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🔬Modern Optics

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5.4 Van Cittert-Zernike theorem

3 min readLast Updated on July 22, 2024

The Van Cittert-Zernike theorem is a key concept in understanding spatial coherence. It links a light source's intensity distribution to its far-field coherence properties, enabling the design of better optical systems like telescopes and microscopes.

This theorem has wide-ranging applications, from analyzing circular apertures to rectangular slits. It helps engineers optimize optical systems by considering source properties and their impact on spatial coherence, ultimately improving resolution and contrast in imaging devices.

Spatial Coherence and the Van Cittert-Zernike Theorem

Van Cittert-Zernike theorem significance

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  • States complex degree of coherence between two far-field points of quasi-monochromatic, incoherent source equals normalized Fourier transform of source's intensity distribution
    • Mathematical expression: γ(x1,x2)=I(x)eik(x1x2)x/zdxI(x)dx\gamma(x_1, x_2) = \frac{\int I(x) e^{-i k (x_1 - x_2) x / z} dx}{\int I(x) dx}
      • γ(x1,x2)\gamma(x_1, x_2) represents complex degree of coherence between points x1x_1 and x2x_2
      • I(x)I(x) represents source's intensity distribution
      • k=2π/λk = 2\pi/\lambda represents wavenumber
      • zz represents distance from source to observation plane
  • Relates spatial coherence properties of light source to its angular intensity distribution
    • Spatial coherence describes correlation between fields at different spatial points
    • Angular intensity distribution describes source's intensity as function of angle
  • Allows determination of source's spatial coherence properties from its intensity distribution
  • Provides foundation for understanding coherence properties of partially coherent sources
  • Enables design of coherent optical systems (telescopes, microscopes) by considering source's spatial coherence

Applications of Van Cittert-Zernike theorem

  • Circular aperture:
    • Intensity distribution: I(r)={I0,ra0,r>aI(r) = \begin{cases} I_0, & r \leq a \\ 0, & r > a \end{cases}
      • aa represents aperture radius
    • Complex degree of coherence: γ(x1,x2)=2J1(ka(x1x2)/z)ka(x1x2)/z\gamma(x_1, x_2) = \frac{2 J_1(k a (x_1 - x_2) / z)}{k a (x_1 - x_2) / z}
      • J1J_1 represents first-order Bessel function of first kind
  • Rectangular slit:
    • Intensity distribution: I(x)={I0,xa0,x>aI(x) = \begin{cases} I_0, & |x| \leq a \\ 0, & |x| > a \end{cases}
      • 2a2a represents slit width
    • Complex degree of coherence: γ(x1,x2)=sinc(ka(x1x2)z)\gamma(x_1, x_2) = \text{sinc}\left(\frac{k a (x_1 - x_2)}{z}\right)
      • sinc(x)=sin(πx)πx\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}

Implications for optical systems

  • Coherent optical systems (telescopes, microscopes) rely on light source's spatial coherence for optimal performance
    • High spatial coherence improves resolution and contrast
    • Low spatial coherence reduces resolution and contrast
  • Allows designers to determine required source properties for desired spatial coherence level
    • Larger sources with uniform intensity distributions produce lower spatial coherence
    • Smaller sources with non-uniform intensity distributions produce higher spatial coherence
  • Telescopes: larger apertures and more uniform primary mirror illumination improve spatial coherence and resolution
  • Microscopes: smaller, more uniform illumination sources (lasers, high-quality LEDs) provide better spatial coherence and imaging performance

Analysis of partially coherent sources

  • Generalized Van Cittert-Zernike theorem extends original theorem to partially coherent sources
    • Describes propagation of mutual intensity function, J(x1,x2)J(x_1, x_2), from source plane to observation plane
    • Mathematical expression: J(x1,x2)=1λ2z2Js(ξ1,ξ2)eik[(x1ξ1)2(x2ξ2)2]/2zdξ1dξ2J(x_1, x_2) = \frac{1}{\lambda^2 z^2} \iint J_s(\xi_1, \xi_2) e^{-i k [(x_1 - \xi_1)^2 - (x_2 - \xi_2)^2] / 2z} d\xi_1 d\xi_2
      • Js(ξ1,ξ2)J_s(\xi_1, \xi_2) represents mutual intensity function in source plane
      • ξ1\xi_1 and ξ2\xi_2 represent coordinates in source plane
  • Mutual intensity function characterizes spatial coherence properties of partially coherent source
    • Defined as cross-correlation of fields at two points: J(x1,x2)=E(x1)E(x2)J(x_1, x_2) = \langle E^*(x_1) E(x_2) \rangle
    • Related to complex degree of coherence by: γ(x1,x2)=J(x1,x2)I(x1)I(x2)\gamma(x_1, x_2) = \frac{J(x_1, x_2)}{\sqrt{I(x_1) I(x_2)}}
  • Analyzing mutual intensity function allows complete description of partially coherent source's spatial coherence properties
    • Diagonal elements, J(x,x)J(x, x), represent intensity distribution
    • Off-diagonal elements, J(x1,x2)J(x_1, x_2), represent correlation between fields at different points


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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.