Chapter 9 Spatial Coherence

Transcription

1 Chapter 9 Spatial Coherence [Continue reading: Goodman, Statistical Optics, Chapter 5 Again we use the interference experiment as a vehicle for discussing spatial coherence Young s double slit experiment ' θ 1 Q source ' θ P We expect an interference fringe at Q under two conditions: a) The time delay difference is less than the coherence time of source (9.1) b) The visibility of the fringe depends on the degree of correlation between light waves at the two pinholes. Here the source size plays a big role. To see a relation between source size and the degree of correlation, we assume a narrowband, point source S 1 fringe due to S S 1 S P screen 1 screen fringe due to S 1 resultant fringe S 1 produces a sinusoidal fringe, with high contrast. Now add a second source, S, nearby, but radiating independently (not coherent with S 1 ). S produces its own fringe, shifted spatially from the S 1 fringe. The fringe period depends on the -P separation. The shift depends on the S 1 -S separation. Spatial_coherence_pt1_post.fm Chapter 9

2 screen 1 screen S 1 S P S fringe S 1 fringe resultant fringe When and P are far apart, the fringe period becomes smaller than the shift. This results in cancellation. When the source actually consists of many such independent point sources (an extended source), then the fringes wash out. For a given source size, the fringe visibility depends on the spacing of -P. this gives a measure of spatial coherence at screen 1. The fringe visibility falls off as the -P separation falls outside of the coherence area. Analytical treatment A generalization of the Huygens-Fresnel principle to broadband and narrowband cases (review Goodman, Fourier Optics, Section 3.8). P 0 r Using the inverse Fourier transform: (9.) (9.3) Let ν' ν, then Spatial_coherence_pt1_post.fm Chapter 9

3 up ( 1, t ) UP ( 1, ν' ) exp( jπν't) dν' (9.4) Expressed this way, (9.5) are linear combinations of monochromatic waves as originally expressed at the beginning of the course. The idea is to use the Huygens-Fresnel integral to relate UP ( o, ν' ) to UP ( 1, ν' ) for each ν' (using the monochromatic result), then to use the above integral superposition to find up ( o in terms of up ( 1. Huygens-Fresnel integral: up ( o UP ( o, ν' ) exp( jπν't) dν up ( 1, up ( o UP ( o, ν' ) j ν' exp( jπν'r - UP ( c 1, ν' ) 01 c) cos( ṋ, ) ds Insert this equation into equation (9.5) above: Σ (9.6) up ( o Differentiate equation (9.4) We can then use this to write Σ cos( ṋ, ) jπν'u( P πc, ν' ) jπν' t 01 exp d - up ( dt 1 jπν'u(, ν' ) exp( jπν't) dν' - dν' ds (9.7) (9.8) (9.9) This is a generalized Huygens-Fresnel integral for broadband light. For narrowband light, ν«ν, re-examine equation (9.7). The width of UPν' (, ) is ν, so we can take the ν' factor out of the integral as c λ. Then equation (9.7) can be written: Spatial_coherence_pt1_post.fm Chapter 9

4 up ( o Σ cos( ṋ, ) UP ( 1, ν' ) jπν' t - exp jλr dν 01 ds (9.10) Using (9.) this is u, t - So, (9.11) This is the Huygens-Fresnel integral for narrowband light. Application to Young s experiment P Q (9.1) K cos( ṋ, ) ds jλr K cos( ṋ, r ) ds jλ P (9.13) We have assumed pinholes small enough so that the field is constant over the whole pinhole. and K are complex constants. (Pure imaginary) K 1 Write IQ ( ) K 1 u(, t c) + K u( P, t c) + K 1 K u(, t c)u ( P, t c) + complex conjugate (9.14) I ( 1) ( Q) K 1 u(, t c) intensity at Q with P blocked. (9.15) Spatial_coherence_pt1_post.fm Chapter 9

5 I ( ) ( Q) K u( P, t c) intensity at Q with blocked. (9.16) We then write the cross correlation of the light at with P as: This is the mutual coherence function. IQ ( ) I ( 1) ( Q) I ( ) ( Q) K 1 K Γ r K 1 K Γ 1 r (9.17) (9.18) Since K 1, K are pure imaginary, K 1 K K 1 K κ 1 κ, and κ 1 K 1 κ K, also Γ 1 ( τ) Γ 1 ( τ). IQ ( ) I ( 1 ) ( Q) I ( ) ( Q) κ 1 κ Re Γ Again we normalize the coherence function using the Schwarz inequality, (9.19) Γ 1 ( τ) Γ 11 ( 0) Γ ( 0) note: Γ 11 ( 0) IP ( 1 ) and Γ ( 0) IP ( ) (9.0) Where Γ 11 ( τ), Γ ( τ) are self-coherence functions. These were defined in the discussion of the Michelson interferometer in chapter 7 of these notes. So we can normalize Γ 1 ( τ) by [ Γ 11 ( 0)Γ ( 0) ] 1. (9.1) This is called the complex degree of mutual coherence. Note: I ( 1) ( Q) κ 1Γ11 ( 0) I ( ) ( Q) κ Γ ( 0) Further simplifying the expression for the fringe pattern, we can write: IQ ( ) I ( 1) ( Q) I ( ) ( Q) I ( 1) ( Q)I ( ) r ( Q )Re γ Anticipating the form of the fringe pattern, we write: γ 1 ( τ) γ 1 ( τ) exp{ j[ πντ α 1 ( τ) ]} (9.) (9.3) (9.4) (9.5) Spatial_coherence_pt1_post.fm Chapter 9

6 IQ ( ) I ( 1) ( Q ) I ( ) ( Q ) I ( 1) ( Q)I ( ) r ( Q)γ r πν r α r cos (9.6) I ( 1) ( Q), I ( ) ( Q) vary spatially as pinhole diffraction patterns. We assume these pinholes are very small so that the intensities are nearly constant over the observation region. On top of this background, we have the fringe pattern, with: - the period in space depends on ν and the geometry ( ). - a slowly varying envelope and phase modulation. Near zero path difference ( 0), the fringe visibility is (9.7) γ 1 ( 0) measures the coherence of up ( 1 and up (. - when P γ 11 ( 0) 1 the pinholes coincide As, P separate, γ 1 ( 0) varies. This indicates the spatial coherence of the light striking the pinhole plane. As the path difference varies, the fringe envelope tapers off. This is an indication of the temporal coherence effect. Spatial_coherence_pt1_post.fm Chapter 9