depth information (optical coherence tomography)
  • Partial coherence can suppress unwanted interference effects (multiple reflections, coherent noise)
    • Using partially coherent sources or introducing controlled path differences reduces coherent noise (speckle reduction, ghost imaging)

Coherence Functions and Their Applications

Analysis using coherence functions

  • Mutual coherence function and complex degree of coherence can be measured experimentally
    • Young's double-slit experiment measures spatial coherence by observing fringe visibility as function of slit separation (star size estimation)
    • Michelson or Mach-Zehnder interferometers measure temporal coherence by varying path difference between arms (coherence length measurement)
  • Wiener-Khinchin theorem relates mutual coherence function to power spectral density of light source
    • Power spectral density S(ω)S(\omega) is Fourier transform of self-coherence function Γ(τ)\Gamma(\tau)
    • Relationship allows coherence properties determination from spectral measurements (spectroscopy)
  • Van Cittert-Zernike theorem describes spatial coherence properties of extended incoherent source
    • Complex degree of coherence between two points given by Fourier transform of source intensity distribution
    • Theorem useful for understanding coherence properties of extended sources (stars, illuminated objects)

Applications in interferometric systems

  • In astronomical interferometry, partial coherence exploited for high angular resolution
    • Multiple telescopes combined to form interferometric array, increasing spatial coherence and resolving power (Very Large Telescope Interferometer)
    • Complex degree of coherence between telescope pairs determines fringe visibility and achievable resolution (stellar diameter measurement)
  • Partial coherence relevant in optical coherence tomography (OCT), a biomedical imaging technique
    • OCT uses low-coherence interferometry for depth-resolved images of biological tissues (retina, skin)
    • Short coherence length of source enables high axial resolution, while lateral resolution determined by focusing optics
  • In holography, light source coherence properties affect quality and sharpness of recorded holograms
    • Highly coherent sources (lasers) typically used for high-quality holograms with high diffraction efficiency
    • Partially coherent sources can reduce speckle noise and improve subjective quality of reconstructed images (reduced granularity, smoother appearance)


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.