*DiffractionSOLO HERMELINUpdated: 16.01.10 4.01.15http://www.solohermelin.com

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*Table of Content SOLOOptics - Diffraction

*SOLODiffractionPhase Approximations Fraunhofer (Near-Field) ApproximationAppendices

*SOLODiffraction The term Diffraction has been defined by Sommerfield as any deviation of light rays from rectilinear paths which cannot be interpreted as reflection or refraction.Sommerfeld, A., Optics, Lectures on Theoretical Physics, vol. IV, Academic Press Inc.,New York, 1954, Chapter V, The Theory of Diffraction, pg. 179, english translation ofSommerfeld, A. , Mathematische Theorie der Diffraction, Math Ann., 1896

*SOLODiffraction - History The Grimaldis description of diffraction was published in1665 , two years after his death: Physico-Mathesis de lumine,Coloribus et iride Francesco M. Grimaldi, S.J. (1613 1663) professor of mathematics and physics at theJesuit college in Bolognia discovered the diffraction of light and gave it the namediffractio, which means breaking up.http://www.faculty.fairfield.edu/jmac/sj/scientists/grimaldi.htm When the light is incident on a smooth white surface it will show an illuminated base IK notable greater than the rays would make which are transmitted in straight lines through the two holes. This is proved as often as the experiment is trayed by observing how great the base IK is in fact and deducing by calculation how great the base NO ought to bewhich is formed by the direct rays. Further it should not beomitted that the illuminated base IK appears in the middlesuffused with pure light, and either extremity its light iscolored.Single SlitDiffractionDouble SlitDiffractionhttp://en.wikipedia.org/wiki/Francesco_Maria_Grimaldi

*SOLOHuygens Principle Christiaan Huygens1629-1695 Every point on a primary wavefront serves the source of spherical secondary wavelets such that the primary wavefront at some later time is the envelope o these wavelets. Moreover, the wavelets advance with a speed and frequency equal to that of the primary wave at each point in space. We have still to consider, in studying the spreading of these waves, that each particle of matter in which a wave proceeds not only communicates its motion to the next particle to it, which is on the straight line drawn from the luminous point, but it also necessarily gives a motion to all the other which touch it and which oppose its motion. The result is that around each particle there arises a wave of which this particle is a center.Huygens visualized the propagation of light in terms of mechanical vibration of an elastic medium (ether). Diffraction - History

Wavefront

Sources

*SOLOJames Gregory (1638 1675) a Scottish mathematician and astronomer professor at the University of St. Andrews and theUniversity of Edinburgh discovered the diffraction grating by passing sunlight through a bird feather and observing the diffraction produced. Diffraction - Historyhttp://en.wikipedia.org/wiki/James_Gregory_%28astronomer_and_mathematician%29http://microscopy.fsu.edu/optics/timeline/people/gregory.html1661

*Diffraction - HistorySOLOM.C. Hutley, Diffraction Gratings, Academic Press., 1982, p. 3Diffraction Gratings1786 The invention of Diffraction Gratings is ascribed to David Rittenhouse who in 1786had been intriged by the effects produced when viewing a distant light source througha fine handkerchief. In order to repeat the phenomenon under controlled conditions, he made up a square of parallel hairs laid across two fine screws made by a watchmaker. When he looked through this at a small opening in the window shutter of a darkened room, he saw three images of approximately equal brightness and several others on either side fainter and growing more faint, coloured and indistinct, the further they were from the main line. He noted that red light was bent more than blue light and ascribed these effects to diffraction.http://experts.about.com/e/d/da/david_rittenhouse.htm

*Optics - HistorySOLOHistory (continue) In 1801 Thomas Young uses constructive and destructive interference of waves to explain the Newtons rings.1801 - 1803 In 1803 Thomas Young explains the fringes at the edges of shadows using the wave theory of light. But, the fact that was belived that the light waves are longitudinal, mad difficult the explanation of double refraction in certain crystals.

*SOLOBetween 1805 and 1815 Laplace, Biot and (in part) Malus created an elaborate mathematical theory of light, based on the notion that light rays are streams of particles that interact with the particles of matter by short range forces. By suitably modifying Newtons original emission theory of light and applying superior mathematical methods, they were able to explain most of the known optical phenomena, including the effect of double refraction which had been the focus of Huyghens work. Diffraction - Historyhttp://microscopy.fsu.edu/optics/timeline/people/gregory.htmlhttp://www.schillerinstitute.org/fid_97-01/993poisson_jbt.htmlIn 1817, expecting to soon celebrate the final triumph of their neo-Newtonian opticsLaplace and Biot arranged for the physics prize of the French Academy of Science to be proposed for the best work on theme of diffraction the apparent bending of light rays at the boundaries between different media.

*SOLOFraunhofers solar dark lines In 1813 Joseph Fraunhofer rediscovered William Hyde Wollastons dark lines in the solar system, which are known as Fraunhofers lines.He began a systematic measurement of the wavelengths of the solar Spectrum, by mapping 570 lines.Diffraction - Historyhttp://www.musoptin.com/spektro1.html1813 Fraunhofer placed a narrow slit in front of a prism and viewed the spectrum of lightpassing through this combination with a small telescope eypiece. By this technique he was able to investigate the spectrum bit by bit, color by color.

*POLARIZATION - History Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that tworays polarized at right angles to each other never interface.SOLOHistory (continue) Arago relayed to Thomas Young in London the resultsof the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillationsin the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed.1816 - 1817

a - M. Born, E. Wolf,"Principles of Optics", pp. xxiiiM.V. Klein, T. Furtak,"Optics", pp.35

*SOLO In 1818 Fresnel, by using Huygens concept of secondary wavelets and Youngs explanation of interface, developed the diffraction theory of scalar waves.1818Diffraction - History

*SOLO In 1818 August Fresnel supported by his friend Andr-Marie Ampre submitted to the French Academy a thesis in which he explained the diffraction by enriching the Huyghens conception of propagation of light by taking in account of the distinct phases within each wavelength and the interaction (interference) between different phases at each locus of the propagation process. Diffraction - Historyhttp://microscopy.fsu.edu/optics/timeline/people/gregory.htmlhttp://www.schillerinstitute.org/fid_97-01/993poisson_jbt.htmlJoseph Louis Guy-Lussac1778-1850JudgingCommitteeofFrenchAcademy 1818

*SOLODiffraction - Historyhttp://microscopy.fsu.edu/optics/timeline/people/gregory.htmlhttp://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html Simon Denis Poisson an Academy member rise the objection that if the Fresnel construction is valid a bright spot would have to appear in the middle of the shadow cast by a spherical or disc-shaped object, when illuminated, and this is absurd. Soon after the meeting, Dominique Francois Arago, one of the judges for the Academy competition, did the experiment and there was the bright spot in the middle of the shadow. Fresnel was awarded the prize in the competition. Poissons or Aragos Spot

*SOLO In 1821 Joseph Fraunhofer build the first diffraction grating, made up of 260 close parallel wires. Latter he built a diffractiongrating using 10,000 parallel lines per inch.Diffraction - HistoryUtzshneider, Fraunhfer, Reichenbach, Mertzhttp://www.musoptin.com/fraunhofer.html1821 - 1823 In 1823 Fraunhofer published his theory of diffraction.

*SOLODffraction Grating Diffraction - History1835By 1835 at the latest, the physicist F. M. Schwerd was able to take exact measurements of the visible spectrum with the aid of such a diffraction grating, and show that red light has a longer wavelength than blue light, and that yellow and blue light lie in the middle of the spectrum. http://colorsystem.com/projekte/engl/16haye.htm1835 - Schwerd developed a "wave" theory of the diffraction grating. http://www.thespectroscopynet.com/Educational/Masson.htmDie Beugungserscheinungen aus den Fundamentalgesetzen der Undulationstheorie http://www.worldcatlibraries.org/wcpa/ow/3e723a9c5ac2a2b2.htmlhttp://193.174.156.247/FMSG/wir_ueber_uns/wer_war_Schwerd.php

*SOLO H.A. Rowland at the John Hopkins University greatly improved diffraction gratings,introducing curved grating.Diffraction - History1882http://thespectroscopynet.com/educational/Kirchhoff.htm American physicist who invented the concave diffraction grating, which replaced prisms and plane gratings in many applications, and revolutionized spectrum analysisthe resolution of a beam of light into components that differ in wavelength. http://www.britannica.com/eb/article-9064251/Henry-Augustus-Rowlandhttp://chem.ch.huji.ac.il/~eugeniik/history/rowland.htmlRowland gratings Rowland invented the ruling machine that can engrave as manyas 20,000 lines to inch for diffraction gratings

*SOLOOptics History Debye-Sears Effect1932Diffraction of light by ultrasonic waves.P. Debye and F. W. Sears, ``On the Scattering of Light by Supersonic Waves'', Proc. Natl. Acad. Sci. U.S.A. 18, 409 (1932). Acousto-optic effect, also known in the scientific literature as acousto-optic interaction or diffraction of light by acoustic waves, was first predicted by Brillouin in 1921 and experimentally revealed by Lucas, Biquard and Debye, Sears in 1932. The basis of the acousto-optic interaction is a more general effect of photoelasticity consisting in the change of the medium permittivity under the action of a mechanical strain a. Phenomenologically, this effect is described as variations of the optical indicatrix coefficients caused by the strain http://www.mt-berlin.com/frames_ao/descriptions/ao_effect.htm

*DiffractionSOLO In 1818 Fresnel, by using Huygens concept of secondary wavelets and Youngs explanation of interface, developed the diffraction theory of scalar waves. According to Huygens Principle second wavelets will start at the aperture and will add at the image point P. Obliquity factor and /2 phase were introduced by Fresnel to explain experiences results.Fresnel Diffraction FormulaFresnel took in consideration the phase of each wavelet to obtain:Fresnel Huygens Diffraction Theory

*SOLOFresnel-Kirchhoff Diffraction Theory In 1882 Gustav Kirchhoff, using mathematical foundation, succeeded to show that the amplitude and phases ascribed to the wavelets by Fresnel, by enhancing the Huyghens Principle, were a consequence of the wave nature of light.For Source lessMedium Maxwell Equations areDiffraction

*Diffraction SOLOFresnel-Kirchhoff Diffraction TheoryDefine the phasor Phasor Scalar Differential Wave EquationThis is the Scalar Helmholtz Differential EquationBoundary Conditions for the Helmholtz Differential Equation: Dirichlet (U given on the boundary) Neumann (dU/dn given on the boundary)

* To find the solution of the Scalar Helmholtz Differential Equation we need to use the following: Scalar Greens Identity Greens Function This Greens Function is a particular solution of the following Helmholtz Non-homogeneous Differential Equation:SOLOFresnel-Kirchhoff Diffraction TheoryDiffraction Free-Space Greens Function

*SOLO Scalar Greens IdentitiesLet start from the Gauss Divergence TheoremThenSubtracting the second equation from the first we obtainFirst Greens IdentitySecond Greens IdentityWe haveFresnel-Kirchhoff Diffraction TheoryDiffraction To find a general solution of the Scalar Helmoltz Differential Equation we need to use the

* Integral Theorem of Helmholtz and Kirchhoff Using:SOLOFresnel-Kirchhoff Diffraction TheoryDiffraction From the left side of the Second Scalar Greens Identity we have: we obtain: Note: This Theorem was developed first by H. von Helmholtz in acoustics.Scalar Helmholtz Differential Equation

* Sommerfeld Radiation ConditionsSOLOFresnel-Kirchhoff Diffraction TheoryDiffraction since the condition that the previous integral be finite is: This is Sommerfeld Radiation Conditions

* Sommerfeld Radiation Conditions (continue)SOLOFresnel-Kirchhoff Diffraction TheoryDiffraction This is Sommerfeld Radiation Conditions This implies that: and the Integral of Helmholtz and Kirchhoff becomes:

*The Kirchhoff Boundary ConditionsSOLOFresnel-Kirchhoff Diffraction TheoryDiffraction Kirchhoff assumed the following boundary conditions:The Integral of Helmholtz and Kirchhoff becomes:Note:However, if the dimensions of the aperture are large relative to the wavelength , the integral agrees well with the experiment.Kirchhoff boundary conditions are not physical since the presence of the screen changes field values on the aperture and on the screen.

*Fresnel-Kirchhoff Diffraction FormulaSOLOFresnel-Kirchhoff Diffraction TheoryDiffraction The Integral of Helmholtz and Kirchhoff:

*Fresnel-Kirchhoff Diffraction FormulaSOLOFresnel-Kirchhoff Diffraction TheoryDiffraction The Integral of Helmholtz and Kirchhoff:

*Fresnel-Kirchhoff Diffraction FormulaSOLOFresnel-Kirchhoff Diffraction TheoryDiffraction Obliquity or Inclination Factor:2.Additional phase /23.The amplitude is scaled by the factor 1/ (not found in Fresnel derivation)

*Reciprocity Theorem of HelmholtzSOLOFresnel-Kirchhoff Diffraction TheoryDiffraction We can see that the Fresnel-Kirchhoff Diffraction Formula is symmetrical with respect to r and rS, i.e. point source and observation point. Therefore we can interchange them and obtain the same relation. This result is called Reciprocity Theorem of Helmholtz.Note:This is similar to Lorentzs Reciprocity Theorem in Electromagnetism.

*Huygens-Fresnel PrincipleSOLOFresnel-Kirchhoff Diffraction TheoryDiffraction The Fresnel Diffraction Formula can be rewritten as:where: The interpretation of this formula is that each pointof a wavefront can be considered as the center of a secondary spherical wave, and those secondary spherical waves interfere to result in the total field, is known as theHuygens-Fresnel Principle.

*SOLODiffraction Consider a diffracting aperture . Suppose that the aperture is divided into two portions 1 and 2 such that = 1 + 2.The two aperture 1 and 2 are said to be complementary.Complementary Apertures. Babinet PrincipleFrom the Fresnel Diffraction Formula: We can see that the result is the added effect of all complimentary apertures. This is known as Babinet Principle. The result can be very helpful when is a very complicatedaperture, that can be decomposed in a few simple apertures.

*SOLODiffraction The Kirchhoff Diffraction Formula is an approximation since for zero field and normal derivative on any finite surface the field is zero everywhere. This was pointed out by Poincare in 1892 and by Sommerfeld in 1894. The first rigorous solution of a diffraction problem was given by Sommerfeld in 1896 for a two-dimensional case of a planar wave incident on an infinitesimally thin,perfectly conducting half plane. This solution is not given here.Sommerfeld, A. : Mathematische Theorie der Diffraction, Math. Ann., 47:317, 1896 translated in english as Optics, Lectures on Theoretical Physics, vol. IV, Academic Press Inc., New York, 1954Rayleigh-Sommerfeld Diffraction Formula

*SOLORayleigh-Sommerfeld Diffraction FormulaDiffraction Let start from the Helmholtz and Kirchhoff Integral:or:We have

*SOLORayleigh-Sommerfeld Diffraction FormulaDiffraction 1. Start from the Helmholtz and Kirchhoff Integral:ChooseThis is Rayleigh-Sommerfeld Diffraction Formula of the first kindwe obtain:

*SOLORayleigh-Sommerfeld Diffraction FormulaDiffraction 2. Start from the Helmholtz and Kirchhoff Integral:ChooseForwe obtain:This is Rayleigh-Sommerfeld Diffraction Formula of the second kind

*SOLODiffraction Start with Fresnel-Kirchhoff (or Rayleigh-Sommerfeld) Diffraction Formula1. If the inclination factor is nearly constant over the apertureExtensions of Fresnel-Kirchhoff Diffraction Theory3. Characterize the aperture by a transfer function to model amplitude or phase changes due to optic system

*SOLODiffraction Phase Approximations Fresnel (Near-Field) ApproximationFresnel Approximation or Near Field Approximation can be used when aperture dimensions are comparable to distance to source rS or image r.Start with Fresnel-Kirchhoff Diffraction FormulaIf the inclination factor is nearly constant over the aperture

*SOLODiffraction Phase Approximations Fraunhofer (Near-Field) ApproximationFraunhofer Approximation or Far Field Approximation can be used when aperture dimensions are very small comparable to distance to source rS or image r.Start with Fresnel-Kirchhoff Diffraction FormulaIf the inclination factor is nearly constant over the aperture

*SOLODiffraction Fresnel and Fraunhofer Diffraction ApproximationsFresnel Approximations at the SourceSpherical wave centered at P0. Lowest order approximation to the phase of a spherical wavefront

*SOLODiffraction Fresnel and Fraunhofer Diffraction ApproximationsFresnel Approximations at the Image planeSpherical wave centered at O. Lowest order approximation to the phase of a spherical wavefront

*SOLODiffraction Fresnel and Fraunhofer Diffraction Approximations (1st way)Fresnel ApproximationFraunhofer ApproximationorIfwe obtainStart with

*SOLODiffraction Fresnel and Fraunhofer Diffraction Approximations (2nd way)Fresnel ApproximationFraunhofer ApproximationIfwe obtainStart with

*SOLODiffraction Fresnel and Fraunhofer Diffraction Approximations

*SOLODiffraction Fraunhofer Diffraction and the Fourier TransformThereforeTwo Dimensional Fourier Transform

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesRectangular ApertureFor a Rectangular ApertureTherefore

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesRectangular Aperture (continue 1)Since U stands for scalar field intensity (E or H), the irradiance I is given bywhere < > is the time average and * is the complex conjugate.ThereforeI (0) is the irradiance at O1 (x1 = y1 = 0).Hecht pg.466

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesSingle Slit ApertureLet substitute in the rectangular aperture 1 0where < > is the time average and * is the complex conjugate.ThereforeI (0) is the irradiance at O1 (x1 = y1 = 0).to obtain the single (vertical) slit diffractionSince U stands for scalar field intensity (E or H), the irradiance I is given by

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesSingle Slit Aperture (continue)I (0) is the irradiance at O1 (x1 = y1 = 0).Define:The extremum of I () is obtained from:The results are given by:The solutions can be obtained graphically as shown in the figure and are:

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesDouble Slit Aperture

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesDouble Slit Aperture (continue -= 1) The factor (sin / )2 that was previously found as the distribution function for a single slit is here the envelope for the interference fringes given by the term cos2.Bright fringes occur for = 0, ,2,The angular separation between fringes is = .

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesDouble Slit Aperture (continue 2)

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesMultiple Slit Aperture The Aperture consists of a large number N of identical parallel slits of width b and separation a.

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesMultiple Slit Aperture (continue 1) The Aperture consists of a large number N of identical parallel slits of width b and separation a.

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesMultiple Slit Aperture (continue 2)

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesMultiple Slit Aperture (continue 2)Sears p.222 Hecht p. 462Sears p.236Interference Irradiation for 1, 2, 3 and 4 slits as function of observation angle.Diffraction Pattern for 1, 2, 3, 4 and 5 slits.

*SOLOResolution of Optical Systems According to Huygens-Fresnel Principle, a differential area dS, within an opticalAperture, may be envisioned as being covered with coherent secondary point sources.Differential area dS, coordinatesImage , coordinatesThe spherical wave that propagates from dS to Image iswhereThe spherical wave at Image, for a Circular Aperture, is

*SOLOResolution of Optical Systems whereBessel Functions (of the first kind)E. Hecht, Optics The spherical wave at Image, for a Circular Aperture, is

*SOLOResolution of Optical Systems IrradianceE. Hecht, Optics Circular Aperture

*SOLOResolution of Optical Systems Distribution of Energy in the Diffraction Pattern at the Focus of a Perfect Circular LensE. Hecht, Optics

*SOLODiffraction Fraunhofer Diffraction Approximations ExamplesCircular Aperture

*SOLOResolution of Optical Systems Airy Rings In 1835, Sir George Biddell Airy, developed the formula for diffraction pattern, of an image of a point source in an aberration-free optical system, using the wave theory.E. Hecht, Optics

*SOLOResolution of Optical Systems E. Hecht, Optics

*Rayleighs Criterion (1902) The images are said to be just resolved when thecenter of one Airy Disk falls on the first minimum of the Airy pattern of the other image. The minimum resolvable angular separation orangular limit is:Sparrows Criterion At the Rayleighs limit there is a central minimumOr saddle point between adjacent peaks. Decreasing the distance between the two point sources cause the central dip to grow shallower and ultimately to disappear. The angular separationcorresponding to that configuration is the SparrowsLimit.SOLOResolution of Optical Systems

*Resolution Diffraction Limit Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error,University of California, BerkleySOLOResolution of Optical Systems

*Diffraction GratingSOLOResolution of Optical Systems

*Diffraction GratingSOLOResolution of Optical Systems

*Diffraction GratingSOLOResolution of Optical Systems

*SOLODiffraction Fresnel Diffraction Approximations ExamplesRectangular AperturedefineFor a Rectangular ApertureTherefore

*SOLODiffraction Fresnel Diffraction Approximations ExamplesRectangular Aperture (continue 1)Define Fresnel IntegralsUsing the Fresnel Integrals we can write

*SOLODiffraction Fresnel Diffraction Approximations ExamplesRectangular Aperture (continue 2)

*SOLODiffraction Fresnel Diffraction Approximations ExamplesRectangular Aperture (continue 3)

*SOLODiffraction Fresnel Diffraction Approximations ExamplesCornu SpiralFresnel Integrals are defined asMarie Alfred Cornu professor at the cole Polytechnique in Paris established a graphical approach, for calculating intensities in Fresnel diffraction integrals.The Cornu Spiral is defined as the plot of S (u) versus C (u)Therefore u may be thought as measuring arc length along the spiral.Mthode nouvelle pour la discussion des problmes de diffraction dans le cas dune onde cylindrique, J.Phys.3 (1874), 5-15,44-52

*SOLODiffraction Fresnel Diffraction Approximations ExamplesCornu Spiral (continue 1)The Cornu Spiral is defined as the plot of S (u) versus C (u)The radius of curvature of Cornu Spiral is The tangent vector of Cornu Spiral is showing that the curve spirals toward the limit points.

*ELECTROMAGNETICSSOLOKIRCHHOFFs SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION GREENs FUNCTION Define the Greens Function as a particular solution of the following Helmholtz Non-homogeneous Differential Equation:where (x) is the Dirac functionLet use the Fourier Transformation to writewhere

*ELECTROMAGNETICSSOLOKIRCHHOFFs SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION GREENs FUNCTION (continue 1)Let use the Fourier Transformation to write Hence or

*ELECTROMAGNETICSSOLOKIRCHHOFFs SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION GREENs FUNCTION (continue 2) Let compute: Therefore: Because this is true for all k and , we obtain

*ELECTROMAGNETICSSOLOKIRCHHOFFs SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION GREENs FUNCTION (continue 3)To find the integral let change by + j where is a small negative number

*ELECTROMAGNETICSSOLOKIRCHHOFFs SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION GREENs FUNCTION (continue 4) In the plane we close the integration path by the semi-circle withand the singular points on the upper side, for > 0 (for t > t)

*ELECTROMAGNETICSSOLOKIRCHHOFFs SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION GREENs FUNCTION (continue 5) We have: Therefore, we can write:

*ELECTROMAGNETICSSOLOKIRCHHOFFs SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION GREENs FUNCTION (continue 6) Let use spherical coordinates relative to vector:

*ELECTROMAGNETICSSOLOKIRCHHOFFs SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION GREENs FUNCTION (continue 7)

*ELECTROMAGNETICSSOLOKIRCHHOFFs SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION GREENs FUNCTION (continue 8)HenceWe shall consider only the progressive wave and use:The other solution is:

*SOLOThe two dimensional Fourier Transform F of the function f (x, y)The Inverse Fourier Transform is Two Dimensional Fourier TransformTwo Dimensional Fourier Transform (FT)Fraunhofer Diffraction and the Fourier TransformIn Fraunhofer Diffraction we arrived two dimensional Fourier Transform of the field within the aperture Using kx = 2 fx and ky = 2 fy we obtain:Diffractions

*SOLO1. Linearity TheoremTwo Dimensional Fourier Transform (FT)Fourier Transform Theorems2. Similarity TheoremIfthen3. Shift TheoremIfthenDiffractions

*SOLOTwo Dimensional Fourier Transform (FT)Fourier Transform Theorems (continue 1)4. Parsevals TheoremIfthen5. Convolution TheoremIfthenand6. Autocorrelation TheoremIfthensimilarlyDiffractions

*SOLOTwo Dimensional Fourier Transform (FT)Fourier Transform Theorems (continue 2)7. Fourier Integral TheoremDiffractions

*SOLOTwo Dimensional Fourier Transform (FT)Fourier Transform for a Circular Symmetric Optical Aperture To exploit the circular symmetry of g (g (r,) = g (r) ) let make the following transformationUse Bessel Function Identityto obtainJ0 is a Bessel Function of the first kind, order zero.Diffractions

*SOLOTwo Dimensional Fourier Transform (FT)Fourier Transform for a Circular Symmetric Optical Aperture For a Circular Pupil of radius a we haveUse Bessel Function IdentityJ1 is a Bessel Function of the first kind, order one.Bessel Functions of the first kindDiffractions

*SOLOE. Hecht, Optics Circular ApertureTwo Dimensional Fourier Transform (FT)Fourier Transform for a Circular Symmetric Optical ApertureDiffractions

*SOLOReferences Diffractions2.Goodman, J.,W., Introduction to Fourier Optics, McGraw-Hill, 1968 1. Sommerfeld, A., Optics, Lectures on Theoretical Physics, vol. IV, Academic Press Inc., New York, 1954, Chapter V, The Theory of Diffraction, 7. M. Born, E. Wolf, Principle of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th Ed., Pergamon Press, 1980, 5. F.A. Jenkins, H.E. White, Fundamentals of Optics, 4th Ed., McGraw-Hill, 19768. M.V.Klein, T.E. Furtak, Optics, 2nd Ed., John Wiley & Sons, 19866. E. Hecht, A. Zajac, Optics, Addison-Wesley, 19793. Elmore, W.C., Heald, M., A., Physics of Waves, Dover Publications, 1969,4. Fowles, G., R., Introduction to Modern Optics, Dover Publications, 1968, 1975,Ch. 5, Diffraction J. Meyer-Arendt, Introduction to Classical & Modern Optics, 3th Ed., Prentince Hall, 1989

*SOLOReferences [1] M. Born, E. Wolf, Principle of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th Ed., Pergamon Press, 1980, [2] C.C. Davis, Laser and Electro-Optics, Cambridge University Press, 1996, OPTICS

*SOLOReferences [3] E.Hecht, A. Zajac, Optics , 3th Ed., Addison Wesley Publishing Company, 1997, [4] M.V. Klein, T.E. Furtak, Optics , 2nd Ed., John Wiley & Sons, 1986OPTICS

**SOLOTechnionIsraeli Institute of Technology1964 1968 BSc EE1968 1971 MSc EEIsraeli Air Force1970 1974RAFAELIsraeli Armament Development Authority1974 2013Stanford University1983 1986 PhD AA

*SOLODiffraction Fresnel Diffraction Approximations ExamplesCircular Aperture

*SOLODiffraction Fresnel Diffraction Approximations ExamplesCircular Obstacles

*SOLODiffraction Fresnel Diffraction Approximations ExamplesFresnel Zone Plate

*SOLODiffraction Fresnel Diffraction Approximations ExamplesFresnel Diffraction by a Slit Hecht p.504 aFresnel Diffraction

*SOLODiffraction Fresnel Diffraction Approximations ExamplesSemi-Infinite Opaque ScreenHecht p.506 a

*SOLODiffraction Fresnel Diffraction Approximations ExamplesSemi-Infinite Opaque ScreenHecht p.506 aHecht p.507

*SOLODiffraction Fresnel Diffraction Approximations ExamplesDiffraction by a Narrow Obstacle

*Optics - DiffractionSOLOElmore p.409

*Optics - DiffractionSOLO

*Optics - DiffractionSOLOHecht p.500

*Optics - DiffractionSOLOJenkins p.320Jenkins p.323Jenkins p.343Klein Furata p.362

*Optics - DiffractionSOLOReynolds 19Reynolds 20 aReynolds 20 b

*Optics - DiffractionSOLOReynolds 62 aReynolds 62 bReynolds 62 c

*Optics - DiffractionSOLOReynolds 63 aReynolds 63 b

*Goodman, J.W., Introduction to Fourier Optics, McGraw-Hill, 1968, pg. 30******M.C Hutley, Diffraction Gratings, Academic Press, 1982, pg.5http://micro.magnet.fsu.edu/optics/timeline/1867-1899.html

*http://www.mt-berlin.com/frames_ao/descriptions/ao_effect.htm

* http://en.wikipedia.org/wiki/Divergence_theoremDivergence Theorem was discovered by Joseph Louis Lagrange in 1762, rediscoveredby Carl Friedrich Gauss in 1813, by George Green in 1825, by Mikhail Vasilievich Ostrogadsky, who also gave the first proof of the Theorem, in 1831.*Kirchhoff, G., Zur Theorie der Lichtstrahlen, Wiedemann Ann., (2) 18:663 (1883)

*Hecht, Optics, 10.2.4, The Rectangular Aperture, pp. 464 - 467*Hecht, Optics, 10.2.4, The Rectangular Aperture, pp. 464 - 467

*Hecht, Optics, 10.2.4, The Single Slit, pp. 452 - 457

*Hecht, Optics, 10.2.4, The Rectangular Aperture, pp. 464 - 467

*Hecht, Optics, 10.2.2, The Double Slit, pp. 457 460Fowles, Introduction to Modern Optics, 1975, The Double Slit, pp.120 122Elmore, Heald, Physics of Waves, The Double Slit, pp. 365 - 368

*Hecht, Optics, 10.2.2, The Double Slit, pp. 457 460Fowles, Introduction to Modern Optics, 1975, The Double Slit, pp.120 122Elmore, Heald, Physics of Waves, The Double Slit, pp. 365 - 368

*Hecht, Optics, pp. 467 - 471*Reynolds, DeVelis, Parrent, Thompson, Physical Optics Notebook: Tutorials in Fourier Optics, SPIE Optical Engineering Press, pg.102*Elmore, Held, Physics of Waves, 10.8, 10.9, 10.10, pp. 375 391Hecht, Optics, 10.2.8 The Diffraction Grating, pp. 476 485

*Hecht, Optics, 10.2.8 The Diffraction Grating, pp. 476 - 485

*Hecht, Optics, 10.2.8 The Diffraction Grating, pp. 476 - 485

*J.C. Wyne, Optics 513*Reference in J. Meyer-Arendt, Introduction to Classical & Modern Optics, 3th Ed.,Prentince Hall, 1989, pg. 258*Reference in J. Meyer-Arendt, Introduction to Classical & Modern Optics, 3th Ed.,Prentince Hall, 1989, pg. 258J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1968*J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1968*J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1968*J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1968*J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1968*J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1968***Hecht, Optics, pp. 490-494*Hecht, Optics, pp. 494-495*Hecht, Optics, pp. 494-495*Hecht, Optics, 10.3.8, Fresnel Diffraction by a Slit, pp. 503-505*Hecht, Optics, 10.3.9, The Semi-Infinite Opaque Screen, pp. 506-507

*Hecht, Optics, 10.3.9, The Semi-Infinite Opaque Screen, pp. 506-507*Hecht, Optics, 10.3.10, Diffraction by a Narrow Obstacle, pp. 507-508