Chapter 12 Effects of Partial coherence on imaging systems

Transcription

1 Chapter 2 Effects of Partial coherence on imaging systems Coherence effects are described by the mutual intensity function J(x, y ; x 2. (under quasimonochromatic conditions We now have the tools to describe the evolution of mutual intensity. The propagation law for mutual intensity: Read Goodman, Statistical Optics, Chapter 7 J i ( x, y ; x J ( λz 2 o ( ξ j----- π [( ξ 2 + η 2 ( ξ + η ] λz e jψ j 2π exp [ x 2 ξ 2 + y 2 η 2 x ξ y η ]dξ λz Transmission through the object: (2. (2.2 Can (2.2 be used to describe the effect of a lens on mutual intensity? Or on another strong phase object? n 2 n n dxy (, d o The delay through this structure, neglecting the constant, is δ [ n 2 ( xy, n ]dxy (, c Assuming a possible amplitude transmission factor B(x, y, u t ( xyt, ; B( x, yu i [ x, y;( t δ( xy, ] Now we evaluate the mutual coherence of the transmitted field: Partially_coherent_imaging_pt_post.fm Chapter 2

2 Here, we assume no deviation of rays, only a phase shift. Then, Thus ; τ BP ( BP ( 2 u, t + τ δ u i ( P 2, t δ( P 2 (2.3 ; τ B B( P 2 Γ i ; τ δ + δ( P 2. (2.4 With narrowband light, we can write: (2.5 So, ( B P2 ; τ BP ( e j2πνδ P ( e j2πνδ P 2 U i, t + τ δ + δ( P 2 U i ( P 2, t e j2πντ ( (2.6 If δ δ( P 2 «τ c -, then the time average is independent of δ( P, so ν δp, ( 2 ; τ BP ( e j2πνδ BP2 ( e j2πνδ ( P 2 Γ i ; τ ; τ t A t A ( P 2 Γ i ; τ (2.7 (2.8 with. This is the same result we had for mutual intensity for a thin aperture. ( t A ( P BP ( e j2πνδ P This is strictly valid only when the delay differences induced by the object are much less than the coherence time. The relation holds even when the time delays through the object fail the quasimonochromatic condition, as long as the condition holds for net time delay differences in the full system. Recall the thin lens transmission function: t l ( xy, exp j π - ( x 2 + y 2 λf P( x, y (2.9 Object-Image coherence relation We assume the quasi-monochromatic condition. Our interest is in the relationship between the mutual intensities in the object and image s. First, assume that the mutual intensity of the object is known. ( Propagate to the lens (2 Transmit through the lens. Partially_coherent_imaging_pt_post.fm Chapter 2

3 (3 Propagate to the object. ( ξ, η ( xy, ( uv, object lens pupil image We use the propagation law for mutual intensity, in the paraxial approximation. J' o ( ξ J l '( x, y ; x 2 : mutual intensity leaving object : mutual intensity leaving lens J l '( x, y ; x ( λ 2 Px (, y P ( x 2 j π λ f [( x 2 + y 2 ( x + y ] J o '( ξ j π [( ξ 2 + η 2 ( ξ + η ] λ j 2π exp [ x 2 ξ 2 + y 2 η 2 x ξ y η ] dξ λ (2.0 Now propagate to the image J i ( u, v, u 2, v 2 dξ J o '( ξ j π [( ξ 2 + η 2 ( ξ + η ] λ dx y Go to the usual image basically the same as before: ( λ ( λ 2 j π [( u 2 + v 2 ( u + v ] λ d dx 2 y 2 P( x, y P ( x 2 j π λ d f [( x 2 + y 2 ( x + y ] u 2 ξ 2 j 2π λ x y v x u v exp y η 2 ξ η - ( Define the amplitude point spread function. This is f Partially_coherent_imaging_pt_post.fm Chapter 2

4 huvξ (, ;, η j π u 2 v 2 ( + j π ξ 2 η 2 exp exp ( + ( λ λz ( o λ λ (2.2 The result now simplifies considerably to look like this: P( x, y j 2π λz u ξ z exp i x v η i + y dxdy J i ( u, v ; u 2, v 2 dξ ( J o '( ξ (2.3 hu (, v ; ξ h ( u 2, v 2 Image intensity I i ( uv, J o '( ξ huvξ (, ; h ( uvξ, ; 2 dξ (2.4 It s easy to verify that for incoherent light coming from the object, we recover our previous result that the image intensity is given by a convolution of the object intensity with the square magnitude of the point spread function. Note that the quasi-monochromatic requirement here is: «l The object field is ; the image field is. 4 4z c L o L o L i L i i For L o, L i is in the centimeter range;, is in the tens of centimeters range, and l c is in the millimeter range. L o 2 L i 2 We now attempt to analyze the complete system for imaging β ( ξ, η ( x, y source illumination optics imaging optics object image This model of an imaging system is applicable to practical applications: - Microlithography - Slide and vugraph projector Partially_coherent_imaging_pt_post.fm Chapter 2

5 - Microscope - Projection TV - Photo-enlarger We need to combine our understanding of propagation of coherence from source to object, with the object-image coherence formulation just completed to get the full theory for this system. We will develop a very powerful linear systems approach for this, which is also very general: - The coherence of the source illumination will be simply characterized by the illuminator pupil function. - The imaging system is characterized by the point spread function, or the imaging pupil function. First - a more conceptually intuitive method, suitable for spatially incoherent source. If the source is incoherent, we can consider the source as a superposition of point sources. Calculate the image intensity for each source point, then add the image intensities. Define f( ξηα;,, β : the point spread function for the illumination optics. This tells us the field at ( ξη, due to the point source at β. Also, huvξ (, ;, η : the point spread function of imaging optics. Assume quasi-monochromatic conditions. The source point at β U S β; t. emits a field phasor Then the field after the object is (We can use coherent optics, since we assume a point source!: U o '( ξηα, ;, β, t f( ξηα, ;, βt o ( ξ, ηu S β; t δ (2.5 Here t o ( ξη, is the object amplitude transmittance, and δ is the time delay from β to ( ξη,. The image field is U i ( uvα, ;, β, t huvξ (, ;, ηt o ( ξ, ηf( ( ξηα, ;, βu S β; t δ δ 2 dξdη Where is the delay from ( ξη, to (u,v. δ 2 (2.6 The partial image intensity at ( uv, is due to the point α, β on source is: (2.7 Partially_coherent_imaging_pt_post.fm Chapter 2

6 I i ( uvα, ;, β huvξ (, ; h ( uvξ, ; 2 t o ( ξ t o ( ξ 2 f( ξ βf ( ξ 2 β (2.8 if the quasi-monochromatic condition applies δ + δ 2 δ ' δ 2 «τ c, the time average inside the integral is simply I S β. To get the full image, we have to sum the contributions from all the source points adding intensities U S β, t δ δ 2 U S β, t δ ' d 2 ' dξ (2.9 I S β huvξ (, ; h ( uvξ, ; 2 f( ξ βf ( ξ 2 β t o ξ ( t o ( ξ 2 dξ dαdβ (2.20 Partially_coherent_imaging_pt_post.fm Chapter 2