C:\Dallas\0_Couses\0_OpSci_330\0 Lectue Notes\04 HfkPopagation.doc: Page of 9 Lectue 04: HFK Popagation Physical Optics II (Optical Sciences 330) (Updated: Fiday, Apil 9, 005, 8:05 PM) W.J. Dallas The Pinhole Wave The popagation of optical waves in fee space is govened by the Helmholt equation which is also known simply as the wave equation. Fo now, we will concentate on a paticulaly useful kind of popagation, that of waves taveling between paallel planes. Fo this discussion, we will use a heuistic, but highly accuate agument. Plane-Paallel Popagation Figue illustates the popagation, in fee space, between two planes of the wave coming fom a pinhole in a flat opaque sceen. We ll call it a pinhole wave. It is paticulaly impotant because the patten in the second plane epesents the PSF of the point in the fist plane. Moving the point lateally just moves the PSF, i.e., wave popagation in fee space is an LSI system. We can completely descibe the system by knowing only the PSF. The pictue in the second plane of Figue is a stylied epesentation of the eal pat of the pinhole wave. It is lateally tuncated and thesholded at a value of eo in that plane. A moe accuate illustation will come afte we have looked into the mathematical epesentation of the pinhole wave. y X =0 = 0 Figue : A pinhole wave popagating fom the pinhole to a second plane. Why the pinhole wave is not the same as the spheical wave The wave fom a point standing fee in space is a spheical wave. A spheical wave has the complex eik amplitude us ( ) =. Fo example, it dies-off lateally in the x-diection as in the plane of the souce: x it cannot be a pinhole wave; that wave is eo except at the oigin.
C:\Dallas\0_Couses\0_OpSci_330\0 Lectue Notes\04 HfkPopagation.doc: Page of 9 How we modify a spheical wave to suit ou needs We can imagine a pinhole-wave candidate by looking at the spheical wave shown in Figue. In that figue we have also de-emphasied the left-popagating wave. Notice that the wave-fonts intesect the x-y plane at ight angles except at the oigin. This behavio suggests that we use u s. x Taking the deivative, with espect to, gives Figue : Spheical wave intesecting the x-y plane ik ik x + y + ik e e e uph( x, y, ) = = = ik π π x + y + π Ou pinhole wave is pecisely the Huygens wavelet. It is closely elated to the Rayleigh-Sommefeld diffaction equation. Nomaliing the pinhole wave We want to be sue that, at the plane = 0 the pinhole wave is a -D delta function. We need to have that ph (,,0) = δ δ u x y x y
C:\Dallas\0_Couses\0_OpSci_330\0 Lectue Notes\04 HfkPopagation.doc: Page 3 of 9 Checking the value at the oigin diectly is not possible because of the delta function s infinite value at that point. Instead, we use the fact that δ ( x) δ ( y) dxdy = of. π Fee-space popagation as an LSI system If we shift the pinhole in the fist plane lateally we get ph (,,0) = δ ( ) δ ( ) u x x y y x x y y 0 0 0 0. It is this integal that gives us the leading facto In the second plane, this shift has the effect of shifting the pinhole wave in that plane, giving ph (,, ) u x x y y 0 0 0 We have a linea shift-invaiant system with a point spead function (,, ) uph x y 0. This fact gives us yet anothe name fo the pinhole wave: the point spead function of fee space. Fom ou knowledge of LSI systems, we know that the popagation is descibed by the convolution u x, y, = u x, y,0 u x, y,. Witten out, this convolution is 0 ph 0 u x y u x y u x x y y dx dy (,, ) = (,,0) (,, ) 0 0 0 ph 0 0 0 0 0 The wave field eveywhee to the ight of the plane = 0 is detemined by the value of the wave in the plane = 0. The exact Huygens-Fesnel-Kichhoff (HFK) fomula Let s look at two abitay planes, not necessaily having one plane at the oigin. In the fom of a convolution, the HFK fomula fo popagation fom the plane = 0 to the plane = is u x, y, = u x, y, u x, y,. Witten out using the explicit wave fomula, this equation is 0 ph 0 Paaxial popagation ( 0) u( x, y, ) = u( x0, y0, 0) ik π i e ( x x ) + ( y y ) + ( ) 0 0 0 ( ) + ( ) + ( ) ik x x0 y y0 0 ( x x ) + ( y y ) + ( ) 0 0 0 dx dy This fomula is highly accuate, but a little cumbesome. It is a common and useful pactice to examine popagation when the obsevation egion is bounded to small distances fom the -axis. The appoximations that go into that calculation lead to paaxial optics. The appoximations ae done in fou steps stating with the pinhole wave fomula. 0 0
C:\Dallas\0_Couses\0_OpSci_330\0 Lectue Notes\04 HfkPopagation.doc: Page 4 of 9 ik e uph( x, y, ) = ik π
Appoximations C:\Dallas\0_Couses\0_OpSci_330\0 Lectue Notes\04 HfkPopagation.doc: Page 5 of 9. π i ik = We note that so that λ is negligible fo λ π the x-y plane of seveal wavelengths. This tem becomesik ik. >>, i.e., a distance away fom. The tem is the angle that the position vecto to a point on the pinhole wave makes with the -axis. y x Figue 3: The angle to the optical axis Fo small angles, i.e. nea the -axis, = cosθ ik ik e e 3. Nea the -axis, so that. ik 4. The appoximation fo the phase, k, must be done caefully because e is apidly vaying. The paabolic appoximation of is done using the Taylo expansion. We begin with the expession x + = x + y + = + y then use the fist-ode Taylo expansion
/ + ε + ε, ε << to get So that: x+ y πi ik ik ik λ C:\Dallas\0_Couses\0_OpSci_330\0 Lectue Notes\04 HfkPopagation.doc: Page 6 of 9 x + y x + y x + y + fo = +. x y iπ + λ e e e = e e. Taking all of the steps togethe we have The paaxial pinhole-wave fomula Tidying up the equation gives πi λ uph ( x, y, ) = e e iλ ( + ) iπ x y λ u ph π i e ( x, y, ) π λ πi λ x + y iπ λ e. The paaxial HFK fomula fo popagation Inseting this expession into the popagation elation we get the paaxial Huygens-Fesnel-Kichhoff (HFK) popagation equation, π ( 0) λ i iπ ( x x0) + ( y y0) e λ( 0) u x y u x y e dx dy,, =,, 0 0 0 0 0 iλ 0 Thin optical elements We will be woking with complex-amplitude tansmitting thin elements. These elements will be chaacteied by a complex-amplitude tansmittance. This tansmittance descibes the element's action on a geneal wave. If we suppose that the element is located in the plane " = 0", then an incident wave slightly to the left of the element will be conveted to an exiting wave slightly to the ight of the element in the manne: ( 0 + ) ( 0 ) u x, y, = u x, y, t x, y
t(x,y) C:\Dallas\0_Couses\0_OpSci_330\0 Lectue Notes\04 HfkPopagation.doc: Page 7 of 9 y x u(x, y, = 0 - ) u(x, y, = 0 + ) Figue 4: A thin optical element The Neutal-Density Filte The thin neutal-density filte is chaacteied by a constant-amplitude attenuation 0 0 t x,y = A whee A Apetues These ae clea holes in opaque sceens. Thee ae seveal common types. Rectangula apetue x x t ( x, y) = ect ect x0 y0 Finite-width infinite-length slit x t ( x, y) = ect x0 Infinitesimal-width infinite-length slit t x,y = δ x Infinitesimal-width finite-length slit y t ( x, y) = δ ( x) ect y0 Cicula apetue
t x,y = cyl d0 C:\Dallas\0_Couses\0_OpSci_330\0 Lectue Notes\04 HfkPopagation.doc: Page 8 of 9 Pinhole t x,y = δ x δ y Double pinhole x0 x0 t x,y = δ x + δ x + δ y Infinite double slit x0 x0 t ( x,y) = δ x + δ x + Finite double slit x0 x0 y t ( x, y) = δ x + δ x + ect y0
The Plane-Paallel Plate C:\Dallas\0_Couses\0_OpSci_330\0 Lectue Notes\04 HfkPopagation.doc: Page 9 of 9 The thin plane-paallel plate is chaacteied by a constant phase shift ove its entie suface. The expession fo the plate's complex amplitude tansmittance is i 0 t( x,y ) = e φ Whee φ0 is a constant. The Pism A pism is a thin element which has a phase shift that vaies linealy with position. The expession fo the pism's complex amplitude tansmittance is i 0 x + 0 t x,y = e π ξ η y whee ξ 0 and η 0 ae constants. The Lens The lens is a focusing element. It essentially convets one pinhole wave into a second pinhole wave. It is consideed to have, in the paabolic appoximation, a quadatic phase facto. The complex-amplitude tansmittance of a lens located on-axis is iπ ( x +y ) λ f t x, y = e ± powe To be examined in subsequent lectues Diffaction gatings Fesnel one plates Chomatic filtes Polaies