EE 430.43.00 06. nd Semester Chapter 5. Diffraction Part 06. 0. 0. Changhee Lee School of Electrical and Computer Engineering Seoul National niv. chlee7@snu.ac.kr /7 Changhee Lee, SN, Korea
5.5 Fresnel diffraction patterns iωt ik r r' ik 0e e [ cos n, r cos n, r ']da 4π rr' P EE 430.43.00 06. nd Semester r r' h R / h' R / h h' R h... h' Fresnel zones: regions bounded by concentric circles, Rconstant, defined such that rr differs by λ/ from one boundary to the next. R λl, R λl,... R nλl n L h h' If R n and R n are the inner and outer radii of the nst zone, the area is π R πr πr λl, independent of n. n n /7 Changhee Lee, SN, Korea
5.5 Fresnel diffraction patterns EE 430.43.00 06. nd Semester The optical disturbance at P is the sum of the contributions from the various Fresnel zones. Since the mean phase changes by exactly 80 o from one zone to the next, p 3... For the case of an infinitely large aperture no aperture at all, the total optical disturbance at P is ½. p 3 4 5 3... n 3/7 Changhee Lee, SN, Korea
5.5 Fresnel diffraction patterns EE 430.43.00 06. nd Semester In the case of an irregular obstacle, If P is in the illuminated region, the presence of the obstacle makes little difference, If it is in the shadow region, the optical disturbance is very small, roughly in agreement with geometrical optics. Diffraction fringes appear around the shadow only if the irregularities at the edge of the obstacle are small compared to the radius of the st Fresnel zone. 4/7 Changhee Lee, SN, Korea
Zone plate EE 430.43.00 06. nd Semester If an aperture is constructed so as to obstruct alternate Fresnel zones, say the evennumbered ones, then the remaining terms in the summation are all of the same sign. Such an aperture is called a zone plate. p 3 5... It is much like a lens, because p is much larger than if there were no aperture. The equivalent focal length is L as given by L R λ st Fresnel zone R λl 5/7 Changhee Lee, SN, Korea
Rectangular aperture iωt ik r r' ik 0e e [ cos n, r cos n, r ']da 4π rr' P EE 430.43.00 06. nd Semester R x y r r' h h' L x y Simplifying assumptions: The obliquity factor and /rr vary so slowly compared to e ikrr /r that they can be taken outside the integral. P C x x y y e ik x y / L dxdy C x x e ikx / L dx y y e iky / L dy 6/7 Changhee Lee, SN, Korea
Rectangular aperture EE 430.43.00 06. nd Semester u x k πl x λl v y k πl y λl P u v iπu / iπv / e dx e dy u, v CπL k s 0 e iπw / dw C s is s Fresnel integrals s cos πw / dw, S s C s sin πw 0 s 0 / dw dc s cos π w / ds, ds s sin πw / ds dc ds ds 7/7 Changhee Lee, SN, Korea
Cornu spiral, a plot of the Fresnel integrals The Cornu spiral is useful for graphical evaluation of the Fresnel integrals. The limit points s and s are marked on the spiral. A straight line segment drawn from s to s gives the value of the integral s i w / e π dw s EE 430.43.00 06. nd Semester The length of the line segment is the magnitude of the integral, and the projections on the C and S axes are the real and imaginary parts, respectively. ds represents an element of arc. dc ds ds 8/7 Changhee Lee, SN, Korea
Changhee Lee, SN, Korea EE 430.43.00 06. nd Semester 9/7 Cornu spiral, a plot of the Fresnel integrals L y y v v s s L x x u u s s λ λ, i S C S C [ ] [ ] 0 v v u u p v is v C u is u C i For the general case in the normalized form
Slits and straightedge EE 430.43.00 06. nd Semester p i [ C v is v ] v 0 v p i i v 0 [ C v is v ] C v is v i 0 F. A. Jenkins and H. E. White, Fundamentals of Optics, 3 rd ed. McGraw-Hill, 957 0/7 Changhee Lee, SN, Korea
Changhee Lee, SN, Korea EE 430.43.00 06. nd Semester /7 Straightedge If the receiving point P is exactly at the geometrical shadow edge, then v 0. [ ] 0 0 0 0 i i i v is v C i v is v C i p v p The highest irradiance occurs just inside the illuminated region at v ~.5, where I p ~.37I 0.
Narrow slits and opaque narrow strips Photographs of a number of Fresnel diffraction patterns for single slits of different widths. As the slit becomes wider, the fringes go through very rapid changes, approaching for a wide slit the general appearance of two opposed straight-edge diffraction patterns. EE 430.43.00 06. nd Semester Babinet's principle is not very useful in dealing with Fresnel diffraction. In Fraunhofer diffraction, the diffraction patterns due to complementary screens are identical. In a typical case of Fresnel diffraction, however, this is not true, as may be seen by comparing two Figs. Fresnel diffraction by narrow opaque strips. F. A. Jenkins and H. E. White, Fundamentals of Optics, 3 rd ed. McGraw-Hill, 957 /7 Changhee Lee, SN, Korea
Fresnel diffraction from a slit EE 430.43.00 06. nd Semester The diffraction pattern from a slit for different Fresnel numbers N F a /λd. corresponding to different distances d from the aperture. At very small distances very large N F, the diffraction pattern is a perfect shadow of the slit. As the distance increases N F decreases, the wave nature of light is exhibited in the form of small oscillations around the edges of the aperture. For very small N F, the Fraunhofer pattern is obtained. This is a sinc function with the first zero subtending an angle λ/d λ/a. Bahaa E. A. Saleh, Malvin Carl Teich, Fundamentals of Photonics 99 3/7 Changhee Lee, SN, Korea
Fresnel integrals e i π X / cos πx / i sin πx / EE 430.43.00 06. nd Semester Bahaa E. A. Saleh, Malvin Carl Teich, Fundamentals of Photonics 99 4/7 Changhee Lee, SN, Korea
5.6 Applications of the Fourier transform to diffraction EE 430.43.00 06. nd Semester Now we consider the general problem of Fraunhofer diffraction by an aperture having not only an arbitrary shape, but also an arbitrary transmission including phase retardation, which may vary over different parts of the aperture. 5/7 Changhee Lee, SN, Korea
5.6 Applications of the Fourier transform to diffraction Path difference δr R ix ˆ ˆjy, δr R nˆ P ~ e ikδr nˆ iˆ α xα yβ x da e ˆj β kˆ γ, X L ik xx yy / L Y y L α, β, γ direction cosines, dxdy L focal length of the lens EE 430.43.00 06. nd Semester 6/7 Changhee Lee, SN, Korea
5.6 Applications of the Fourier transform to diffraction For a nonuniform aperture we introduce an aperture function gx,y. X, Y µ, ν g x, y e g x, y e ik xx yy / L i µ x νy dxdy, dxdy µ kx L Spatial frequency, ν ky L Diffraction pattern is a Fourier resolution of the aperture function. g y g g cos ν 0 y g cosν 0 y..., ν 0 0 π h EE 430.43.00 06. nd Semester 7/7 Changhee Lee, SN, Korea
Apodization EE 430.43.00 06. nd Semester Apodization literally to remove the feet is any process by which the aperture function is altered in such a way as to produce a redistribution of energy in the diffraction pattern. It is an optical filtering technique, primarily used to remove Airy disks caused by diffraction around an intensity peak, improving the focus. Consider a single slit. gy for b/ <y< b/ and gy0 otherwise. b sin νb iνy e dy b b νb Suppose now that aperture function is altered by apodizing in such a way that the resultant aperture transmission is a cosine function: g y cos πy / b for b / < y < b / Apodization suppresses the higher spatial frequencies. In this way, it is possible to apodize the circular aperture of a telescope so as to reduce greatly the relative intensities of the diffraction rings that appear around the images of stars. This enhances the ability of the telescope to resolve the image of a dim star near that of a bright one. b b πy cos e b iνy dy νb cos ν π / b ν π / b 8/7 Changhee Lee, SN, Korea
Spatial filtering EE 430.43.00 06. nd Semester The xy plane represents the location of some coherently illuminated object. This object is imaged by an optical system, the image appearing in the x y plane. The diffraction pattern µ,ν of the object function gx,y appears in the µν plane. µ, ν g x, y e i µ x νy dxdy µ,ν is the Fourier transform of gx,y. The image function g x,y that appears in the x y plane is, in turn, the Fourier transform of µ,ν. 9/7 Changhee Lee, SN, Korea
Spatial filtering ' µ, ν T µ, ν µ, ν EE 430.43.00 06. nd Semester The finite size of the aperture at the µν plane limits the spatial frequencies that are transmitted by the optical system. And there are lens defects, aberrations, etc., which result in a modification of the function µ,ν. All of these effects can be incorporated into the transfer function Tµ,ν of the optical system, defined as follows: g' x', y' T µ, ν µ, ν e i µ x' νy' dµ dν The image function g x,y is the Fourier transform of the product of Tµ,ν µ,ν. The transfer function can be modified by placing various screens and apertures in the µν plane. This is known as spatial filtering. 0/7 Changhee Lee, SN, Korea
Spatial filtering EE 430.43.00 06. nd Semester Low-pass spatial filtering /7 Changhee Lee, SN, Korea
Spatial filtering EE 430.43.00 06. nd Semester High-pass spatial filtering /7 Changhee Lee, SN, Korea
Phase contrast and Phase gratings EE 430.43.00 06. nd Semester The method of phase contrast was invented by Zernike, and it is used to render visible a transparent object whose index of refraction differs slightly from that of a surrounding transparent medium. Phase contrast is particularly useful in microscopy for examination of living organisms. This method consists of the use of a special type of spatial filter. For example, consider a phase grating consisting of alternate strips of high- and low-index material, all strips being perfectly transparent. g y e iφ y iφ y ν b / b / e iνy ν i iφ y e dy i b / b / ν iνy dy φ y e iνy dy and are 80 o out of phase. The phase-contrast method is inserting a phase plate which shifts the phase of i by an additional 90 o. 3/7 Changhee Lee, SN, Korea
Phase contrast and Phase gratings EE 430.43.00 06. nd Semester The phase plate is just a transparent-glass plate having a small section whose optical thickness is λ/4 greater than the remainder of the plate. This thicker section is located in the central part of the µν plane, that is, in the region of low spatial frequencies. After inserting phase plate, ν i ν ν ν iνy' iνy' g' y' ν e dν ν e dν g y' g y' The g is the image function of the whole object aperture. It represents the constant background. The g the image function for a regular grating of alternate transparent and opaque strips. Thus, the phase grating has been rendered visible. It appears in the image plane as alternate bright and dark strips. 0 phase shift of 90 to the carrier freq. phase - modulated signal amplitude- modulated signal 4/7 Changhee Lee, SN, Korea
5.7 Reconstruction of the wave front by diffraction, Holography EE 430.43.00 06. nd Semester Holography is the science and practice of making holograms. Dennis Gabor was awarded the Nobel Prize in Physics in 97 "for his invention and development of the holographic method. Typically, a hologram is a photographic recording of a light field, rather than of an image formed by a lens, and it is used to display a fully three-dimensional image of the holographed subject, which is seen without the aid of special glasses or other intermediate optics. The hologram is an encoding of the light field as an interference pattern in the photographic medium. When suitably lit, the interference pattern diffracts the light into a reproduction of the original light field and the objects that were in it appear to still be there, exhibiting visual depth cues such as parallax and perspective that change realistically with any change in the relative position of the observer. Recording a hologram Reconstructing a hologram https://en.wikipedia.org/wiki/holography 5/7 Changhee Lee, SN, Korea
5.7 Reconstruction of the wave front by diffraction, Holography EE 430.43.00 06. nd Semester 6/7 Changhee Lee, SN, Korea
EE 430.43.00 06. nd Semester Homework set #4. Due date: 06.. 3 목 Problems in Chapter 5., 4, 8,, 5, 9, 0,. Midterm Exam 06.. 8 화 :00-:5 시험범위 : Chapter 3~ Chapter 5 sec. 3.~3.5 제외 7/7 Changhee Lee, SN, Korea