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 ( x 1 , x 2 ) = ∫ I ( x ) e − i k ( x 1 − x 2 ) x / z d x ∫ I ( x ) d x \gamma(x_1, x_2) = \frac{\int I(x) e^{-i k (x_1 - x_2) x / z} dx}{\int I(x) dx} γ ( x 1 , x 2 ) = ∫ I ( x ) d x ∫ I ( x ) e − ik ( x 1 − x 2 ) x / z d x
γ ( x 1 , x 2 ) \gamma(x_1, x_2) γ ( x 1 , x 2 ) represents complex degree of coherence between points x 1 x_1 x 1 and x 2 x_2 x 2
I ( x ) I(x) I ( x ) represents source's intensity distribution
k = 2 π / λ k = 2\pi/\lambda k = 2 π / λ represents wavenumber
z z z 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 ) = { I 0 , r ≤ a 0 , r > a I(r) = \begin{cases} I_0, & r \leq a \\ 0, & r > a \end{cases} I ( r ) = { I 0 , 0 , r ≤ a r > a
a a a represents aperture radius
Complex degree of coherence: γ ( x 1 , x 2 ) = 2 J 1 ( k a ( x 1 − x 2 ) / z ) k a ( x 1 − x 2 ) / z \gamma(x_1, x_2) = \frac{2 J_1(k a (x_1 - x_2) / z)}{k a (x_1 - x_2) / z} γ ( x 1 , x 2 ) = ka ( x 1 − x 2 ) / z 2 J 1 ( ka ( x 1 − x 2 ) / z )
J 1 J_1 J 1 represents first-order Bessel function of first kind
Rectangular slit:
Intensity distribution: I ( x ) = { I 0 , ∣ x ∣ ≤ a 0 , ∣ x ∣ > a I(x) = \begin{cases} I_0, & |x| \leq a \\ 0, & |x| > a \end{cases} I ( x ) = { I 0 , 0 , ∣ x ∣ ≤ a ∣ x ∣ > a
2 a 2a 2 a represents slit width
Complex degree of coherence: γ ( x 1 , x 2 ) = sinc ( k a ( x 1 − x 2 ) z ) \gamma(x_1, x_2) = \text{sinc}\left(\frac{k a (x_1 - x_2)}{z}\right) γ ( x 1 , x 2 ) = sinc ( z ka ( x 1 − x 2 ) )
sinc ( x ) = sin ( π x ) π x \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} sinc ( x ) = π x s i n ( π 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 ( x 1 , x 2 ) J(x_1, x_2) J ( x 1 , x 2 ) , from source plane to observation plane
Mathematical expression: J ( x 1 , x 2 ) = 1 λ 2 z 2 ∬ J s ( ξ 1 , ξ 2 ) e − i k [ ( x 1 − ξ 1 ) 2 − ( x 2 − ξ 2 ) 2 ] / 2 z d ξ 1 d ξ 2 J(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 J ( x 1 , x 2 ) = λ 2 z 2 1 ∬ J s ( ξ 1 , ξ 2 ) e − ik [( x 1 − ξ 1 ) 2 − ( x 2 − ξ 2 ) 2 ] /2 z d ξ 1 d ξ 2
J s ( ξ 1 , ξ 2 ) J_s(\xi_1, \xi_2) J s ( ξ 1 , ξ 2 ) represents mutual intensity function in source plane
ξ 1 \xi_1 ξ 1 and ξ 2 \xi_2 ξ 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 ( x 1 , x 2 ) = ⟨ E ∗ ( x 1 ) E ( x 2 ) ⟩ J(x_1, x_2) = \langle E^*(x_1) E(x_2) \rangle J ( x 1 , x 2 ) = ⟨ E ∗ ( x 1 ) E ( x 2 )⟩
Related to complex degree of coherence by: γ ( x 1 , x 2 ) = J ( x 1 , x 2 ) I ( x 1 ) I ( x 2 ) \gamma(x_1, x_2) = \frac{J(x_1, x_2)}{\sqrt{I(x_1) I(x_2)}} γ ( x 1 , x 2 ) = I ( x 1 ) I ( x 2 ) J ( x 1 , 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) J ( x , x ) , represent intensity distribution
Off-diagonal elements, J ( x 1 , x 2 ) J(x_1, x_2) J ( x 1 , x 2 ) , represent correlation between fields at different points