The van Cittert–Zernike theorem, named after physicists Pieter Hendrik van Cittert and Frits Zernike, is a formula in coherence theory that states that under certain conditions the Fourier transform of the intensity distribution function of a distant, incoherent source is equal to its complex visibility. This implies that the wavefront from an incoherent source will appear mostly coherent at large distances. Intuitively, this can be understood by considering the wavefronts created by two incoherent sources. If we measure the wavefront immediately in front of one of the sources, our measurement will be dominated by the nearby source. If we make the same measurement far from the sources, our measurement will no longer be dominated by a single source; both sources will contribute almost equally to the wavefront at large distances.
This reasoning can be easily visualized by dropping two stones in the center of a calm pond. Near the center of the pond, the disturbance created by the two stones will be very complicated. As the disturbance propagates towards the edge of the pond, however, the waves will smooth out and will appear to be nearly circular. The van Cittert–Zernike theorem has important implications for radio astronomy. With the exception of pulsars and masers, all astronomical sources are spatially incoherent. Nevertheless, because they are observed at distances large enough to satisfy the van Cittert–Zernike theorem, these objects exhibit a non-zero degree of coherence at different points in the imaging plane. By measuring the degree of coherence at different points in the imaging plane (the so-called "visibility function") of an astronomical object, a radio astronomer can thereby reconstruct the source's brightness distribution and make a two-dimensional map of the source's appearance.
Statement of the theorem Consider two very distant parallel planes, both perpendicular to the line of sight, and let's call them source plane and observation plane; If Γ 12 ( u , v , 0 ) {\displaystyle \Gamma _{12}(u,v,0)} is the mutual coherence function between two points in the observation plane, then
Γ 12 ( u , v , 0 ) = ∬ I ( l , m ) e − 2 π i ( u l + v m ) d l d m {\displaystyle \Gamma _{12}(u,v,0)=\iint I(l,m)e^{-2\pi i(ul+vm)}\,dl\,dm}
where l {\displaystyle l} and m {\displaystyle m} are the direction cosines of a point on a distant source in the source plane, u {\displaystyle u} and v {\displaystyle v} are respectively the x-distance and the y-distance between the two observation points on the observation plane in unit of wavelength and I {\displaystyle I} is the intensity of the source. This theorem was first derived by Pieter Hendrik van Cittert in 1934 with a simpler proof provided by Frits Zernike in 1938. This theorem will remain confusing to some engineers or scientists because of its statistical nature and difference from simple correlation or even covariance processing methods. A good reference (which still might not clarify the issue for some users, but does have a great sketch to drive the method home) is Goodman, starting on page 207.
The mutual coherence function The mutual coherence function for some electric field E ( t ) {\displaystyle E(t)} measured at two points in a plane of observation (call them 1 and 2), is defined to be
Γ 12 ( τ ) = lim T → ∞ 1 2 T ∫ − T T E 1 ( t ) E 2 ∗ ( t − τ ) d t {\displaystyle \Gamma _{12}(\tau )=\lim _{T\to \infty }{\frac {1}{2T}}\int _{-T}^{T}E_{1}(t)E_{2}^{*}(t-\tau )dt}
where τ {\displaystyle \tau } is the time offset between the measurement of E ( t ) {\displaystyle E(t)} at observation points 1 and 2. The mutual coherence of the field at two points may be thought of as the time-averaged cross-correlation between the electric fields at the two points separated in time by τ {\displaystyle \tau } . Thus, if we are observing two fully incoherent sources we should expect the mutual coherence function to be relatively small between the two random points in the observation plane, because the sources will interfere destructively as well as constructively. Far away from the sources, however, we should expect the mutual coherence function to be relatively large because the sum of the observed fields will be almost the same at any two points. Normalization of the mutual coherence function to the product of the square roots of the intensities of the two electric fields yields the complex degree of (second-order) coherence (correlation coefficient function):
γ 12 ( τ ) = Γ 12 ( τ ) I 1 I 2 {\displaystyle \gamma _{12}(\tau )={\frac {\Gamma _{12}(\tau )}{{\sqrt {I_{1}}}{\sqrt {I_{2}}}}}}
Proof of the theorem Let X Y {\displaystyle XY} and x y {\displaystyle xy} be respectively the cartesian coordinates of the source plane and the observation plane. Suppose the electric field due to some point from the source in the source plane is measured at two points, P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} , in the observation plane. The position of a point in the source may be referred to by its direction cosines ( l , m ) {\displaystyle (l,m)} . (Since the source is distant, its direction should be the same at P 1 {\displaystyle P_{1}} as at P 2 {\displaystyle P_{2}} .) The electric field measured at P 1 {\displaystyle P_{1}} can then be written using phasors:
E 1 ( l , m , t ) = A ( l , m , t − R 1 c ) e − i ω ( t − R 1 c ) R 1 {\displaystyle E_{1}(l,m,t)=A\left(l,m,t-{\frac {R_{1}}{c}}\right){\frac {e^{-i\omega \left(t-{\frac {R_{1}}{c}}\right)}}{R_{1}}}}
where R 1 {\displaystyle R_{1}} is the distance from the source to P 1 {\displaystyle P_{1}} , ω {\displaystyle \omega } is the angular frequency of the light, and A {\displaystyle A} is the complex amplitude of the electric field. Similarly, the electric field measured at P 2 {\displaystyle P_{2}} can be written as
E 2 ( l , m , t ) = A ( l , m , t − R 2 c ) e − i ω ( t − R 2 c ) R 2 {\displaystyle E_{2}(l,m,t)=A\left(l,m,t-{\frac {R_{2}}{c}}\right){\frac {e^{-i\omega \left(t-{\frac {R_{2}}{c}}\right)}}{R_{2}}}}
Let us now calculate the time-averaged cross-correlation between the electric field at P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} :
⟨ E 1 ( l , m , t ) E 2 ∗ ( l , m , t ) ⟩ = ⟨ A ( l , m , t − R 1 c ) A ∗ ( l , m , t − R 2 c ) ⟩ × e i ω R 1 c R 1 × e − i ω R 2 c R 2 {\displaystyle {\big \langle }E_{1}(l,m,t)E_{2}^{*}(l,m,t){\big \rangle }={\Bigg \langle }A\left(l,m,t-{\frac {R_{1}}{c}}\right)A^{*}\left(l,m,t-{\frac {R_{2}}{c}}\right){\Bigg \rangle }\times {\frac {e^{i\omega {\frac {R_{1}}{c}}}}{R_{1}}}\times {\frac {e^{-i\omega {\frac {R_{2}}{c}}}}{R_{2}}}}
Because the quantity in the angle brackets is time-averaged an arbitrary offset to the temporal term of the amplitudes may be added as long as the same offset is added to both. Let us now add R 1 c {\displaystyle {\frac {R_{1}}{c}}} to the temporal term of both amplitudes. The time-averaged cross-correlation of the electric field at the two points therefore simplifies to
⟨ E 1 ( l , m , t ) E 2 ∗ ( l , m , t ) ⟩ = ⟨ A ( l , m , t ) A ∗ ( l , m , t − R 2 − R 1 c ) ⟩ × e i ω ( R 1 − R 2 c ) R 1 R 2 {\displaystyle {\big \langle }E_{1}(l,m,t)E_{2}^{*}(l,m,t){\big \rangle }={\Bigg \langle }A(l,m,t)A^{*}\left(l,m,t-{\frac {R_{2}-R_{1}}{c}}\right){\Bigg \rangle }\times {\frac {e^{i\omega \left({\frac {R_{1}-R_{2}}{c}}\right)}}{R_{1}R_{2}}}}
But if the source is in the far field then the difference between R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} will be small compared to the distance light travels in time t {\displaystyle t} . ( t {\displaystyle t} is on the same order as the inverse bandwidth.) This small correction can therefore be neglected, further simplifying our expression for the cross-correlation of the electric field at P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} to
⟨ E 1 ( l , m , t ) E 2 ∗ ( l , m , t ) ⟩ = ⟨ A ( l , m , t ) A ∗ ( l , m , t ) ⟩ × e i ω ( R 1 − R 2 c ) R 1 R 2 {\displaystyle \langle E_{1}(l,m,t)E_{2}^{*}(l,m,t)\rangle =\langle A(l,m,t)A^{*}(l,m,t)\rangle \times {\frac {e^{i\omega \left({\frac {R_{1}-R_{2}}{c}}\right)}}{R_{1}R_{2}}}}
Now, ⟨ A ( l , m , t ) A ∗ ( l , m , t ) ⟩ {\displaystyle \langle A(l,m,t)A^{*}(l,m,t)\rangle } is simply the intensity of the source at a particular point, I ( l , m ) {\displaystyle I(l,m)} . So our expression for the cross-correlation simplifies further to
⟨ E 1 ( l , m , t ) E 2 ∗ ( l , m , t ) ⟩ = I ( l , m ) e i ω ( R 1 − R 2 c ) R 1 R 2 {\displaystyle \langle E_{1}(l,m,t)E_{2}^{*}(l,m,t)\rangle =I(l,m){\frac {e^{i\omega \left({\frac {R_{1}-R_{2}}{c}}\right)}}{R_{1}R_{2}}}}
To calculate the mutual coherence function from this expression, simply integrate over the entire source.
Γ 12 ( u , v , 0 ) = ∬ source I ( l , m ) e i ω ( R 1 − R 2 c ) R 1 R 2 d S {\displaystyle \Gamma _{12}(u,v,0)=\iint _{\textrm {source}}I(l,m){\frac {e^{i\omega \left({\frac {R_{1}-R_{2}}{c}}\right)}}{R_{1}R_{2}}}\,dS}
Note that cross terms of the form ⟨ A 1 ( l , m , t ) A 2 ∗ ( l , m , t ) ⟩ {\displaystyle \langle A_{1}(l,m,t)A_{2}^{*}(l,m,t)\rangle } are not included due to the assumption that the source is incoherent. The time-averaged correlation between two different points from the source will therefore be zero. Next rewrite the R 2 − R 1 {\displaystyle R_{2}-R_{1}} term using u , v , l {\displaystyle u,v,l} and m {\displaystyle m} . To do this, let P 1 = ( x 1 , y 1 ) {\displaystyle P_{1}=(x_{1},y_{1})} and P 2 = ( x 2 , y 2 ) {\displaystyle P_{2}=(x_{2},y_{2})} . This gives
R 1 = R 2 + x 1 2 + y 1 2 {\displaystyle R_{1}={\sqrt {R^{2}+x_{1}^{2}+y_{1}^{2}}}\,}
R 2 = R 2 + x 2 2 + y 2 2 {\displaystyle R_{2}={\sqrt {R^{2}+x_{2}^{2}+y_{2}^{2}}}\,}
where R {\displaystyle R} is the distance between the center of the plane of observation and the center of the source. The difference between R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} thus becomes
R 2 − R 1 = R 1 + x 2 2 R 2 + y 2 2 R 2 − R 1 + x 1 2 R 2 + y 1 2 R 2 {\displaystyle R_{2}-R_{1}=R{\sqrt {1+{\frac {x_{2}^{2}}{R^{2}}}+{\frac {y_{2}^{2}}{R^{2}}}}}-R{\sqrt {1+{\frac {x_{1}^{2}}{R^{2}}}+{\frac {y_{1}^{2}}{R^{2}}}}}}
But because x 1 , x 2 , y 1 {\displaystyle x_{1},x_{2},y_{1}} and y 2 {\displaystyle y_{2}} are all much less than R {\displaystyle R} , the square roots may be Taylor expanded, yielding, to first order,
R 2 − R 1 = R ( 1 + 1 2 ( x 2 2 + y 2 2 R 2 ) ) − R ( 1 + 1 2 ( x 1 2 + y 1 2 R 2 ) ) {\displaystyle R_{2}-R_{1}=R\left(1+{\frac {1}{2}}\left({\frac {x_{2}^{2}+y_{2}^{2}}{R^{2}}}\right)\right)-R\left(1+{\frac {1}{2}}\left({\frac {x_{1}^{2}+y_{1}^{2}}{R^{2}}}\right)\right)}
which, after some algebraic manipulation, simplifies to
R 2 − R 1 = 1 2 R ( ( x 2 − x 1 ) ( x 2 + x 1 ) + ( y 2 − y 1 ) ( y 2 + y 1 ) ) {\displaystyle R_{2}-R_{1}={\frac {1}{2R}}\left((x_{2}-x_{1})(x_{2}+x_{1})+(y_{2}-y_{1})(y_{2}+y_{1})\right)}
Now, 1 2 ( x 2 + x 1 ) {\displaystyle {\frac {1}{2}}(x_{2}+x_{1})} is the midpoint along the x {\displaystyle x} -axis between P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} , so 1 2 R ( x 2 + x 1 ) {\displaystyle {\frac {1}{2R}}(x_{2}+x_{1})} gives us l {\displaystyle l} , one of the direction cosines to the sources. Similarly, m = 1 2 R ( y 2 + y 1 ) {\displaystyle m={\frac {1}{2R}}(y_{2}+y_{1})} . Moreover, recall that u {\displaystyle u} was defined to be the number of wavelengths along the x {\displaystyle x} -axis between P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} . So
u = ω 2 π c ( x 1 − x 2 ) {\displaystyle u={\frac {\omega }{2\pi c}}(x_{1}-x_{2})}
Similarly, v {\displaystyle v} is the number of wavelengths between P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} along the y {\displaystyle y} -axis, so
v = ω 2 π c ( y 1 − y 2 ) {\displaystyle v={\frac {\omega }{2\pi c}}(y_{1}-y_{2})}
Hence
R 2 − R 1 = 2 π c ω ( u l + v m ) {\displaystyle R_{2}-R_{1}={\frac {2\pi c}{\omega }}(ul+vm)}
Because x 1 , x 2 , y 1 , {\displaystyle x_{1},x_{2},y_{1},} and y 2 {\displaystyle y_{2}} are all much less than R {\displaystyle R} , R 1 ≃ R 2 ≃ R {\displaystyle R_{1}\simeq R_{2}\simeq R} . The differential area element, d S {\displaystyle dS} , may then be written as a differential element of solid angle of R 2 d l d m {\displaystyle R^{2}\,dl\,dm} . Our expression for the mutual coherence function becomes
Γ 12 ( u , v , 0 ) = ∬ source I ( l , m ) e − i ω c 2 π c ω ( u l + v m ) d l d m {\displaystyle \Gamma _{12}(u,v,0)=\iint _{\textrm {source}}I(l,m)e^{-{\frac {i\omega }{c}}{\frac {2\pi c}{\omega }}(ul+vm)}\,dl\,dm}
Which reduces to
Γ 12 ( u , v , 0 ) = ∬ source I ( l , m ) e − 2