Chapter 2 Effect Of A Lens

Transcription

1 EE90F Chapter Eect O A Lens [Reading assignment: Goodman, Fourier Optics Ch. 5; 5.4 is optional ( xy, ) o 3 We consider the model o a thin lens, where we introduce a phase delay at (x, y) due to the lens material. We assume a negligible translation o rays inside the lens. input plane output plane The thickness unction is ( xy, ). o is the maximum thickness. φ( xy, ) kn ( xy, ) + k[ o ( xy, )] (.) phase delay in lens material The transmission unction o the lens is thereore: phase delay in ree space between the input and output planes Let s calculate, assuming the lens suraces are spherical. The convention is (.) positive radius negative radius ( xy, ) R R R x y is independent o (x, y), so we can drop it as a simple constant phase. Lenses_post.m Chapter

2 EE90F A simple analysis then gives ( xy, ) o R x + y R R x + y R (.3) For a biconvex lens R < 0 The paraxial approximation assumes ( x + y ) max «R, (slow lens or low NA). Then, In this approximation, x + y x + y R, R, (.4) ( xy, ) x y + o So now we can write (neglecting the constant phase): R R t l ( xy, ) jk( n ) x + y exp R R (.5) (.6) The ocal length, is given by the lens makers ormula : (.7) The transmission unction is now: t l ( xy, ) exp j k - ( x + y ) (.8) This is the paraxial approximation to the spherical phase Note: the incident plane-wave is converted to a spherical wave converging to a point at behind the lens ( positive) or diverging rom the point at in ront o lens ( negative). Lenses_post.m Chapter

3 EE90F Fourier Transorm property o lenses object Suppose we place a transmitting object directly against a lens with ocal length. This is illuminated by a plane wave, amplitude A. The object amplitude transmission is t A ( xy, ). The lens pupil unction is Pxy (, ). We deine U l ( xy, ) At A ( xy, )Pxy (, ) Here the object is windowed by the lens pupil. The eect o the lens is (.9) U l '( xy, ) U l ( xy, ) exp j k - ( x + y ) (.0) We now use the Fresnel ormula to ind the amplitude at the back ocal plane z U exp j k - ( u + v ) e jkz jλ (.) dx The phase terms that are quadratic in dyu l '( xy, ) exp j k - ( x + y ) exp j π ( xu + yv) λ x + y cancel each other. (.) This is precisely the Fraunhoer diraction pattern o apply here. U exp j k - ( u + v ) dx yu jλ d l ( xy, ) exp j π ( xu + yv) λ (.3)! Note that a large z criterion does not The ocal plane amplitude distribution is a Fourier transorm o the widowed input distribution, multiplied by a quadratic phase term. However, the intensity distribution is exactly U l Lenses_post.m Chapter

4 EE90F I A λ F[ U l ( xy, )] x u - λ y v - λ (.4) When the input is somewhere in ront o the lens d Let F o ( x, y ) F[ At A ] F l ( x, y ) F[ U l ] (the object Fourier Transorm) (the Fourier transorm o light is incident on the lens). We use the Fresnel ormula (in transer unction orm) to propagate the distance o d rom the object to the lens. F l ( x, y ) F o ( x, y ) exp[ jπλd( x + y )] Now use equation (.3) (neglect the inite lens pupil): (.5) U exp j k - ( u + v ) F u jλ l -, v - λ λ (.6) j k - d - exp ( u + v ) F u jλ o -, v - λ λ (.7) at d. (.8) This is the exact Fourier transorm relationship or d! Image Formation For now we assume a purely monochromatic radiation (ully coherent temporally). It can be shown (we will discuss in detail later) that this also implies complete spatial coherence. Under these assumptions, the imaging system is linear in ield amplitude. Lenses_post.m Chapter

5 EE90F Formation o a real image by a positive lens: U o U l U' l U i ( ξ, η) plane (u, v) plane u, v: image coordinates. This is a pure phase in image space. It actors out o the convoz z (x, y) plane Since the system is linear, we want to ind the impulse response. Then we can use superposition to ind the general response. Deine: h(u, v; ξ, η) : response at (u, v) in the output plane to a point source at (ξ, η) in the input plane. Then, the image will be given by: (.9) Outline o the calculation:. A point source gives rise to a spherical wave. Use the paraxial approximation (the quadratic phase) to ind U l ( xy, ), at the propagation distance z.. Multiply by the lens quadratic phase shit and pupil Pxy (, ) unctions (ocal length ) to get U' l. 3. Use the Fresnel integral to propagate the distance to get. The result is: huvξ (, ;, η) exp j k ( u + v ) exp j k ( ξ + η ) λ z z z z dx exp jk (.0) z U i dyp( xy, ) j k exp ( x + y ) z z ---- ξ u x η v y z z z z The quadratic phase inside the integrand broadens out the result. Otherwise we would have a simple Fourier transorm relation. Choose to eliminate. This gives us the lens law. There are quadratic phase actors in ront o the integral. Lenses_post.m Chapter

6 EE90F lution integral, and this does not contribute to the image intensity pattern. ξ, η: object coordinates. This actor will appear inside the convolution integral. There is one interesting way to eliminate this term: z Illuminate the object with a converging wave, with radius z. Kohler illumination Another argument or dropping this term is given in Goodman, Fourier Optics, nd edition section We can argue that at any image point (u, v) only a small patch in ( ξη, ) object space contributes. The quadratic phase does not vary much over this patch, so it can actor out o the convolution integral. Then, again, this does not aect the intensity o the image. Having dropped the phase actors, we deine M z z huvξ (, ;, η) λ dx yp( xy, ) j π d exp [( u Mξ )x+ ( v Mη)y] z z λz (.) This is the impulse response o the imaging system. It is the Fraunhoer diraction pattern o the lens pupil, centered at the image point u Mξ ; v Mη, which is essentially a Fourier transorm. The eect o diraction is to broaden the system impulse response. We reer to this response as the point spread unction. In geometric optics, the image is (.) The eect o the diraction by the lens is to smooth this ideal image by the convolution with the point spread unction. Where a straightorward set o coordinate transormations gives: h dx dỹ P( λz x, λz ỹ) exp[ jπ( ux + vỹ) ] The Fourier transorm o the scaled pupil unction yields the point spread unction. (.3) (.4) Lenses_post.m Chapter

7 EE90F Example: a circular lens, with radius w w Θ z P circ q w - ( q ξ + η ) hr ( ) F[ P( λz q) ] F circ λz q w ( r u + v ) (.5) (.6) hr () A J ( πωr λz ) λ z πωr λz (.7) The spot diameter paraxial approximation As we will see, the Rayleigh resolution or incoherent illumination is d 0.66λ Θ, which is hal o the spot diameter. For a large Θ, d λ 0.66 λ - sinθ NA Lenses_post.m Chapter