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Fraunhofer Diffraction Last Lecture • Dichroic Materials • Polarization by Scattering • Polarization by Reflection from Dielectric Surfaces • Birefringent Materials • Double Refraction • The Pockel’s Cell This Lecture • Fraunhofer versus Fresnel Diffraction • Diffraction from a Single Slit • Beam Spreading • Rectangular and Circular Apertures Optical Diffraction • Diffraction is any deviation from geometric optics that results from the obstruction of a light wave, such as sending a laser beam through an aperture to reduce the beam size. Diffraction results from the interaction of light waves with the edges of objects. • The edges of optical images are blurred by diffraction, and this represents a fundamental limitation on the resolution of an optical imaging system. • There is no physical difference between the phenomena of interference and diffraction, both result from the superposition of light waves. Diffraction results from the superposition of many light waves, interference results from the interference of a few light waves. Optical Diffraction Hecht, Optics, Chapter 10 Fraunhofer versus Fresnel Diffraction • The passage of light through an aperture or slit and the resulting diffraction patterns can be analyzed using either Fraunhofer or Fresnel diffraction theory. In Fraunhofer (far-field) diffraction theory the source is far enough from the aperture that the wavefronts are planar at the aperture, and the image plane is far enough from the aperture that the wavefronts are planar at the image plane. • If the curvature of the optical waves must be taken into account at the aperture or image plane, then we must use Fresnel (near-field) diffraction theory. • The Huygens-Fresnel principle is used in diffraction theory, in that every point of a given wavefront of light can be considered as a source of secondary wavelets. To analyze two-slit interference, we assumed that the slits were point sources. To analyze diffraction, we need to consider the generation of wavelets at different spatial positions within the slit. Fraunhofer versus Fresnel Diffraction • We can move to the Fraunhofer regime by placing lenses on each side of the aperture. Lens 1 is place a focal length away from the point source so that the wavefronts are planar at the aperture. The observation screen is in the focal plane of lens 2 so that the diffraction pattern is imaged at infinity. Fraunhofer Diffraction from a Single Slit • Consider the geometry shown below. Assume that the slit is very long in the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction. Fraunhofer vs. Fresnel diffraction • In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away) • If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction S P 7 Hecht 10.2 Hecht 10.3 Fraunhofer Vs. Fresnel Diffraction Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it d '2 h d 2 h d '2 h'2 d 2 h 2 2 2 1 h' 2 1 h 2 1 h' 2 1 h 2 d 1 d ' 1 d 1 d ' 1 2 d' 2 d 2 d' 2 d 11 1 2 h' h 2 d d' d' d 8 Fraunhofer diffraction limit • If aperture is a square - X • The same relation holds in azimuthal plane and 2 ~ measure of the area of the aperture • Then we have the Fraunhofer diffraction if, 2 d or , d area of aperture Fraunhofer or far field limit 9 Fraunhofer, Fresnel limits • The near field, or Fresnel, limit is 2 d 10 Fraunhofer Diffraction from a Single Slit The contribution to the electric field amplitude at point P due to the wavelet emanating from the element ds in the slit is given by dE dEP 0 exp i kr t r Let r r0 for the source element ds at s 0. Then for any element dE0 dEP exp i k r0 t r0 We can neglect the path difference in the amplitude term, but not in the phase term. We let dE0 EL ds, where EL is the electric field amplitude, assumed uniform over the width of the slit . The path difference s sin . Substituting we obtain E ds dEP L exp i k r0 s sin t r0 E EP L exp i kr0 t r0 exp i k s sin E EP L exp i kr0 t i k sin r0 b / 2 b/2 Integrating we obtain b/2 b / 2 exp i k s sin ds Fraunhofer Diffraction from a Single Slit Evaluating with the integral limits we obtain exp i exp i E EP L exp i kr0 t i k sin r0 where 1 k b sin 2 Rearranging we obtain E b EP L exp i kr0 t exp i exp i 2i r0 E E b b sin L exp i kr0 t 2 i sin L exp i kr0 t 2i r0 r0 The irradiance at point P is given by 2 E b sin 2 1 1 sin 2 I = 0 c EP EP* 0 c L I I 0 sinc 2 0 2 2 2 2 r 0 Fraunhofer Diffraction from a Single Slit The irradiance at point P is given by 1 1 I = 0 c EP EP* 0 c 2 2 2 EL b sin 2 sin 2 I0 I 0 sinc 2 2 2 r0 The sinc function is 1 for 0, lim sinc lim 0 0 The zeroes of irradiance occur when sin 0, or when sin 1 1 k b sin m 2 m 1, 2, Fraunhofer Diffraction from a Single Slit In terms of the length y on the observation screen, y f sin , and in terms of wavelength 2 / k , we can write 1 2 y by b 2 f f Zeroes in the irradiance pattern will occur when by m f y m f b The maximum in the irradiance pattern is at β = 0. Secondary maxima are found from d sin cos sin cos sin 0 d 2 2 sin tan cos Fraunhofer Diffraction from a Single Slit Note: x- and y-axes switched in book, Figs. 165a (here) and Fig. 16-1 do not match. Beam Spreading The angular width of the central maximum is defined as the angular separation Δθ between the first minima on either side of the central maximum, y sin f The first min ima in the irradiance pattern will occur when y 1 f m f b b Δθ 2 b The width W of the diffraction pattern thus increases linearly with distance from the slit, in the regions far from the slit where Fraunhofer diffraction applies W = L Δθ 2L b Rectangular Apertures When the length a and width b of the rectangular aperture are comparable, a diffraction pattern is observed in both the x - and y - dimensions, governed in each dimension by the formula we have already developed. The irradiance pattern is x I I 0 sinc 2 sinc 2 where y 1 k a sin 2 Zeroes in the irradiance pattern are observed when y m f b or x m f a Square Apertures Fraunhofer Diffraction from General Apertures For the general aperture dEP EA exp i t kr dA r where dA dy dz 1/ 2 2 2 r X 2 Y y Z z 1/ 2 R X 2 Y 2 Z 2 Fraunhofer Diffraction from General Apertures We can combine the relations for r and R to obtain r R 1 y 2 z2 R2 1/ 2 2 Yy Zz R2 In the far field R is very large compared to the aperture dimensions, and y we can neglect the term 2 Yy Zz r R 1 R2 1/ 2 2 + z2 R2 , and we can write Yy Zz R Yy Zz R 1 R2 R Therefore, the total electric field at point P is given by E EP A exp i t kR exp ik Yy Zz dA R aperture EA exp i t kR exp ik Yy Zz dy dz R aperture Fraunhofer Diffraction from Circular Apertures Now we specialize to a circular aperture of radius a. At this point we switch to cylindrical coordinates z cos y sin Z q cos Y q sin dA d d Substituting into our expression for E P we obtain EP a 2 ik q sin sin q cos cos EA exp i t kR exp d d 0 0 R R a 2 EA i k q exp i t kR exp cos d d 0 0 R R Fraunhofer Diffraction from Circular Apertures Because of symmetry our solution will be the same for any angle . Choosing Φ = 0, we obtain EP But a 2 EA i k q exp i t kR exp cos d d 0 0 R R 2 J 0 (u ) 0 kq exp i cos d R 1 2 v 2 v 0 is in the form of a Bessel function. exp i u cos v dv is a Bessel function of the order zero. We can write EP a EA k q exp i t kR 2 J 0 d 0 R R In general , a Bessel function of the order m is given by im J m (u ) 2 v 2 v 0 exp i mv u cos v dv Fraunhofer Diffraction from Circular Apertures A useful recurrence relation for Bessel functions is d u m J m u u m J m 1 u du When m = 1, we can integrate the expression to find u 0 J 0 u du u J1 u Define w R k q , then d dw gives us R kq a R E E k q EP A exp i t kR 2 J 0 d A exp i t kR 2 0 R R R kq From the recurrence relation we obtain EP R kaq EA exp i t kR 2 a 2 J1 R kaq R The irradiance is given by 1 I c 0 EP EP* 2 2 E A 2 R kaq J1 A R kaq R 2 2 2 w kaq / R 0 J 0 w w dw Fraunhofer Diffraction from Circular Apertures Assuming that R is essentially constant over the observation screen and recognizing that sin q / R, we can write the irradiance as 2 J kaq R 2 J1 k a sin I I 0 1 I 0 kaq R k a sin 2 where we have used the relation that lim u 0 J1 u 1 u 2 recognizing that u 0 when q 0. 2 Fraunhofer Diffraction from Circular Apertures: Bessel Functions Fraunhofer Diffraction from Circular Apertures: The Airy Pattern First minimum in the Airy pattern is at 2 D k a sin k a 3.83 min 2 min 1.22 D where D 2a. Circular Apertures