Or les différents systèmes philosophiques, économiques et politiques qui régissent les hommes sont tous d accord sur un point: bernons-les Roger

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1 Or les différents systèmes philosophiques, économiques et politiques qui régissent les hommes sont tous d accord sur un point: bernons-les Roger Waters (Pink Floyd, 1974) 1

2 Brief summary of main results obtained during the last lecture: ρ = R / z T eff = (F/σ) 1/4 = (f / σ ρ 2 ) 1/4 E = A(z) exp[i2πνt] E = A(z, t) exp[i2πνt] τ =1/ Δν λ eff = λ 2 / Δλ I = A A* = A 2 = a 2. 2

3 If Δ λ / (2B), fringe disappearance! I = I + I + 2 I γ q 12(0) cos ( ) β 2Πντ 12 γ 12 V ( τ ) = ( t ) ( t τ ) / 1 2 V I Fringe visibility: I I max min υ = = max + I I min γ 12 (0) 3

4 2.2. The Huygens-Fresnel principle 1st experiment! 4

5 4 Interferometry with two independent telescopes b) Fizeau the father of stellar interferometry (1868) 2nd experiment! 5

6 5 Light coherence 5.3 Spatial light coherence?? S γ 12 S = ΣdS i for i = 1, N ds i V P 1 P V ( τ = 0) = ( t ) ( t) / 1 2 V 1 (t) V 2 (t) I N V ( t) = V ( t) 1 i1 i= 1 (5.3.1) N (5.3.2) V ( t) = V ( t) 2 i2 i= 1 I q? + q N N γ ( 0) = 12 V 1V + i2 V i1v j2 i= 1 / i i j I (5.3.3) 6

7 5 Light coherence 5.3 Spatial light coherence N N γ ( 0) = 12 V 1V + i2 V i1v j2 V V V ( a ( t r / c) / r ) exp{ i2π ( t r / i i1 i1 i )} ( a ( t r / c) / r ) exp{ i2π ( t r / i i2 i2 i )} ( t) = ν c 1 1 i i i= 1 ( t) = ν c 2 2 (5.3.4) ( 2 t) V ( t) = i ai t ri1 c /( ri r )exp 2 ( / 1 i2 1 i1 ) as long as: / i i j r i1 I (5.3.3) r c / Δν = 2 i2 λ / Δλ = { i2πν ( r r ) / c} i2 i (5.3.5) (5.3.6) 7

8 5 Light coherence 5.3 Spatial light coherence I ( s) ds = ( t r / c) ai 2 (5.3.7) γ I( s) ( 0) = exp λ 12 1 S r 1 r 2 { i2π( r r )/ } ds / I 2 (5.3.8)!!! Theorem of Zernicke-van Cittert!!! 8

9 5 Light coherence 5.3 Spatial light coherence P i z ds i (X, Y, Z ) y P 1 (X, Y, 0) + x + P 2 (0, 0, 0) r 2 - r 1 = P 2 P i - P 1 P i = -(X 2 + Y 2 ) / 2 Z + (X ζ + Y η) (5.3.9) where ζ = X / Z and η = Y / Z (5.3.10) 9

10 5 Light coherence 5.3 Spatial light coherence γ (0, X / λ, Y / λ) = exp( i φ ) { i2π( Xζ Yη )/ λ} I( ζ, η)exp + S I( ζ ', η') dζ ' 12 X, Y d η ' S I '( ζ, η) I'( ζ, η) = = I( ζ, η) / γ (0, u, v)exp( iφ )exp + 12 Setting: u = X/λ, v = Y/λ: S I( ζ ', η') dζ ' dη' γ (0, u, v) = exp( iφ ) 2 12 u, v u, v { i Π( uζ vη) } I'( ζ, η)exp + S dζdη { i2π( ζu ηv) } d( u) d( v) dζdη (5.3.11) (5.3.12) (5.3.13) (5.3.14) 10

11 5.4 Fourier transform (cf. Léna 1996) Definitions: TF _ f ( s) = f ( x) e 2 i π sx dx, (5.4.1) f ( x) = TF _ f ( s) e 2 i π sx ds, (5.4.2) 2 f ( x) dx. (5.4.3) 11

12 5.4 Fourier transform (cf. Léna 1996) Definitions: Generalisation: TF _ f ( w) = f ( r) e 2iπ r w dr. (5.4.4) Some properties: a) Linearity: TF_(af) = a TF_f, a R, a being a constant, (5.4.5) TF_(f+g) = TF_f + TF_g. (5.4.6) 12

13 5.4 Fourier transform (cf. Léna 1996) Some properties: b) Symmetry & parity: f(x) = P(x) + I(x), (5.4.7) TF _ f ( s) = 2 0 P( x)cos(2πxs) dx 2i 0 I( x)sin(2πxs) dx. (5.4.8) Illustration of TF_f(s): f(x) is a real fonction. The real and imaginary parts of TF_f(s) are shown. f(s) 13

14 5.4 Fourier transform (cf. Léna 1996) c) Similitude: TF_(f(x/a))(s) = a TF_(f(x))(sa), (5.4.9) where a R, is a constant. d) Translation: TF_(f(x - a))(s) = e -2iπas TF_(f(x))(s). (5.4.10) 14

15 5.4 Fourier transform (cf. Léna 1996) e) Derivation: TF_(df/dx)(s) = 2iπs TF_f(s), TF_(d n f/dx n )(s) = (2iπs) n TF_f(s). (5.4.11) Some important cases (one dimension): a) Door function: Π(x) = 1 if x ]-1/2, 1/2[, = 0 if x ]-, -1/2] or x [1/2, [. (5.4.12) 15

16 5.4 Fourier transform (cf. Léna 1996) The door function and its Fourier transform (cardinal sine) TF_ (Π(x))(s) = sinc(s) = sin(πs) / πs. (5.4.13) TF_ (Π(x/a))(s) = a sinc(as) = a sin(πas) / πas. (5.4.14) 16

17 5.4 Fourier transform (cf. Léna 1996) b) Dirac distribution: 2 i π δ ( x) = e sx ds. (5.4.15) its Fourier transform is thus unity (= 1) in the interval ]-, [. 17

18 5.4 Fourier transform (cf. Léna 1996) c) Dirac comb: (x) = δ(x-n), for n in the interval ]-, [. (5.4.16) TF_ (s) = (s). (5.4.17) (x) f(x) = f(x) δ(x-n), for n in the interval ]-, [, (5.4.18) 18

19 5.4 Fourier transform (cf. Léna 1996) c) Dirac comb: (x) * f(x) = f(x-n), for n in the interval ]-, [, (5.4.19) (a) Sampling of the function f(x) by means of the Dirac comb; (b) Replication of the function f(x) after convolution by the Dirac comb. 19

20 5 Light coherence 5.5 Aperture synthesis ( ΠBb / λz' ) sin υ γ (0, B / λ) = 12 ΠBb / λz' = (5.5.2) Π Bb / λ z' = Π (5.5.3) Δ ~ λ / B, for a rectangular source. (5.5.4) Δ ~ 1.22 λ / B, for a circular source! (5.5.5) 20

21 5 Light coherence 5.5 Aperture synthesis Exercises ( ): point-like source?, double pointlike source with a flux ratio = 1?, gaussian-like source?, uniform disk source?, 21