Chapter 10 Spatial Coherence Part 2

Transcription

1 EE90F Chapte0 Spatial Coherence Part Geometry of the fringe pattern: η ( ξ, η ) ξ y Qxy (, ) x ρ ρ r ( ξ, η ) P z pinhole plane viewing plane Define: z + ( x) +( η y) ξ r z + ( x) +( η y) ξ (0.) (0.) ρ ξ + η ρ ξ + η (0.3) as the raial istances of the pinholes from the optic axis. The pinhole spacings are given by: We efine the pinhole spacing vector as: (0.4) (0.5) an Q ( xy, ) is the observation point in x, y plane Uner the paraxial approximation z» Q, ρ, ρ Assuming that α r r - [ ρ z ρ ξx ηy] [ ρ z ρ P Q] is constant, the fringe moulation has the form (0.6) Spatial_coherence_pt_post.fm Chapte0

2 EE90F (0.7) This has the form of a plane wave with a wavevector in the x-y plane: P π Defining P, as the pinhole spacing, then the fringe perio is The fringes are straight, an run perpenicular to P. y line joining P fringes x' x IQ ( ) I ( ) ( Q) + I ( ) ( Q) I ( ) ( Q)I ( ) r ( Q)γ r c cos π ρ λ z ρ P Q z Let s look at a cut through the fringe pattern along the x' axis. Assume that I ( ) ( Q), I ( ) ( Q) are nearly constant (tiny pinholes) an axis). ρ ρ 0 (where the pinholes are equiistant from the Ix' ( ) z c ν I ( ) + I ( ) half-with of fringe packet, (similar to Michelson) z c ν x' The total number of fringes observable is roughly: Spatial_coherence_pt_post.fm Chapte0

3 EE90F (0.8) Each fringe represents a path ifference of λ ] The coherence length is cτ c c ν c ν λ ν - ν ν ν [ fractional banwith] λ Quasimonochromatic conition We wish to concentrate on the spatial coherence effects an make the temporal coherence effects negligible. Assume that the light is narrowban. Not just ν «ν, but for all source points an observation points, the path lengths are the same within. τ c S r' Q r' P r Thus, for all source points an all Q. Also ( r ) c, ( r' r' ) c «τ c (0.9) If these conitions are satisfie, the fringe contrast is constant over the observing region. The coherence function simplifies The mutual intensity is: (0.0) (0.) J Γ ( 0) u( u ( P (0.) µ γ ( 0) J [ IP ( )IP ( )] 0 ũ Uner these conitions, the intensity in the x-y plane becomes (0.3) Spatial_coherence_pt_post.fm Chapte0

4 EE90F Ixy (, ) I ( ) I + ( ) + κ κ J π cos P Q+ φ + ( ) + I ( ) I ( ) µ π cos P Q + φ I ( ) I (0.4) (0.5) (0.6) (0.7) The fringe visibility is given by: V I ( ) I ( ) I ( ) + I ( ) µ (0.8) an V µ if I ( ) I ( ). µ 0 µ tells us there are no fringes. The two waves are mutually incoherent. tells us that the waves are perfectly correlate, an are mutually coherent. For 0 < µ < the waves are partially coherent. The above cases refer to two particular pinhole points, P. This is a limiting case of illumination over the full object plane. Then we are concerne with all the pairs of, P over the full object plane. Various measures of coherence Symbol Definition Name Temporal or Spatial Γ ( τ) up (, t τ)u ( γ ( τ) Γ ( τ ) Γ ( 0) Γ ( τ) up (, t τ)u ( P Self coherence function Complex egree of self-coherence Mutual coherence function Temporal Temporal Spatial an Temporal Spatial_coherence_pt_post.fm Chapte0

5 EE90F Symbol Definition Name Temporal or Spatial γ ( τ) Γ ( τ) Γ ( 0)Γ ( 0) J Γ ( 0) µ γ ( 0) Propagation of Mutual coherence Complex egree of coherence Mutual intensity Complex coherence factor Spatial an Temporal Spatial quasimonochromatic Spatial quasimonochromatic θ Q As light propagates from to Σ, the mutual coherence function changes (propagates). P θ r Q Given, P Γ(, P ; τ) for all pairs, how to fin Γ( Q, Q ; τ)? Σ Start with narrowban light. Later we will specialize to quasi-monochromatic light. (0.9) The generalize Huygens-Fresnel integral relates the fiels on Σ to fiels on. uq (, t + τ) u, t + τ ---- jλr c cosθ s (0.0) Then u ( Q r u P, t ---- jλr c cosθ s (0.) Spatial_coherence_pt_post.fm Chapte0

6 EE90F Thus u, t + τ ---- c u P, t ---- c Γ( Q, Q, τ) s s ( λ) cosθ cosθ r r (0.) (0.3) As a sie note, using the generalize Huygens-Fresnel integral for broaban light given earlier, we obtain the propagation law for mutual coherence of broaban light: (0.4) Now make a quasimonochromatic approximation; that is, the (maximum ifference in pathlengths) << (the coherence of the length of light). At, Σ Γ( Q, Q ; τ) s s Γ r P τ r, ; cosθ c cosθ λ λr Γ( Q, Q ; τ) s s τ Γ r P τ r, ; cosθ c cosθ πc πcr The integran in equation () with τ 0, (0.5) (0.6) Γ r P r, ; c J (, P ) exp j π ( r ) λ This leas us to the propagation law for mutual intensity (0.7) J ( Q, Q ) s s J (, P ) exp j π ( λ r ) cosθ cosθ λ λr (0.8) In general, the intensity istribution can be foun by letting Q, Q merge in the mutual intensity. (0.9) Limiting forms of the coherence function Fully coherent fiel in the quasi-monochromatic limit µ for all pairs (, P ). Spatial_coherence_pt_post.fm Chapte0

7 EE90F This means Apply the Schwarz inequality J There is equality if an only if gt () kf() t [ IP ( )IP ( )] UP (, t )U ( P [ UP ( UP ( ] ft ()g ()t t [ ft () t gt () t] with k a complex constant. (0.30) (0.3) (0.3) Thus UP ( K UP ( K epens on, P but not t. Choose some reference point P o. We can then write Thus UP ( UP ( ) UP o ( ; UP ( (0.33) [ IP ( o )] UP ( ) UP o ( [ IP ( o )] (0.34) corresponingly where µ exp{ j[ φ( ) φ( P )]} φ( ) arg[ UP ( )] φ( P ) arg[ UP ( )] (0.35) (0.36) The incoherent limit: For two pinholes µ 0 represents incoherent waves. A fully coherent fiel has µ for all pairs (, P ). A logical extension might be to say that for an incoherent fiel: Γ ( τ) 0 for all P an for all τ However, if we examine the consequence of this in equation (), the first integration over except at P where the value is finite (with a finite singularity IP ( )). Hence, the result of integration is precisely zero. Thus Γ( Q, Q ; τ) 0 for an incoherent fiel at, but IQ ( ) Γ( Q, Q ; 0) 0 also. 0 Spatial_coherence_pt_post.fm Chapte0

8 EE90F An incoherent wave fiel on oes not propagate! Why is this? An incoherent fiel in this sense has an infinitesimally fine spatial structure. Recall our iscussion of the spatial frequency cutoff. The spatial frequencies A perfectly incoherent surface oes not raiate. f xy, > - on t propagate. For a propagating wavefiel, coherence must exist over a linear imension of at least λ. For the quasimonochromatic case, the mutual intensity representing a propagating incoherent wave fiel can be represente by: λ where J is the Bessel function. J(, P ) IP ( )IP ( ) J ( k x + y ) k x + y, (0.37) This is somewhat cumbersome for calculations. If an optical system has a resolution coarser than λ at ( xy, ) plane, the exact shape of J(, P ) is not important. This will become clearer as we go along. Uner this assumption, it is reasonable to approximate (0.38) (0.39) If coherence extens over more than λ, but is still not resolve, then the δ-function is still vali, but the value of κ changes. For a wavefiel that is incoherent in this sense, the mutual intensity propagates as follows: JQ (, Q ) J(, P ) j π cosθ ( r ) exp cosθ s s λ λ λr Using equation (0.38): k ( λ) IP ( ) j π λ r cosθ ( ) cosθ exp s λ λr Σ (0.40) (0.4) Spatial_coherence_pt_post.fm Chapte0

9 EE90F The geometry: η ξ θ Q y x Σ source P θ z r Q observation region Uner the conitions of the Fresnel approximation: (0.4) (0.43) r ( x z ξ) + ( y η) z (0.44) Then ( x z ξ ) + ( y η) z (0.45) r [ x z + y ( x + y ) + xξ + yη ] (0.46) (the pure ξ, η terms cancel). Let: (0.47) I( ξ, η) J( x, y) ke j ψ is efine so that it is zero outsie Σ ( ) I( ξ, η) exp j π ( xξ + yη) ξ η (0.48) (0.49) Van Cittert - Zernike theorem The normalize form of the propagation law gives the complex coherence factor: Spatial_coherence_pt_post.fm Chapte0

10 EE90F µ ( x, y) e jψ I( ξ, η) exp j π ( xξ+ yη) ξη I( ξ, η) ξη (0.50) Interpretation e jψ Asie from the phase factor, - J( x, y) is foun from a two-imensional Fourier Transform of intensity I( ξ, η) across the source. - The mutual intensity is analogous to Fraunhofer iffraction pattern of source aperture. - But it is vali in the Fresnel region. - Moulus of the coherence factor µ epens only on coorinate ifferences. With of µ efines a coherence area A c. We efine this by analogy to τ c. (0.5) - The coherence area grows with the istance z from the source like ivergence of a iffraction pattern. (0.5) where A s is the source area Example A circular source, with raius a, which is uniformly bright, quasimonochromatic, an incoherent. I( ξη, ) I o circ ξ + η a Then the mutual intensity has the form of an Airy pattern (0.53) Jxyx (, ;, y x ) πa I o κ e jψ ( ) J πa - x + y πa x + y (0.54) Spatial_coherence_pt_post.fm Chapte0

11 EE90F Recall that ψ µ ( x, y ; x, y ) e jψ π [( x + y ) ( x + y )] J πa - x + y πa x + y (0.55) Note: this is a coherence factor, not a fiel or intensity. It relates to coherence at points. It can then be use to preict the result of the interference experiment. incoherent source P pinhole plane viewing plane The Van Cittert - Zernike theorem gives the coherence between, P as a function of the istance between, P. In turn we can then preict the fringe visibility at the viewing plane. Also, a careful measurement of the fringe phase an visibility gives µ ( x, y ; x, y ) at the pinhole plane. Young s experiment can be use to measure the coherence. For an incoherent source, what is the intensity of the pattern at the pinhole plane? (0.56) Ieally the incoherent source has a fine spatial structure of ~ λ. This iffracts out to all angles. Spatial_coherence_pt_post.fm Chapte0