Chapter 6 SCALAR DIFFRACTION THEORY

Chapter 6 SCALAR DIFFRACTION THEORY [Reading assignment: Hect 0..4-0..6,0..8,.3.3] Scalar Electromagnetic theory: monochromatic wave P : position t : time : optical frequency u(p, t) represents the E or H field strength for a particular transverse polarization component U(P) : represents the complex field amplitude UP up e jp up Diffraction: : real P aperture P o screen Approximations:. We impose the boundary condition on U, that U 0 on the screen.. The field in the aperture is not affected by the presence of the screen. UP o r 0» ---- UP j expjkr 0 --------------------------ds r 0 expanding spherical 9 Jeffrey Bokor, 000, all rights reserved

This equation expresses the Huygens-Fresnel principle: The observed field is expressed as a superposition of point sources in the aperture, with a weighting factor. Fresnel approximation Huygens-Fresnel integral in rectangular coordinates: y P o x r 0 z P The Fresnel approximation involves setting: r 0 z in the denominator, and r 0 z -- x + ------------------ in exponent z + -- ------------------- y z This is equivalent to the paraxial approximation in ray optics. Uxy --------------------- expjkz jk d U ---- x + y jz d exp z (A) Let s examine the validity of the Fresnel approximation in the Fresnel integral. The next higher order term in exponent must be small compared to. So the valid range of the Fresnel approximation is: z 3 ----- x + y» max 4 For field sizes of cm, 0.5m, we find z» 5 cm. Actually we should look at the effect on the total integral. Upon closer analysis, it is found that the Fresnel approximation holds for a much closer z. This is referred to as the near-field region. Farther out in z, we can approximate the quadratic phase as flat 9 Jeffrey Bokor, 000, all rights reserved

k + z» ---------------------------------- max This region is referred to as the far-field or Fraunhofer region. Uxy e jkz j---- k x + y z e ----------------------------------- d U j ----- jz d exp x + y z FU Now this is exactly the Fourier transform of the aperture distribution with f x x ----- y f z y ----- z The Fraunhofer region is farther out. For the field size of cm, and range of z» 50 meters! Again, examining the full integral, Fraunhofer is actually accurate and usable to much closer distances. 0.5m, we find the valid Examples A rectangular aperture, illuminated by a normally incident plane wave: w y w x t A rect-------- rect -------- w x w y With plane wave illumination, we have: U t A Ux y z e jkz j---- k x + y z e ----------------------------------- FU jz f x f y ----- x z ----- y z 93 Jeffrey Bokor, 000, all rights reserved

jk z x y + ---------------- + z e ------------------------------- Asinc w x ----------- x sinc w y ------------ y jz z z A 4w x w y Recall. The observable is intensity I U. I A ----------sinc w x x z ----------- sinc w y ------------ y z z The width of the central lobe of the diffraction pattern is The diffraction half angle x ----- -------- w x x For a circular aperture with radius w : t A circ q w --- q + radial coordinates In circular coordinates, we use the Fourier - Bessel transform: B{U(q)} gives immediately: Ir A ----- J kwr z z ------------------------- kwr z Airy pattern d Airy disk area inside d diameter of Airy disk 94 Jeffrey Bokor, 000, all rights reserved

Diffraction grating (transmission) w sinusoidal amplitude w t A m: peak to peak amplitude change f 0 : grating spatial frequency -- m + --- cosf 0 rect------ rect ------ w w 0 m t A L grating period L --- f 0 By convolution, the diffracted amplitude is F -- m + --- cosf 0 F rect------rect ------ w w I(x) z ----- w f 0 z f 0 z x A Ixy -------- sinc wy --------- sinc --------- wx m z z + ------sinc ------ w x + f z 4 z 0 z m ------sinc w + ------ x f 4 z 0 z We have neglected interference terms between orders. Compared to the square aperture, which has the central peak with intensity I o, we now have: 95 Jeffrey Bokor, 000, all rights reserved

The resolving power of the grating DIFFRACTION THEORY OF A LENS We have previously seen that light passing through a lens experiences a phase delay given by: jkn ----------------- + exp ----- R The focal length, f is given by: ----- R (neglecting the constant phase) The lens makers formula The transmission function is now: 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 f behind the lens (f positive) or diverging from the point at f in front of lens (f negative). f f Diffraction from the lens pupil Suppose the lens is illuminated by a plane wave, amplitude A. The lens pupil function is P. The full effect of the lens is U l ' P 96 Jeffrey Bokor, 000, all rights reserved

We now use the Fresnel formula to find the amplitude at the back focal plane z f U f x y exp j---- k x + y f ------------------------------------------- e jkf jf The phase terms that are quadratic in U f x y U l ' P exp j---- k + f d + du l ' exp j---- k + exp j ----- x + y f f cancel each other. exp j---- k x + y f ------------------------------------------- e jkf ddp exp j ----- x + y jf f This is precisely the Fraunhofer diffraction pattern of P! Note that a large z criterion does not apply here. The focal plane amplitude distribution is a Fourier transform of the lens pupil function P(), multiplied by a quadratic phase term. However, the intensity distribution is exactly (B) I f x y A --------- f FP f x ---- x f f y ---- y f Example: a circular lens, with radius w w z P circ q w --- q + let hr FPz q F circ z ----------- q r x + y w A ------- J wr z ------------------------------------ z wr z hr ---------- A J wr z ------------------------------------ z wr z 97 Jeffrey Bokor, 000, all rights reserved