TITULO XI La disciplina deportiva
DISPOSICIONES TRANSITORIAS Primera
A monochromatic electric field can be represented by
E(r, t) =<
n
Ae−i(kr−kct) o
, (1.6)
whereAis the amplitude of the wave andk = 2λπ. By taking the real part of the electric field in one dimension, this may be written as:
herexis the optical path length andφis the phase of the wavefront. A detector measures the time averaged modulus for N beams,
S(k) = 1 T Z T 0 N X i=1 E(t) 2 dt (1.8)
InsertingEquation 1.7intoEquation 1.8and assuming two-beam combination this becomes
S(k) = 1 T Z T 0 A1cos (kx1−kct+φ1) +A2cos (kx2−kct+φ2) 2 dt, (1.9)
and expanding this yields
S(k) =1 T Z T 0 A21cos2(kx1−kct+φ1)+ 2A1A2cos (kx1−kct+φ1) cos (kx2−kct+φ2)+ A22cos2(kx2 −kct+φ2) dt. (1.10)
A basic trigonometric relation derived from the unit circle,
cos (β−α) = cosαcosβ+ sinαsinβ, allows this the cross term to be rewritten
cos (kx1−kct+φ1) cos (kx2 −kct+φ2) =
cos (kx1+φ1) cos (kct) + sin (kx1 +φ1) sin (kct)
cos (kx2+φ2) cos (kct) + sin (kx2 +φ2) sin (kct) .
(1.11)
Plugging this intoEquation 1.10and expanding gives
S(k) =1 T Z T 0 A21cos2(kx1−kct+φ1)+
2A1A2 cos (kx1+φ1) cos (kct) + sin (kx1 +φ1) sin (kct)
cos (kx2+φ2) cos (kct) + sin (kx2+φ2) sin (kct)
+
A22cos2(kx2−kct+φ2)
dt.
The integrals: 21πR02πcos2xdx= 21πR2π 0 sin 2xdx= 1 2 and 1 2π R2π
0 cosxsinxdx= 0allow us to greatly simplify the equation.
S(k) =1 2A 2 1 + 1 2A 2 2+ A1A2
cos (kx1+φ1) cos (kx2+φ2) + sin (kx1+φ1) sin (kx2+φ2)
(1.13)
SubstitutingI =A2 and condensing the trigonometry gives
S(k) = 1 2I1+ 1 2I2+ p I1I2cos (k(x1−x2) + (φ1−φ2)). (1.14)
The visibility amplitudes are normalized so that the results are between 0 and 1 and the signal is divided by the mean intensity.
S(k) = 1 + 2
√
I1I2
I1+I2
cos (k(x1−x2) + (φ1−φ2)). (1.15)
Equation 1.15is known as the monochromatic fringe equation and introduces the transfer function, T = 2 √ I1I2 I1+I2 . (1.16)
The full N-beam derivation of this is found inten Brummelaar (2014).
In reality, light from a physical source is never purely monochromatic. Light from a real source has some bandwidth,∆λ, and its fringes are finite and only exist near the zero Optical Path Difference (OPD) position.
To describe the wave field produced by a polychromatic source it is helpful to introduce the concept of correlation and coherence. In interferometry, we are essentially exploiting the nature of how light interferes with itself and then measuring the cross-correlation. The
cross-correlation of two signalsE1 andE2 is defined as
Γ(τ) =
Z +∞
−∞
E1(t)×E2∗(t+τ) dt, (1.17)
whereE2∗ is the complex conjugate of the electric field.
Born & Wolf(1999) present a full derivation of discussion of correlation functions of light beams. For our purposes here, we will introduce the mutual coherence function for our detected electric field, which is a specific form of the cross-correlation function,
Γ12(τ) =
S1(t+τ)S2∗(t)
. (1.18)
The mutual coherence function can be normalized to the autocorrelation functions, yielding the complex degree of coherence,
γ12(τ) = Γ√12(τ)
I1I2
. (1.19)
From this we can obtain the general interference formulation,
I =I1+I2+ 2pI1I2γ12ek(s2−s1), (1.20)
where s is the path length. Taking only the real part of the complex degree of coherence, this can be written as
I =I1+I2+ 2
p
I1I2|γ12|cos (2πσ∆x+ Φ12), (1.21)
whereσis the wavenumber andσ = 1/λ=k/2π. If0< γ12<1then the waves are partially coherent. It is this degree of coherence, along with the phase term, which we measure with
an interferometer. Bringing this together withEquation 1.15andEquation 1.16, gives us
S(k) = 1 +T12|γ12|cos [2πσ∆x+ ∆φ+ Φ12]. (1.22)
Because light is a wave and is capable of exhibiting interference, it must then possess the property of coherence. Waves are perfectly coherent if they have the same frequency and phase. The degree to which two waves, or portions of the same wave, match can be
measured by its degree of correlation. For a partially coherent source, the cross-correlation is between 0 and 1. A wave may maintain a certain degree of coherence over a distance of its propagation. This is the coherence length, L,
L= λ
2 0
∆λ =R×λ0, (1.23)
whereR = ∆λλ is the spectral resolving power,λ0 is the central wavelength, and∆λis the bandwidth. This is related to the coherence time by
tc=
L
c, (1.24)
where c is the speed of light. A polychromatic wave, thanks to Fourier, can be expanded as a sum of monochromatic waves.
E(r, t) =
Z ∞
0
A(λ, t)e−i2π(λr−ctλ) dλ. (1.25)
If we restrict the possible range of wavelengths toλ0±∆λ/2, or in terms of wave number
σ0±∆σ/2, where wavenumber isσ= 2π/λ, we develop the quasi-monochromatic case with a finite bandwidth. Let’s assume the bandpass is a top-hat function centered on the central
wavelength. IntegratingEquation 1.22across the bandwidth gives S(k) = 1 ∆σ Z σ0+∆2σ σ0−∆2σ 1 +T12|γ12|cos [2πσ∆x+ ∆φ+ Φ12] dσ. (1.26)
Evaluating the integral leads to
S(k) = 1 ∆σ σ| σ0+∆2σ σ0−∆2σ T12|γ12| sin 2πσ∆x+ ∆φ+ Φ12 2π∆x σ0+∆2σ σ0−∆2σ (1.27)
and using the trigonometric identitysin (a+b)−sin (a−b) =
sin (a) cos (b) + sin (b) cos (a)−sin (a) cos (b) + sin (b) cos (a) = 2 sin (b) cos (a)with
a= 2πσ0∆x+ ∆φ+ Φ12andb =π∆σ∆xto evaluate the limit conditions results in
S(k) = 1 +T12|γ12|
sin (π∆σ∆x) cos (2πσ0∆x+ ∆φ+ Φ12)
π∆σ∆x (1.28)
which can be further simplified to
S(k) = 1 +T12|γ12|sinc (π∆σ∆x) cos (2πσ0∆x+ ∆φ+ Φ12) (1.29)
This is the fringe equation for the quasi-monochromatic case. The fringe envelope is the Fourier transform of the optical bandpass. In this case, the bandpass was a top-hat function and the resulting envelope is a sinc function. As one restricts the bandwidth, the
monochromatic case is approached and the fringe envelope, or packet, widens; on the contrary, with a broad bandwidth, a narrower fringe packet is produced (SeeFigure 1.2).
Today’s work in interferometry relies upon the concepts of the coherence function developed by van Cittert in 1934 (van Cittert 1934) and Zernike in 1938 (Zernike 1938). Recalling Young’s double-slit experiment or any setup where light from a distant source passes through
a mask to project upon a screen, if multiple sources are present then a superposition of fringes will be observed on the screen. If a source is large enough, a blurring of the fringes will be seen. The larger the source, the greater the blurring. While the fringes are distinct, the wavefront of the light can be considered coherent. The distance over which light is spatially coherent is the coherence length (Equation 1.23) and depends upon the wavelength and the bandwidth. If the source size (or slit separation) is increased beyond this coherence distance fringes will no longer be seen.
A thought experiment to illustrate spatial coherence is to consider the analogy of a pond where a large object drops into the still water. Near to the object, the water’s surface is chaotic but as the waves travel to the far shore they are orderly ripples or it could be stated; they are spatially coherent.
If the field emitted by one source is sampled at two points such that the light paths are within the coherence length, the self-interference will not necessarily be zero. Distinct astronomical sources are spatially incoherent; however, point sources are self-coherent. In between these two cases there exists a range where partial coherence is displayed. The relationship
between partial coherence and the object is the basis of the van Cittert-Zernike theorem.
To put this another way: A spherical electric field wave from a light source radiates outward at the speed, c. A photon does not exist until detected, instead there is the probability wave and a certain chance that the photon will interact and collapse the wave. It is these waves that can interfere if they are coherent. It is also possible for these waves to interfere with themselves. In stellar interferometry, photons originate from a distance source. Due to the
vast distance involved, the source is effectively a point source. The photons may originate from different regions of the surface of a star and are not coherent with each other. But the probability waves of each photon pass through multiple apertures and the photons in the form of this wave interfere with themselves. Upon striking a detector, the wave function collapses into a photon and this results in the fringe pattern that we can measure.
The van Cittert-Zernike theorem formally relates the fringe contrast to the angular distribution of the source on the sky. Proof of this theorem is available inBorn & Wolf(1999). The van Cittert-Zernike theorem states that for a monochromatic incoherent source, the Fourier
transform of the complex spatial coherence function yields the angular intensity distribution of the source. This relation is described in terms of the complex visibility,γ, the intensity
distribution of the source,S(−→α), and the spatial frequency in the Fourier plane,→−u(u, v) = −→Bλ, whereuandv are components of the baseline vector projected on the sky:
γ(−→u) =
Z Z
S(−→α)e2iπ−→α·−→u d−→α =|γ|eiφ. (1.30)
More simply as the mutual coherence is related the Fourier transform of the target intensity map by
γ(u, v) = F(I(α, β)). (1.31)
Five assumptions are made with van Cittert-Zernike theorem. First, the source is assumed to be distant and the in the far field condition. Second, the source is assumed to be small in angle but extended in two-dimensions. Astronomical sources are, of course,
As discussed earlier, the quasi-monochromatic case is assumed, where the light is filtered through a finite bandpass. Next, it is assumed that the source is spatially incoherent, which is the case for most astronomical sources. Finally, space is assumed to be a homogeneous medium, so that light from each region of the source is not differentially refracted in
comparison to light from other regions of the source. This holds true to all but the smallest degree, until the light enters Earth’s atmosphere, where variations due to turbulence impart distortions on the wavefront. This results in the need for careful calibration in order to accurately interpret interferometric observations.
From this theorem, it can be shown that an incoherent source observed from a great distance may display spatial coherence. This means it is possible to measure fringes from
astronomical sources, and with a multiple aperture interferometer each baseline gives one component of the Fourier transform of the source. The visibility is the modulus of the degree of coherence,|γ|, and the phase is the argument,arg(γ).
This demonstrates the basic relationship between coherence and the visibility of fringes. The visibility modulus for a uniform disk as a source is given by
V = 2J1 πBθ λ πBθ λ (1.32)
where J is the Bessel function of the first kind and first order, and B is the separation between two apertures. This function is the Fourier transform of the uniform disk.Figure 1.4shows the visibility curves for stars of various angular diameter.
0 100 200 300 400 500 baseline (m) 0.0 0.2 0.4 0.6 0.8 1.0 V 1 2 4 8 16
Figure 1.4: Visibility curves for stars of 1, 2, 4, 8, and 16mas in K band. A source is said to be unresolved where V is close to 1, resolved if V = 0, and over-resolved after the first null.
visibilty term. However, from van Cittert-Zernike theorem there is also the phase term,Φ, which is a measurable quantity. However, much of the phase information is corrupted by the effects of the turbulent atmosphere. If a patch of turbulent air crosses into the path of a single telescope of an interferometer, this will change the amount of delay and induce a shift in the interference pattern. Large shifts can cause phase wrapping, making it impossible to
unambiguously measure phase. Imaging of complex or asymmetric objects relies on complex phase information. Thankfully, there exists a way to recover at least some of this information. The shift in delay over one telescope has a corresponding shift in the opposite direction of the
other telescope. So if one were to form a triangle of three telescopes one could express, φ12 =Φ12+1−2 φ23 =Φ23+2−3 φ31 =Φ31+3−1 CP =φ12+φ23+φ31. (1.33)
Here,is the unknown phase error; andφand Φrepresent, respectively, the measured and real phase of a baseline pair. The sum of these phases is the closure phase, CP, which is an observable quantity where individual phase delays introduced by the atmospheric turbulence over a specific telescope cancel out. This technique was first introduced for radio
interferometry by Roger Jennison in 1958 (Jennison 1958).