Diffraction

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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.
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For more presentations on different subjects visit my website at http://www.solohermelin.com. This presentation is in the Optics folder.

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  • Goodman, J.W., “Introduction to Fourier Optics”, McGraw-Hill, 1968, pg. 30
  • M.C Hutley, “Diffraction Gratings”, Academic Press, 1982, pg.5
    http://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_theorem
    Divergence Theorem was discovered by Joseph Louis Lagrange in 1762, rediscovered
    by 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 – 460
    Fowles, “Introduction to Modern Optics”, 1975, “The Double Slit”, pp.120 – 122
    Elmore, Heald, “Physics of Waves”, § The Double Slit”, pp. 365 - 368
  • Hecht, “Optics”, § 10.2.2, “The Double Slit”, pp. 457 – 460
    Fowles, “Introduction to Modern Optics”, 1975, “The Double Slit”, pp.120 – 122
    Elmore, 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 – 391
    Hecht, “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. 258
  • 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
  • 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
  • Diffraction

    1. 1. 1 Diffraction SOLO HERMELIN Updated: 16.01.10 4.01.15 http://www.solohermelin.com
    2. 2. 2 Table of Content SOLO Optics - Diffraction James Gregory Diffraction Grating History of Diffraction Francesco Grimaldi Huygens Principle Thomas Young Interference Laplace, Biot, Poisson, Arago, Fresnel - Particle versus Wave Theory Fresnel – Huygens’ Diffraction Theory Fraunhofer Diffraction Theory Fresnel-Kirchhoff Diffraction Theory Complementary Apertures. Babinet Principle Rayleigh-Sommerfeld Diffraction Formula Extensions of Fresnel-Kirchhoff Diffraction Theory Fresnel and Fraunhofer Diffraction Approximations
    3. 3. 3 SOLO Diffraction Fraunhofer Diffraction and the Fourier Transform Phase Approximations – Fresnel (Near-Field) Approximation Phase Approximations – Fraunhofer (Near-Field) Approximation Fresnel and Fraunhofer Diffraction Approximations Fraunhofer Diffraction Approximations Examples Resolution of Optical Systems Fresnel Diffraction Approximations Examples Kirchhoff’s Solution of the Scalar Helmholtz Nonhomogeneous Differential Equation Two Dimensional Fourier Transform (FT) Appendices
    4. 4. 4 SOLO Diffraction 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 of Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 Sommerfeld, A. , “Mathematische Theorie der Diffraction”, Math Ann., 1896 Table of Content
    5. 5. 5 SOLO Diffraction - History The Grimaldi’s description of diffraction was published in 1665 , 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 the Jesuit college in Bolognia discovered the diffraction of light and gave it the name diffractio, 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 be which is formed by the direct rays. Further it should not be omitted that the illuminated base IK appears in the middle suffused with pure light, and either extremity its light is colored.” Single Slit Diffraction Double Slit Diffraction http://en.wikipedia.org/wiki/Francesco_Maria_Grimaldi Table of Content
    6. 6. 6 SOLO Huygens Principle Christiaan Huygens 1629-1695 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 Table of Content
    7. 7. 7 SOLO James Gregory (1638 – 1675) a Scottish mathematician and astronomer professor at the University of St. Andrews and the University of Edinburgh discovered the diffraction grating by passing sunlight through a bird feather and observing the diffraction produced. Diffraction - History http://en.wikipedia.org/wiki/James_Gregory_%28astronomer_and_mathematician%29 http://microscopy.fsu.edu/optics/timeline/people/gregory.html Table of Content 1661
    8. 8. 8 Diffraction - History SOLO M.C. Hutley, “Diffraction Gratings”, Academic Press., 1982, p. 3 Diffraction Gratings 1786 The invention of Diffraction Gratings is ascribed to David Rittenhouse who in 1786 had been intriged by the effects produced when viewing a distant light source through a 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 David Rittenhouse 1732 - 1796
    9. 9. 9 Optics - History SOLO History (continue) In 1801 Thomas Young uses constructive and destructive interference of waves to explain the Newton’s rings. Thomas Young 1773-1829 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. Table of Content
    10. 10. 10 SOLO “Between 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 Newton’s 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 Huyghen’s work. Diffraction - History http://microscopy.fsu.edu/optics/timeline/people/gregory.html http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html Pierre-Simon Laplace (1749-1827) In 1817, expecting to soon celebrate the final triumph of their neo-Newtonian optics’ Laplace 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.”
    11. 11. 11 SOLO Fraunhofer’s solar dark lines In 1813 Joseph Fraunhofer rediscovered William Hyde Wollaston’s dark lines in the solar system, which are known as Fraunhofer’s lines. He began a systematic measurement of the wavelengths of the solar Spectrum, by mapping 570 lines. Diffraction - History http://www.musoptin.com/spektro1.html 1813 Fraunhofer Telescope. Fraunhofer placed a narrow slit in front of a prism and viewed the spectrum of light passing through this combination with a small telescope eypiece. By this technique he was able to investigate the spectrum bit by bit, color by color.
    12. 12. 12 POLARIZATION - History Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that two rays polarized at right angles to each other never interface. SOLO History (continue) Dominique François Jean Arago 1786-1853 Augustin Jean Fresnel 1788-1827 Arago relayed to Thomas Young in London the results of the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillations in the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young 1773-1829 1816-1817 longitudinal waves transversal waves
    13. 13. 13 SOLO Augustin Jean Fresnel 1788-1827 In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves. 1818Diffraction - History
    14. 14. 14 SOLO In 1818 August Fresnel supported by his friend André-Marie Ampère 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 - History http://microscopy.fsu.edu/optics/timeline/people/gregory.html http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html André-Marie Ampère (1775-1836) Dominique François Jean Arago 1786-1853 Siméon Denis Poisson 1781-1840 Pierre-Simon Laplace (1749-1827) Joseph Louis Guy-Lussac 1778-1850 Judging Committee of French Academy 1818
    15. 15. 15 SOLO Diffraction - History http://microscopy.fsu.edu/optics/timeline/people/gregory.html http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html Dominique François Jean Arago 1786-1853 Siméon 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. Siméon Denis Poisson 1781-1840 Poisson’s or Arago’s Spot
    16. 16. 16 SOLO In 1821 Joseph Fraunhofer build the first diffraction grating, made up of 260 close parallel wires. Latter he built a diffraction grating using 10,000 parallel lines per inch. Diffraction - History Utzshneider, Fraunhfer, Reichenbach, Mertz http://www.musoptin.com/fraunhofer.html 1821-1823 In 1823 Fraunhofer published his theory of diffraction. Table of Content
    17. 17. 17 SOLO Dffraction Grating Diffraction - History 1835 By 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.htm 1835 - Schwerd developed a "wave" theory of the diffraction grating. http://www.thespectroscopynet.com/Educational/Masson.htm “Die Beugungserscheinungen aus den Fundamentalgesetzen der Undulationstheorie” http://www.worldcatlibraries.org/wcpa/ow/3e723a9c5ac2a2b2.html Friedrich Magnus Schwerd 1792 - 1871 http://193.174.156.247/FMSG/wir_ueber_uns/wer_war_Schwerd.php
    18. 18. 18 SOLO H.A. Rowland at the John Hopkins University greatly improved diffraction gratings, introducing curved grating. Diffraction - History 1882 http://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 analysis—the resolution of a beam of light into components that differ in wavelength. http://www.britannica.com/eb/article-9064251/Henry-Augustus-Rowland Henry Augustus Rowland 1848 - 1901 http://chem.ch.huji.ac.il/~eugeniik/history/rowland.html Rowland gratings Rowland invented the ruling machine that can engrave as many as 20,000 lines to inch for diffraction gratings
    19. 19. 19 SOLO Optics History Debye-Sears Effect 1932 Nobel Prize in Chemistry 1936 Peter Josephus Wilhelmus Debye 1884 – 1966 Diffraction of light by ultrasonic waves. Francis Weston Sears 1898 - 1975 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
    20. 20. 20 DiffractionSOLO Augustin Jean Fresnel 1788-1827 In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves. P 0P Q 1x 0x 1y 0y η ξ Fr  Sr  ρ  r  O 'θ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen F rPP  =0 SrQP  =0 rQP  = From a source P0 at a distance from a aperture a spherical wavelet propagates toward the aperture: ( ) ( )Srktj S source Q e r A tU − = ' ' ω According to Huygens Principle second wavelets will start at the aperture and will add at the image point P. ( ) ( ) ( )( ) ( ) ( )( ) ∫∫ Σ ++− Σ +−− == dre rr A Kdre r U KtU rrktj S sourcerkttjQ P S 2/2/' ',', πωπω θθθθ where: ( )',θθK obliquity or inclination factor ( ) ( )SSS nrnr 11cos&11cos' 11 ⋅=⋅= −− θθ ( ) ( )   === === 0',0 max0',0 πθθ θθ K K Obliquity factor and π/2 phase were introduced by Fresnel to explain experiences results. Fresnel Diffraction Formula Fresnel took in consideration the phase of each wavelet to obtain: Fresnel – Huygens’ Diffraction Theory Table of Content Fresnel –Kirchoff Diffraction Formula
    21. 21. 21 SOLO Fresnel-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 Huyghen’s Principle, were a consequence of the wave nature of light. HBED  µε == & For an Homogeneous, Linear and Isotropic Medium where are constant scalars, we have µε, t E t D H t t H t B E ED HB ∂ ∂ = ∂ ∂ =×∇ ∂ ∂ ∂ ∂ −= ∂ ∂ −=×∇×∇ = =       εµ µ ε µ Since we have also tt ∂ ∂ ×∇=∇× ∂ ∂ t D H ∂ ∂ =×∇   t B E ∂ ∂ −=×∇   For Source less Medium ( ) ( ) ( )                   =⋅∇= ∇−⋅∇∇=×∇×∇ = ∂ ∂ +×∇×∇ 0& 0 2 2 2 DED EEE t E E     ε µε 02 2 2 = ∂ ∂ −∇ t E E   µε Maxwell Equations are ( ) eJ t D HA    + ∂ ∂ =×∇ mBGM ρ=⋅∇  )( ( ) mJ t B EF    − ∂ ∂ −=×∇ ( ) eDGE ρ=⋅∇  James C. Maxwell (1831-1879) Gustav Robert Kirchhoff 1824-1887 Diffraction
    22. 22. 22 DiffractionSOLO Fresnel-Kirchhoff Diffraction Theory 0 1 2 2 2 2 =      ∂ ∂ −∇ U tv Scalar Differential Wave Equation For a monochromatic wave of frequency f ( ω = 2πf ) a solution is: ( ) ( ) ( )[ ] ( ) ( )[ ] ( ){ }tjPjPUPtPUtPU ωφφω expexpRecos, −=+= Define the phasor ( ) ( ) ( )[ ]PjPUPU φ−= exp U v U tv 2 2 2 2 2 1 ω −= ∂ ∂ λ π π ω 2 2 === v f v k ( ) 022 =+∇ UkPhasor Scalar Differential Wave Equation This is the Scalar Helmholtz Differential Equation Hermann von Helmholtz 1821-1894 Boundary Conditions for the Helmholtz Differential Equation: • Dirichlet (U given on the boundary) • Neumann (dU/dn given on the boundary) Johann Peter Gustav Lejeune Dirichlet 1805-1859 Franz Neumann 1798-1895 εµ 1 0 11 2 2 2 2 2 2 2 2 2 ==      ∂ ∂ −∇= ∂ ∂ −∇ vE tvt E v E    Vector Differential Wave Equation
    23. 23. 23 To find the solution of the Scalar Helmholtz Differential Equation we need to use the following: • Scalar Green’s Identity ( ) ( )∫∫ → ⋅∇−∇=∇−∇ SV dSGUUGdVGUUG 22 • Green’s Function ( ) ( ) SF SF FS rr rrkj rrG    − − = exp ; This Green’s Function is a particular solution of the following Helmholtz Non-homogeneous Differential Equation: ( ) ( ) ( )SFFSFSS rrrrGkrrG  −=+∇ πδ4;; 22 SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction provided that and are continuous in volume V UUU 2 ,, ∇∇ GGG 2 ,, ∇∇ Free-Space Green’s Function  n i iSS 1= = iS nS dV dSnS → 1 V Fr  Sr  F 0r SF rrr  −= PositionSourcerS  PositionFieldrF  ( ) 022 =+∇ Uk Scalar Helmholtz Differential Equation
    24. 24. 24 SOLO • Scalar Green’s Identities ( ) ( )∫∫ → ⋅∇−∇=∇−∇ SV dSGUUGdVGUUG 22 Let start from the Gauss’ Divergence Theorem ∫∫ → ⋅=⋅∇ SV dSAdVA  Karl Friederich Gauss 1777-1855 where is any vector field (function of position and time) continuous and differentiable in the volume V bounded by the enclosed surface S. Let define . A  UGA ∇=  ( ) UGUGUGA 2 ∇+∇⋅∇=∇⋅∇=⋅∇  Then ( ) ( ) ∫∫∫ → ⋅∇=∇+∇⋅∇=∇⋅∇ S Gauss VV dSUGdVUGUGdVUG 2 ( ) ( ) ∫∫∫ → ⋅∇=∇+∇⋅∇=∇⋅∇ S Gauss VV dSGUdVGUUGdVGU 2 Subtracting the second equation from the first we obtain First Green’s Identity Second Green’s Identity We have GEORGE GREEN 1793-1841 Fresnel-Kirchhoff Diffraction Theory Diffraction To find a general solution of the Scalar Helmoltz Differential Equation we need to use the If we interchange with we obtainG U
    25. 25. 25 Integral Theorem of Helmholtz and Kirchhoff ( ) ( )( ) ( )F V sF V SS rUdVUrrUGkUGkdVGUUG  πδπ 442222 −=−−+−=∇−∇ ∫∫ Using: ( ) ( ) ( )SFFSFSS rrrrGkrrG  −=+∇ πδ4;; 22  n i iSS 1= = iS nS dV dSnS → 1 V Fr  Sr  F 0r SF rrr  −= PositionSourcerS  PositionFieldrF  SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction ( ) ( ) 0,22 =+∇ SFS rrUk  From the left side of the Second Scalar Green’s Identity we have: ( ) ∫∫       ∂ ∂ − ∂ ∂ =⋅∇−∇ → SS SS dS n G U n U GdSGUUG ( ) ( ) SF SF FS rr rrkj rrG    − − = exp ;Using: we obtain: ( ) ( ) ( ) ∫                 − − ∂ ∂ − ∂ ∂ − − −= S SF SF SF SF F dS rr rrkj n U n U rr rrkj rU      expexp 4 1 π This is the Integral Theorem of Helmholtz and Kirchhoff that enables to calculate as function of the values of and on the enclosed surface S.nU ∂∂ /UU Note: This Theorem was developed first by H. von Helmholtz in acoustics. Hermann von Helmholtz 1821-1894 Gustav Robert Kirchhoff 1824-1887 From the right side of the Second Scalar Green’s Identity, using we have:dS n U dSnUdSU SSS ∂ ∂ =⋅∇=⋅∇ →→ 1 Scalar Helmholtz Differential Equation
    26. 26. 26 Sommerfeld Radiation Conditions SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction ( ) ∫∫ ∫ ∞+Σ+       ∂ ∂ − ∂ ∂ −=       ∂ ∂ − ∂ ∂ −= SS S F dS n G U n U G dS n G U n U GrU 1 4 1 4 1 π π  P Fr  Sr  r  Σ 1S ∞S R Screen Aperture ωd → Sn1 → Sn1 since the condition that the previous integral be finite is: ( ) ( ) R Rkj rrG SFS exp ; → ∞  Consider the surface of integration ∞+Σ+= SSS 1 1S - on the screen ∞S - hemisphere with radius ∞→R ( ) Gkj R Rkj R kj n G ≅      −= ∂ ∂ exp1 ∫∫∫ Ω       − ∂ ∂ =      ∂ ∂ − ∂ ∂ ∞ ωdRUkj n U GdS n G U n U G S 2 ( ) 1 exp limlim == ∞→∞→ R Rkj RGR RR 0lim =      − ∂ ∂ ∞→ Ukj n U R R This is Sommerfeld Radiation Conditions Σ - on the aperture
    27. 27. 27 is known as optical disturbance. Being a scalar quantity, it cannot accurately represent an electromagnetic field. However, the square of this scalar quantity can be regarded as a measure of the irradiance at a given point. U Sommerfeld Radiation Conditions (continue) SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction ( ) ∫∫ ∫ ∞+Σ+       ∂ ∂ − ∂ ∂ −=       ∂ ∂ − ∂ ∂ −= SS S F dS n G U n U G dS n G U n U GrU 1 4 1 4 1 π π  0lim =      − ∂ ∂ ∞→ Ukj n U R R This is Sommerfeld Radiation Conditions This implies that: 0 4 1 =      ∂ ∂ − ∂ ∂ ∫∫ ∞S dS n G U n U G π and the Integral of Helmholtz and Kirchhoff becomes: ( ) ∫∫Σ+       ∂ ∂ − ∂ ∂ −= 1 4 1 S F dS n G U n U GrU π  P Fr  Sr  r  Σ 1S ∞S R Screen Aperture ωd→ Sn1 → Sn1 0P Q Arnold Johannes Wilhelm Sommerfeld 1868 - 1951
    28. 28. 28 The Kirchhoff Boundary Conditions SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction Kirchhoff assumed the following boundary conditions: ( ) ∫∫Σ       ∂ ∂ − ∂ ∂ −= dS n G U n U GrU F π4 1 1. The field distribution and its derivative , across the aperture , are the same as in the absence of the screen. U nU ∂∂ / Σ 2. On the shadowed part of the screen and0 1 =S U 0/ 1 =∂∂ S nU The Integral of Helmholtz and Kirchhoff becomes: The field at point P is the superposition of the aperture values 0=Σ U 0/ =∂∂ Σ nU Note: Moreover, mathematically the condition implies0/&0 11 =∂∂= SS nUU 0/&0 =∂∂= ΣΣ nUU However, if the dimensions of the aperture are large relative to the wavelength λ, the integral agrees well with the experiment. P Fr  Sr  r  Σ 1S ∞S R Screen Aperture 0P 0,0 1 1 = ∂ ∂ = S S n U U Σ Σ ∂ ∂ n U U , Q → Sn1 → Sn1 Sθ θ Kirchhoff boundary conditions are not physical since the presence of the screen changes field values on the aperture and on the screen.
    29. 29. 29 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction ( ) ∫∫Σ       ∂ ∂ − ∂ ∂ −= dS n G U n U GrU F π4 1 The Integral of Helmholtz and Kirchhoff: ΣAssume that the aperture is illuminated by a single spherical wave: ( ) ( ) S Ssource S r rkjA rU exp =  ( ) ( ) ( ) ( )       ⋅      −= ⋅      ∇=⋅∇= ∂ ∂ →→ →→ SS S Ssource S S S Ssource SSSS S nr r rkjA r kj n r rkjA nrU n rU 11 exp1 1 exp 1   ( ) ( ) SF SF FS rr rrkj rrG    − − = exp ; ( ) ( ) ( ) ( )       ⋅      −−= ⋅         − − ∇=⋅∇= ∂ ∂ →→ →=−→ S S SF SF S rrr SFSS FS nr r rkj r kj n rr rrkj nrrG n rrG SF 11 exp1 1 exp 1, ,      P Fr  Sr  r  Σ 1S ∞S R Screen Aperture 0P 0,0 1 1 = ∂ ∂ = S S n U U Σ Σ ∂ ∂ n U U , Q → Sn1 → Sn1 Sθ θ
    30. 30. 30 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction ( ) ∫∫Σ       ∂ ∂ − ∂ ∂ −= dS n G U n U GrU F π4 1 The Integral of Helmholtz and Kirchhoff: ΣAssume that the aperture is illuminated by a single spherical wave, and: Srr,<<λ ( ) ( )       ⋅≅ ∂ ∂ →→ SS S SsourceS nr r rkjA j n rU 11 exp2 λ π  ( ) ( ) r rkj rrG FS exp ; =  ( ) ( )       ⋅−≅ ∂ ∂ →→ S FS nr r rkj j n rrG 11 exp2, λ π  Srr k 1 , 12 >>= λ π ( ) ( )( ) ∫∫Σ →→→→                   ⋅−      ⋅−       + = dS nrnr rr rrkjA jrU SSS s ssource F 2 1111 exp λ  ( ) ( ) S Ssource S r rkjA rU exp =  P Fr  Sr  r  Σ 1S ∞S R Screen Aperture 0P 0,0 1 1 = ∂ ∂ = S S n U U Σ Σ ∂ ∂ n U U , Q → Sn1 → Sn1 Sθ θ
    31. 31. 31 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction ( ) ( )( ) ( ) ( )∫∫ ∫∫ Σ Σ →→→→                   ++ =                   ⋅−      ⋅−       + = dSK rr rrkj A dS nrnr rr rrkjA jrU S s s source SSS s ssource F θθ π λ λ , 2 exp 2 1111 exp ( )       ⋅−=      ⋅−= + =       ⋅−      ⋅− = →→→→ →→→→ SSSS S SSS S nrnr nrnr K 11cos&11cos 2 coscos 2 1111 , θθ θθ θθ 1. Obliquity or Inclination Factor: ( ) ( ) 0,0&10,0 ====== πθθθθ SS KK 2. Additional phase π/2 3. The amplitude is scaled by the factor 1/λ (not found in Fresnel derivation) P Fr  Sr  r  Σ 1S ∞S R Screen Aperture 0P 0,0 1 1 = ∂ ∂ = S S n U U Σ Σ ∂ ∂ n U U , Q → Sn1 → Sn1 Sθ θ We recovered Fresnel Diffraction Formula with:
    32. 32. 32 Reciprocity Theorem of Helmholtz SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction ( ) ( )( ) ( ) ( )∫∫ ∫∫ Σ Σ →→→→                   ++ =                   ⋅−      ⋅−       + = dSK rr rrkj A dS nrnr rr rrkjA jrU S s s source SSS s ssource F θθ π λ λ , 2 exp 2 1111 exp 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. P Fr  Sr  r  Σ 1S ∞S R Screen Aperture 0P 0,0 1 1 = ∂ ∂ = S S n U U Σ Σ ∂ ∂ n U U , Q → Sn1 → Sn1 Sθ θ Hermann von Helmholtz 1821-1894 Note: This is similar to Lorentz’s Reciprocity Theorem in Electromagnetism.
    33. 33. 33 Huygens-Fresnel Principle SOLO Fresnel-Kirchhoff Diffraction Theory Diffraction ( ) ( ) ( )∫∫Σ                   ++ = dSK rr rrkj A rU S s s source F θθ π λ , 2 exp  The Fresnel Diffraction Formula can be rewritten as: ( ) ( ) ( ) ∫∫Σ = dS r rkj QVrU F exp where: ( ) ( ) s s S source r rkj K A QV       + = 2 exp , π θθ λ The interpretation of this formula is that each point of 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 the Huygens-Fresnel Principle. P Fr  Sr  r  Σ 1S ∞S R Screen Aperture 0P 0,0 1 1 = ∂ ∂ = S S n U U Σ Σ ∂ ∂ n U U , Q → Sn1 → Sn1 Sθ θ Table of Content
    34. 34. 34 SOLO Diffraction Consider a diffracting aperture Σ. Suppose that the aperture is divided into two portions Σ 1 and Σ 2 such that Σ = Σ1 + Σ2. The two aperture Σ1 and Σ2 are said to be complementary. Complementary Apertures. Babinet Principle From the Fresnel Diffraction Formula: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫∫∫ ∫∫ ΣΣ Σ += = 21 dS r rkj QVdS r rkj QV dS r rkj QVrU F expexp exp P Fr  1S r  1r  2 Σ 1S Screen Apertures 0P 1 Q→ Sn1 2S r  2 r 1 Σ 2 Q We can see that the result is the added effect of all complimentary apertures. This is known as Babinet Principle. Table of Content The result can be very helpful when Σ is a very complicated aperture, that can be decomposed in a few simple apertures.
    35. 35. 35 SOLO Diffraction 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. Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 Jules Henri Poincaré 1854-1912 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, 1954 Rayleigh-Sommerfeld Diffraction Formula
    36. 36. 36 SOLO Rayleigh-Sommerfeld Diffraction Formula Diffraction Let start from the Helmholtz and Kirchhoff Integral: P Fr  Sr  r  Σ 1S ∞S R Screen Aperture 0P 0,0 1 1 = ∂ ∂ = S S n U U Σ Σ ∂ ∂ n U U , Q → Sn1 → Sn1 Sθ θ 0'PFr'  →→       ⋅−= SSSSS nnrrr 112'  ( ) ∫∫+Σ       ∂ ∂ − ∂ ∂ −= 1 4 1 S F dS n G U n U GrU π  Suppose that the Scalar Green Function is generated not only by P0 located at , but also by a point P’0 located symmetric relative to the screen at →→       ⋅−= SSSSS nnrrr 112'  Sr  G ( ) ( ) ( ) SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,_      − − − − − = ( ) ( ) ( ) SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,      − − + − − =+ or: We have 11 ,, ' SSFSSF rrrr ΣΣ −=−  ( ) ( ) → Σ → Σ ⋅−−=⋅− SSSFSSSF nrrnrr 1'1 11 ,,  0 1, = Σ− S G ( ) 011 exp1 2 11 ,, _ ≠            ⋅      −−= ∂ ∂ Σ →→ Σ S S S nr r rkj r kj n G 0 1, = ∂ ∂ Σ + S n G( ) 0 exp 2 1 1 , , ≠    = Σ Σ+ S S r rkj G
    37. 37. 37 SOLO Rayleigh-Sommerfeld Diffraction Formula Diffraction 1. Start from the Helmholtz and Kirchhoff Integral: P Fr  Sr  r  Σ 1S ∞S R Screen Aperture 0P 0,0 1 1 = ∂ ∂ = S S n U U Σ Σ ∂ ∂ n U U , Q → Sn1 → Sn1 Sθ θ 0'PFr'  →→       ⋅−= SSSSS nnrrr 112'  ( ) ∫∫+Σ       ∂ ∂ − ∂ ∂ −= 1 4 1 S F dS n G U n U GrU π  ( ) ( ) ( ) SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,_      − − − − − =Choose 0 1, = Σ− S G ( ) 011 exp1 2 11 ,, _ ≠            ⋅      −−= ∂ ∂ Σ →→ Σ S S S nr r rkj r kj n G On the shadowed part of the screen and0 1 =S U 0/ 1 ≠∂∂ S nU ( ) ( ) ( ) ∫∫∫∫ Σ →→= >>Σ       ⋅−≅         ∂ ∂ = dSnr r rkj rU j dS n G UrU SS k r kj F 11 exp 4 1 /2 1 _  λπ λπ This is Rayleigh-Sommerfeld Diffraction Formula of the first kind SF rrr  −= Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 ( ) ( ) S Ssource S r rkjA UrU exp == Σ  John William Strutt Lord Rayleigh (1842-1919) ( ) ( ) ( ) ∫∫Σ →→       ⋅−= dSnr r rkj r rkjAj rU S S Ssource F 11 expexp λ  we obtain:
    38. 38. 38 SOLO Rayleigh-Sommerfeld Diffraction Formula Diffraction 2. Start from the Helmholtz and Kirchhoff Integral: P Fr  Sr  r  Σ 1S ∞S R Screen Aperture 0P 0,0 1 1 = ∂ ∂ = S S n U U Σ Σ ∂ ∂ n U U , Q → Sn1 → Sn1 Sθ θ 0'PFr'  →→       ⋅−= SSSSS nnrrr 112'  ( ) ∫∫+Σ       ∂ ∂ − ∂ ∂ −= 1 4 1 S F dS n G U n U GrU π  ( ) ( ) ( ) SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,      − − + − − =+ Choose On the shadowed part of the screen and0 1 ≠S U 0/ 1 =∂∂ S nU SF rrr  −=0 1, = ∂ ∂ Σ + S n G( ) 0 exp 2 1 1 , , ≠      = Σ Σ+ S S r rkj G ( ) ( ) ∫∫∫∫ ΣΣ + ∂ ∂ −=      ∂ ∂ −= dS n U r rkj dS n U GrU F exp 2 1 4 1 ππ  ( ) ( ) S Ssource S r rkjA UrU exp == Σ  ( ) ( )       ⋅≅ ∂ ∂ →→ SS S SsourceS nr r rkjA j n rU 11 exp2 λ π  ( ) ( ) ( ) ∫∫Σ →→       ⋅−= dSnr r rkj r rkjAj rU SS S Ssource F 11 expexp λ  For we obtain: Table of Content This is Rayleigh-Sommerfeld Diffraction Formula of the second kind
    39. 39. 39 P 0P Q 1x 0x 1y 0y η ξ Sr'  Sr  ρ  r  O Sθ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn θcos11 =⋅ θcos11 =⋅ rnS z Sn1 'r  Fr  F rPP  =0 S rQP  =0 rQP  = SrOP '0  = '1 rOO  = SOLO Diffraction ( ) ( ) ( ) ∫∫Σ + = dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp θθ λ  Start with Fresnel-Kirchhoff (or Rayleigh-Sommerfeld) Diffraction Formula 1. If the inclination factor is nearly constant over the aperture ( ) constKK S ==θθ , Extensions of Fresnel-Kirchhoff Diffraction Theory ( ) ( ) ( ) ∫∫Σ ≈ dS r rkj r rkjAKj rU S Ssource F expexp λ  ( ) ( ) ( ) ∫∫Σ = dS r rkj rU Kj rU SF exp λ 2. Replace the incident point source wavefront with a general waveform ( ) S S r rkjexp ( )Sinc rU  3. Characterize the aperture by a transfer function τ to model amplitude or phase changes due to optic system ( ) ( ) ( ) ( ) ∫∫Σ = dS r rkj rrU j rU SSF exp τ λ Table of Content
    40. 40. 40 SOLO Diffraction Phase Approximations – Fresnel (Near-Field) Approximation Fresnel Approximation or Near Field Approximation can be used when aperture dimensions are comparable to distance to source rS or image r. ( ) ( ) ( ) ∫∫Σ + = dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp θθ λ  Start with Fresnel-Kirchhoff Diffraction Formula If the inclination factor is nearly constant over the aperture ( ) constKK S ==θθ , ( ) ( ) ( ) ( ) ( ) ∫∫∫∫ ΣΣ == dS r rkj rU Kj dS r rkj r rkjAKj rU S S Ssource F expexpexp  λλ P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1 Σ 1r  z Sn1 'r  rQP  = '1 rOO  = P 0P Q 1x 0x 1y 0y η ξ Sr'  Sr  ρ  r  O Sθ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn θcos11 =⋅ θcos11 =⋅ rnS z Sn1 'r  Fr  ( ) ( ) ( ) 1 ''2 ' ' 1''2' ' 2 2 1 2 1 2 11 2/1 2 2 1 2/1 11 0 1 2 1 << − + − +≈         − +=         −⋅−+−⋅+= −+= ++=+ r r k r r r r r rrrrrrr rrr x x ρρ ρ ρρρ ρ          ( ) ( )                 − ≈ '2 exp ' 'expexp 2 1 r r kj r rkj r rkj ρ  ( ) ( ) ( ) 2 max 2 1 2 1 ' '2 exp ' 'exp rrk dS r r kjrU r rkjKj rU SF <<−         − ≈ ∫∫Σ ρ ρ λ    Augustin Jean Fresnel 1788-1827
    41. 41. 41 SOLO Diffraction Phase Approximations – Fraunhofer (Near-Field) Approximation Fraunhofer Approximation or Far Field Approximation can be used when aperture dimensions are very small comparable to distance to source rS or image r. ( ) ( ) ( ) ∫∫Σ + = dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp θθ λ  Start with Fresnel-Kirchhoff Diffraction Formula If the inclination factor is nearly constant over the aperture ( ) constKK S ==θθ , ( ) ( ) ( ) ( ) ( ) ∫∫∫∫ ΣΣ =≈ dS r rkj rU Kj dS r rkj r rkjAKj rU S S Ssource F expexpexp  λλ P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1 Σ 1r  z Sn1 'r  rQP  = '1 rOO  = P 0P Q 1x 0x 1y 0y η ξ Sr'  Sr  ρ  r  O Sθ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn θcos11 =⋅ θcos11 =⋅ rnS z Sn1 'r  Fr  ( ) ( )       ⋅ −≈ ' exp ' 'expexp 1 r r kj r rkj r rkj ρ  ( ) ( ) ( ) ( ) 2 max 22 1 1 ' 2 ' exp ' 'exp rr k dS r r kjrU r rkjKj rU SF <<+       ⋅ −≈ ∫∫Σ ρ ρ λ   ( ) ( ) ( ) 1 '2' ' '2' ' ' 2 1''2' ' 2 22 11 2 22 11 2 11 2/1 2 2 1 2 1 2/1 11 0 1 2 1 << +⋅ −≈+ + + ⋅ −≈       +⋅− +=         −⋅−+−⋅+= −+= ++=+ r rk r r r r r r r r r rr rrrrrrr rrr x x ρρρρ ρρ ρρρ ρ          Table of Content
    42. 42. 42 0P Q 0x 0y η Sr'  Sr  ρ  O Sθ ScreenSource plane 0O Σ Sn10r  S rQP  =0 SrOP '0  = SOLO Diffraction Fresnel and Fraunhofer Diffraction Approximations Fresnel Approximations at the Source [ ] ( )          + ⋅ −+⋅+≈               +⋅+= +⋅+= += +−+=+ S S SS S S xx x SS S S SSS SS r r rr r r rr r r rrr rr '2 '1 '2' ' ' '' ' 21' '2' ' 2 282 11 2/12 2 2/122 2 ρρ ρ ρ ρ ρρ ρ ( ) ( ) ( ) ( )         ⋅− ⋅= S S S S S S S r r kjrkj r rkj r rkj '2 '1 exp'1exp ' 'expexp 2 2 ρρ ρ   ( ) S S r rkj ' 'exp ( )ρ  ⋅Srkj '1exp ( )         ⋅− S S r r kj '2 '1 exp 2 2 ρρ  Spherical wave centered at P0. Lowest order approximation to the phase of a spherical wavefront Planar wave propagating in directionSr'1 P 0P Q 1x 0x 1y 0y η ξ Sr'  Sr  ρ  r  O Sθ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn θcos11 =⋅ θcos11 =⋅ rnS z Sn1 'r  Fr 
    43. 43. 43 SOLO Diffraction Fresnel and Fraunhofer Diffraction Approximations P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1 Σ 1r  z Sn1 'r  rQP  = '1 rOO  = ( ) ( ) ( )         + ⋅ − + +≈       +⋅− +=         −⋅−+−⋅+= −+= ++=+ ''2 ' ' 2 1' '2' ' 1 22 1 2 11 2/1 2 2 1 2 1 2/1 11 0 1 2 1 r r r r r r rr r rrrrrr rrr x x ρρ ρρ ρρρ ρ ( ) ( )               ⋅ −                               ≈ ' exp '2 exp '2 exp ' 'expexp 1 22 1 r r kj r kj r r kj r rkj r rkj ρρ  Fresnel Approximations at the Image plane P 0P Q 1x 0x 1y 0y η ξ Sr'  Sr  ρ  r  O Sθ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn θcos11 =⋅ θcos11 =⋅ rnS z Sn1 'r  Fr  ( ) ' 'exp r rkj ( )ρ  ⋅− '1exp rkj ( )         ⋅− '2 '1 exp 2 2 r r kj ρρ  Spherical wave centered at O. Lowest order approximation to the phase of a spherical wavefront Planar wave propagating in direction'1r
    44. 44. 44 SOLO Diffraction Fresnel and Fraunhofer Diffraction Approximations (1st way) ( ) ( ) ( ) ( ) ∫∫Σ + = dS r rkj r rkjAj rU SK S S Ssource F     θθ θθ λ , 2 coscosexpexp Fresnel Approximation ( ) ( )[ ] [ ] ( )[ ] ( ) ( ) ∫∫Σ                       −⋅−− + ⋅− ⋅−⋅ + ≈ dS r rrr r r kjrrkjrrkj rr rrkjKAj rU S S S S Ssource F '2 '1 '2 '1 exp'1'1exp'1exp '' ''exp 2 1 2 1 2 2 1 ρρρρ ρ λ   Fraunhofer Approximation  ( ) ( ) 1 '2 '1 '2 '12 2 1 2 1 2 2 << −⋅−− + ⋅− S S k r rrr r r ρρρρ λ π  or S MAX rr ',' 2 << λ ρ ( ) ( )[ ] [ ] ( )[ ]∫∫Σ ⋅−⋅ + ≈ dSrrkjrrkj rr rrkjKAj rU S S Ssource F ρ λ  '1'1exp'1exp '' ''exp 1 If ( ) ( ) 1 '2 '1 '2 '1 exp 2 1 2 1 2 2 ≈                     −⋅−− + ⋅− S S r rrr r r kj ρρρρ  we obtain Augustin Jean Fresnel 1788-1827 ( ) constKK S =≈θθ , Start with '1'1 rrq S −=  P 0P Q 1x 0x 1y 0y η ξ Sr'  Sr  ρ  r  O Sθ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn θcos11 =⋅ θcos11 =⋅ rnS z Sn1 'r  Fr  F rPP  =0 SrQP  =0 rQP  = S rOP '0  = '1 rOO  =
    45. 45. 45 SOLO Diffraction Fresnel and Fraunhofer Diffraction Approximations (2nd way) Fresnel Approximation ( ) ( ) ( ) ( ) ( ) ( ) ∫∫Σ       −⋅− = dS r rr kjrrU r rkjj rU SSF '2 exp ' 'exp 11 ρρ τ λ   Fraunhofer Approximation  ( ) '1 '2 2 max 2 1 22 1 2 r r r r k << + ⇒<< + λ ρρ λ πIf we obtain Augustin Jean Fresnel 1788-1827 Start with ( ) ( ) ( ) ( ) ∫∫Σ = dS r rkj rrU j rU SSF exp τ λ - aperture optical transfer function( )Sr  τ - disturbance at the aperture( )SrU  ( ) ( ) ( ) ( )∫∫Σ             ⋅ −= dS r r kjrrU r rkjj rU SSF ' exp ' 'exp 1 ρ τ λ   ( ) ( )             ⋅ −                 + ≈ ' exp '2 exp ' 'expexp 1 2 1 2 r r kj r r kj r rkj r rkj ρρ  P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1 Σ 1r  z Sn1 'r  rQP  = '1 rOO  = ρ  −+= 1 ' rrr ( ) ( ) ( ) ( ) ( ) ( ) ( )       −⋅− +≅      −⋅− +=         −⋅−+−⋅+= 2 11 2/1 2 11 2/1 11 0 1 2 '2 1' ' 1''2' r rr r r rr rrrrrrr ρρρρ ρρρ     ( ) ( ) ( ) ( )∫∫Σ             ⋅ −      + = dS r r kj r r kjrrU r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 2 ρρ τ λ   ( ) ( ) ( )       + + + ⋅ −≈      +⋅− +=         −⋅−+−⋅+= ++=+ '2' ' ' 2 1''2' 22 11 2 112/1 2 2 1 2 1 2/1 11 0 1 2 r r r r r r rr rrrrrrr x x ρρρρ ρρρ
    46. 46. 46 SOLO Diffraction Fresnel and Fraunhofer Diffraction Approximations Augustin Jean Fresnel 1788-1827 1x 1yη ξ max ρ=D Screen 1O 1r  z λ 2 D R < Fresnel Region Fraunhofer Region λ 2 D R > R Σ O  ( ) '1 '2 2 max 2 1 22 1 2 r r r r k << + ⇒<< + λ ρρ λ π Fraunhofer Approximation If Table of Content
    47. 47. 47 SOLO Diffraction Fraunhofer Diffraction and the Fourier Transform ( ) ( ) ( ) ( )∫∫Σ             ⋅ −= dS r r kjrrU r rkjj rU SSF ' exp ' 'exp 1 ρ τ λ   ( )ηξ λ πρ 11 1 ' 2 ' yx rr r k +=      ⋅  ( ) ( ) ( ) ( ) ( )∫∫Σ       +−= ηξηξ λ π τ λ ddyx r jrrU r rkjj rU SSF 11 ' 2 exp ' 'exp  The integral is the two dimensional Fourier Transform of the field within the aperture ( ) ( )SS rrU  τ ( ) ( ) ( ) ( )[ ] { }Σ Σ =+−= ∫∫ fFTddkkjfkkF yxyx ηξηξηξ π exp, 2 1 :, 2 ( ) ( ) ( ) ( ) ( ){ }SSF rrUFT r rkjj rU  τπ λ 2 2 ' 'exp =Therefore P 0P Q 1x 0x 1y 0y η ξ Sr'  Sr  ρ  r  O Sθ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn θcos11 =⋅ θcos11 =⋅ rnS z Sn1 'r  Fr  F rPP  =0 SrQP  =0 rQP  = SrOP '0  = '1 rOO  = Two Dimensional Fourier Transform Table of Content
    48. 48. 48 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Rectangular Aperture P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1Σ 1r  z Sn1 'r 1ξ 2ξ 1η 2η ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ −− = Σ       ⋅−      ⋅−=       ⋅ −      ⋅ −      = 1 1 1 1 11 0 2 11 2 10 ' 2 exp ' 2 exp '2 'exp ' exp ' exp '2 exp ' 'exp η η ξ ξ λ π ηη λ π ξξ λ π π ηξ ηξ λ dy r jdx r j r rkjUkj dd r y kj r x kj r r kj r rkjUj rU k F  ( ) ( )    ≤≤≤≤ = elsevere U rrU SS 0 & 21110 ηηηξξξ τ  For a Rectangular Aperture Therefore ( ) ( ) ( ) ( )∫∫Σ             ⋅ −      = dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 ρ τ λ         ⋅       ⋅ =       −       ⋅+−      ⋅− =      ⋅−∫− 11 11 1 1 1111 1 ' 2 ' 2 sin 2 ' 2 ' 2 exp ' 2 exp ' 2 exp 1 1 ξ λ π ξ λ π ξ λ π ξ λ π ξ λ π ξξ λ π ξ ξ x r x r x r j x r jx r j dx r j       ⋅       ⋅ =       −       ⋅+−      ⋅− =      ⋅−∫− 11 11 1 1 1111 1 ' 2 ' 2 sin 2 ' 2 ' 2 exp ' 2 exp ' 2 exp 1 1 η λ π η λ π η λ π η λ π η λ π ξη λ π ξ ξ y r y r y r j y r jy r j dy r j ( ) ( )        ⋅       ⋅       ⋅       ⋅ = 11 11 11 11 4/ 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp2 η λ π η λ π ξ λ π ξ λ π ηξπ y r y r x r x r r rkjUkj rU A F 
    49. 49. 49 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Rectangular Aperture (continue – 1) Since U stands for scalar field intensity (E or H), the irradiance I is given by where < > is the time average and * is the complex conjugate. ( ) ( )       ⋅       ⋅       ⋅       ⋅ = 11 11 11 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp8 η λ π η λ π ξ λ π ξ λ π π y r y r x r x r Ar rkjUkj rU F  ( ) ( ) ( ) >⋅< ∗ FFF rUrUrI  ~ Therefore ( ) ( ) 2 11 11 2 2 11 11 2 ' 2 ' 2 sin ' 2 ' 2 sin 0       ⋅       ⋅       ⋅       ⋅ = η λ π η λ π ξ λ π ξ λ π y r y r x r x r IrI F  I (0) is the irradiance at O1 (x1 = y1 = 0). Hecht pg.466
    50. 50. 50 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Single Slit Aperture Let substitute in the rectangular aperture ξ1 → 0 where < > is the time average and * is the complex conjugate. ( ) ( )       ⋅       ⋅       ⋅       ⋅ = 11 11 11 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp8 η λ π η λ π ξ λ π ξ λ π π y r y r x r x r Ar rkjUkj rU F  ( ) ( ) ( ) >⋅< ∗ FFF rUrUrI  ~ Therefore ( ) ( ) 2 11 11 2 ' 2 ' 2 sin 0       ⋅       ⋅ = η λ π η λ π y r y r IrI F  I (0) is the irradiance at O1 (x1 = y1 = 0). to obtain the single (vertical) slit diffraction ( ) ( )       ⋅       ⋅ = 11 11 0 ' 2 ' 2 sin ' 'exp2 η λ π η λ π π y r y r Ar rkjUkj rU FSLITSINGLE  Since U stands for scalar field intensity (E or H), the irradiance I is given by Hecht, pg. 453 Hecht, pg. 456
    51. 51. 51 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Single Slit Aperture (continue) ( ) ( ) 2 11 11 2 ' 2 ' 2 sin 0       ⋅       ⋅ = η λ π η λ π y r y r IrI F  I (0) is the irradiance at O1 (x1 = y1 = 0). Hecht, pg. 456 Hecht 455 Define: 11 ' 2 : η λ π β ⋅= y r ( ) ( ) 2 2 sin 0 β β β II = The extremum of I (β) is obtained from: ( ) ( ) ( ) 0 sincossin2 0 3 = − = β ββββ β β I d Id The results are given by: minimum,3,2,0sin πππββ ±±±=⇒= maximumββ =tan The solutions can be obtained graphically as shown in the figure and are: ,4707.3,4590.2,4303.1 πππβ ±±±=
    52. 52. 52 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Double Slit Aperture ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               ⋅ −+      ⋅ −      = ∫∫ + − +− −− ξ ξ ξ ξ λ d r x kjd r x kj r r kj r rkjUj rU ba ba ba ba F 2/ 2/ 1 2/ 2/ 1 2 10 ' exp ' exp '2 exp ' 'exp P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1Σ 1r  z Sn1 'r  1η b b a( ) ( ) ( ) ( )∫∫Σ             ⋅ −      = dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 ρ τ λ   ( ) ( ) ( ) ( )       ⋅       ⋅       ⋅ =       −       −−⋅−−      +−⋅− =      ⋅−∫ +− −− ax r j bx r bx r b x r j bax r jbax r j dx r j ba ba 1 1 1 1 112/ 2/ 1 ' exp ' ' sin 1 ' 2 ' exp ' exp ' 2 exp λ π λ π λ π λ π λ π λ π ξξ λ π ( ) ( ) ( ) ( )       ⋅−       ⋅       ⋅ =       −       −⋅−−      +⋅− =      ⋅−∫ + − ax r j bx r bx r b x r j bax r jbax r j dx r j ba ba 1 1 1 1 112/ 2/ 1 ' exp ' ' sin 1 ' 2 ' exp ' exp ' 2 exp λ π λ π λ π λ π λ π λ π ξξ λ π ( ) ( )       ⋅       ⋅       ⋅       −= ax r bx r bx r br r kj r rkjUj rU F 1 1 12 10 ' cos ' ' sin 2 '2 exp ' 'exp λ π λ π λ π λ  ( ) ( )       ⋅       ⋅       ⋅ = ax r bx r bx r IrI F 1 2 2 1 1 2 ' cos ' ' sin 0 λ π λ π λ π  ( ) ( ) ( ) >⋅< ∗ FFF rUrUrI  ~ ( ) ( ) ( ) ( ) ( ) ( )    +≤≤−+−≤≤−− = elsevere babababaU rrU SS 0 2/2/&2/2/0 ξξ τ 
    53. 53. 53 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Double Slit Aperture (continue -= 1) Hecht p.458 ( ) ( ) ( ) γ β β λ π λ π λ π 2 2 2 12 2 1 12 cos sin 0 ' cos ' ' sin 0 I a r x b r x b r x IrI F =      ⋅       ⋅       ⋅ =  P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1Σ 1r  z Sn1 'r  1η b b a The factor (sin β/ β)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 γ. Bright fringes occur for γ = 0,±π ,±2π,… The angular separation between fringes is Δγ = π.
    54. 54. 54 Hecht 459 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Double Slit Aperture (continue – 2)
    55. 55. 55 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture P 1y r  Image plane 1O 1r  Q η ξ ρ  O θ Screen Σ Sn1 'r  b b a a b b a b a The Aperture consists of a large number N of identical parallel slits of width b and separation a. ( ) ( ) ( ) ∑ ∫ − = + −               ⋅ −      = 1 0 2/ 2/ 1 2 10 ' exp '2 exp ' 'exp N k bak bak F d r x kj r r kj r rkjUj rU ξ ξ λ  ( ) ( ) ( ) ( )∫∫Σ             ⋅ −      = dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 ρ τ λ         ⋅−       ⋅       ⋅ =       −             −⋅−−            +⋅− =      ⋅−∫ + − akx r j bx r bx r b x r j b kax r j b kax r j dx r j bka bka 1 1 1 1 112/ 2/ 1 ' 2 exp ' ' sin 1 ' 2 2' 2 exp 2' 2 exp ' 2 exp λ π λ π λ π λ π λ π λ π ξξ λ π ( ) ( ) ( ) ( )       ⋅       ⋅       ⋅       ⋅       =       ⋅−−       ⋅−−       ⋅       ⋅       =                     ⋅−       ⋅       ⋅       = ∑ − = ax r aNx r bx r bx r br r kj r rkjUj ax r j aNx r j bx r bx r br r kj r rkjUj akx r j bx r bx r br r kj r rkjUj rU N k F 1 1 1 12 10 1 1 1 12 10 1 0 1 1 12 10 ' sin ' sin ' ' sin 1 '2 exp ' 'exp ' 2 exp1 ' 2 exp1 ' ' sin 1 '2 exp ' 'exp ' 2 exp ' ' sin 1 '2 exp ' 'exp λ π λ π λ π λ π λ λ π λ π λ π λ π λ λ π λ π λ π λ  ( ) ( )    −=+≤≤+ = elsevere NkbkabkaU rrU SS 0 1,,1,02/2/0  ξ τ
    56. 56. 56 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue – 1) P 1y r  Image plane 1O 1r  Q η ξ ρ  O θ Screen Σ Sn1 'r  b b a a b b a b a The Aperture consists of a large number N of identical parallel slits of width b and separation a. ( ) ( )       ⋅       ⋅       ⋅       ⋅       = ax r aNx r bx r bx r br r kj r rkjUj rU F 1 1 1 12 10 ' sin ' sin ' ' sin 1 '2 exp ' 'exp λ π λ π λ π λ π λ  ( ) ( ) ( ) α α β β λ π λ π λ π λ π 22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F =       ⋅       ⋅       ⋅       ⋅ =  ( ) ( ) ( ) >⋅< ∗ FFF rUrUrI  ~
    57. 57. 57 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue – 2) Hecht p.462 Hecht p.463 ( ) ( ) ( ) α α β β λ π λ π λ π λ π 22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F =       ⋅       ⋅       ⋅       ⋅ = 
    58. 58. 58 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue – 2) ( ) ( ) ( ) α α β β λ π λ π λ π λ π 22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F =       ⋅       ⋅       ⋅       ⋅ =  Sears p.222 Hecht p. 462 Sears p.236 Interference Irradiation for 1, 2, 3 and 4 slits as function of observation angle. Diffraction Pattern for 1, 2, 3, 4 and 5 slits.
    59. 59. 59 SOLO Resolution of Optical Systems According to Huygens-Fresnel Principle, a differential area dS, within an optical Aperture, may be envisioned as being covered with coherent secondary point sources. φρφρ sincos == yz Φ=Φ= sincos qYqZ Differential area dS, coordinates Image , coordinates φρρ dddS = ( ) dSe r E dE rktiA −       = ω The spherical wave that propagates from dS to Image is where ( ) ( )[ ] ( )[ ] ( )[ ]22/122/1222 /1/21 RZzYyRRZzYyRzZyYXr +−≈+−≈−+−+= [ ] 2/1222 ZYXR ++= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )RkaqJkaqRe R E ddee R E dSee R E dEE RktiA a RkpqiRktiA Aperture RzZyYkiRktiA Aperture // 1 0 2 0 cos// − = = Φ−−+−       =      =      == ∫ ∫∫∫∫ ω ρ π φ φωω φρρ The spherical wave at Image, for a Circular Aperture, is
    60. 60. 60 SOLO Resolution of Optical Systems ( ) ( ) ( )RkaqJkaqRe R E E RktiA // 1 −       = ω where ( ) ( ) ∫ + − = π π 2 0 cos 2 dve i uJ vuvmi m m Bessel Functions (of the first kind) E. Hecht, “Optics” The spherical wave at Image, for a Circular Aperture, is
    61. 61. 61 SOLO Resolution of Optical Systems Irradiance ∗ = ∗ ==×== EEHEHESI EH µ εµ ε 2 1 2 1 2 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 2 22 / /2 0 / /22 2 1       =      == ∗ Rkaq RkaqJ I Rkaq RkaqJ R aE EEI A π µ ε µ ε D nnn λ θθ 44.22 == E. Hecht, “Optics” Circular Aperture ( ) Daaak RkaquuJ n Rq λλ λ π θ θ 22.1 2 22.1 2 83.383.3 sin83.3/0 sin/ 1 ====⇒==⇒= =
    62. 62. 62 SOLO Resolution of Optical Systems Distribution of Energy in the Diffraction Pattern at the Focus of a Perfect Circular Lens E. Hecht, “Optics” Ring f/(λf#) Peak Energy in ring Illumination (%) Central max 0 1 83.9 1st dark ring 1.22 0 1st bright ring 1.64 0.017 7.1 2nd dark 2.24 0 2nd bright 2.66 0.0041 2.8 3rd dark 3.24 0 3rd bright 3.70 0.0016 1.5 4th dark 4.24 0 4th bright 4.74 0.00078 1.0 5th dark 5.24 0
    63. 63. 63 SOLO Diffraction Fraunhofer Diffraction Approximations Examples Circular Aperture Hecht p.469
    64. 64. 64 SOLO Resolution 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”
    65. 65. 65 SOLO Resolution of Optical Systems E. Hecht, “Optics”
    66. 66. 66 Rayleigh’s Criterion (1902) The images are said to be just resolved when the center of one Airy Disk falls on the first minimum of the Airy pattern of the other image. The minimum resolvable angular separation or angular limit is: D nnn λ θθ 44.22 == Sparrow’s Criterion At the Rayleigh’s limit there is a central minimum Or 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 separation corresponding to that configuration is the Sparrow’s Limit. SOLO Resolution of Optical Systems
    67. 67. 67 Resolution – Diffraction Limit Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Resolution of Optical Systems
    68. 68. 68 Diffraction Grating SOLO Resolution of Optical Systems Hecht 478
    69. 69. 69 Diffraction Grating SOLO Resolution of Optical Systems Hecht 480
    70. 70. 70 Diffraction Grating SOLO Resolution of Optical Systems Hecht 479a Hecht 479b Hecht 485 Table of Content
    71. 71. 71 SOLO Diffraction Fresnel Diffraction Approximations Examples Rectangular Aperture ( ) ( ) ( ) ( ) ( ) ( ) ∫∫Σ       −⋅− = dS r rr kjrrU r rkjj rU SSF '2 exp ' 'exp 11 ρρ τ λ   define Augustin Jean Fresnel 1788-1827 P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1Σ 1r  z Sn1 'r 1ξ 2ξ 1η 2η ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫ ∫∫       −       − =       −       − = = Σ 2 1 2 1 ' 2 2 exp ' 2 2 exp '2 'exp '2 exp '2 exp ' 'exp 2 1 2 10 2 2 1 2 10 η η ξ ξ λ π η λ ηπ ξ λ ξπ π ηξ ηξ λ d r y jd r x j r rkjUkj dd r y kj r x kj r rkjUj rU k F  ( ) ( )    ≤≤≤≤ = elsevere U rrU SS 0 & 21110 ηηηξξξ τ  For a Rectangular Aperture ( ) ξ λ α λ ξ α d r d r x ' 2 ' 2 : 2 12 = − = ( ) ∫∫       =      − 2 1 2 1 2 2 1 2 exp 2 ' ' 2 2 exp α α ξ ξ αα πλ ξ λ ξπ dj r d r x j ( ) ( )212111 ' 2 & ' 2 ξ λ αξ λ α −=−= x r x r Therefore ( ) η λ β λ η β d r d r y ' 2 ' 2 : 2 12 = − = ( ) ( )212111 ' 2 & ' 2 η λ βη λ β −=−= y r y r ( ) ∫∫       =      − 2 1 2 1 2 2 1 2 exp 2 ' ' 2 2 exp β β η η ββ πλ η λ ηπ dj r d r y j
    72. 72. 72 SOLO Diffraction Fresnel Diffraction Approximations Examples Rectangular Aperture (continue – 1) Augustin Jean Fresnel 1788-1827 P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1Σ 1r  z Sn1 'r 1ξ 2ξ 1η 2η ( ) ( ) ( ) ( ) ( ) ∫∫ ∫∫             =       −       − = 2 1 2 1 2 1 2 1 220 2 1 2 10 2 exp 2 exp 2 'exp ' 2 2 exp ' 2 2 exp ' 'exp β β α α η η ξ ξ ββ π αα π η λ ηπ ξ λ ξπ λ djdj rkjUj d r y jd r x j r rkjUj rU F  Define Fresnel Integrals ( ) ( ) ∫ ∫       =       = α α αα π α αα π α 0 2 0 2 2 sin: 2 cos: dS dC ( ) ( )αααα πα SjCdj +=      ∫0 2 2 exp ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] 2 1 2 1 2 'exp0 β β α α ββαα SjCSjC rkjUj rU F ++=  Using the Fresnel Integrals we can write ( ) ( ) 5.0±=∞±=∞± SC
    73. 73. 73 SOLO Diffraction Fresnel Diffraction Approximations Examples Rectangular Aperture (continue – 2) Augustin Jean Fresnel 1788-1827 Hecht p.499
    74. 74. 74 SOLO Diffraction Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 P Q 1x 1y η ξ ρ  r  O θ Screen Image plane 1O Sn1Σ 1r  z Sn1 'r 1ξ 2ξ 1η 2η Rectangular Aperture (continue – 3)
    75. 75. 75 SOLO Diffraction Fresnel Diffraction Approximations Examples Cornu Spiral Fresnel Integrals are defined as ( ) ( ) ∫∫       =      = uu duuuSduuuC 0 2 0 2 2 sin:& 2 cos: ππ ( ) ( )uSjuCduuj u +=      ∫0 2 2 exp π ( ) ( ) 5.0±=∞±=∞± SC Marie Alfred Cornu professor at the École 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) duuSd duuCd       =       = 2 2 2 sin 2 cos π π ( ) ( ) duSdCd =+ 22 Therefore u may be thought as measuring arc length along the spiral. “Méthode nouvelle pour la discussion des problèmes de diffraction dans le cas d’une onde cylindrique”, J.Phys.3 (1874), 5-15,44-52
    76. 76. 76 SOLO Diffraction Fresnel Diffraction Approximations Examples Cornu Spiral (continue – 1) ( ) ( ) ∫∫       =      = uu duuuSduuuC 0 2 0 2 2 sin:& 2 cos: ππ ( ) ( )uSjuCduuj u +=      ∫0 2 2 exp π ( ) ( ) 5.0±=∞±=∞± SC The Cornu Spiral is defined as the plot of S (u) versus C (u) duuSdduuCd       =      = 22 2 sin& 2 cos ππ ( ) ( ) duSdCd =+ 22       =             = 2 2 2 2 tan 2 cos 2 sin u u u Cd Sd π π π Therefore every point on the curve makes the angle with the real ( C ) axis. 2 2 u π The radius of curvature of Cornu Spiral is The tangent vector of Cornu Spiral is SuCuT 1 2 sin1 2 cos 22       +      = ππ ( ) ( ) u SuCuu udTdSdCdTd πππ π ρ 1 1 2 cos1 2 sin 1 / 1 / 1 22 22 =             +      − == + =  showing that the curve spirals toward the limit points. ∫       2 1 2 2 cos u u duu π ∫       2 1 2 2 sin u u duu π ∫       2 1 2 2 exp u u duuj π Table of Content
    77. 77. 77 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION Define the Green’s Function as a particular solution of the following Helmholtz Non-homogeneous Differential Equation: ( ) ( ) ( )SFFSFSS rrrrG v rrG  −−=+∇ πδ ω 4;; 2 2 2 where δ (x) is the Dirac function ( ) ( )          = =∞ ≠ = ∫ ∞ ∞− 1 0 00 dxx x x x δ δ Let use the Fourier Transformation to write ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]{ } ( ) ( )[ ]∫ ∫ ∫ ∫ ∫∫∫ −⋅−= −+−+−−=         −−        −−        −−= −−−=− ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− 3 3 3 exp 2 1 exp 2 1 exp 2 1 exp 2 1 exp 2 1 dkrrkj dkdkdkzzkyykxxkj dkzzjkdkyyjkdkxxjk zzyyxxrr SF zyxSFzSFySFx zSFzySFyxSFx SFSFSFSF   π π πππ δδδδ zyx zyx dkdkdkdk zkykxkk = ++= →→→ 3 111  where Paul Dirac 1902-1984 Joseph Fourier 1768-1830
    78. 78. 78 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 1) Let use the Fourier Transformation to write ( ) ( ) ( )[ ]∫ −⋅−= SFFS rrkjkgdkrrG  exp,; 3 ω Hence ( ) ( )[ ] ( ) ( )[ ]∫∫ −⋅−−=−⋅−      +∇ SFSFS rrkjdkrrkjkgdk v  exp 2 4 exp, 3 3 3 2 2 2 π π ω ω or ( ) ( )[ ] ( )[ ]{ } ( ) ( )[ ]∫∫ −⋅−−=−⋅−+−⋅∇ SFSFSFS rrkjdkrrkjkrrkjkgdk  exp 2 4 expexp, 3 3 223 π π ω  n i iSS 1= = iS nS dV dSn → 1 V Fr  Sr  F 0r SF rrr  −= PositionSourcerS  PositionFieldrF 
    79. 79. 79 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 2) Let compute: ( )[ ] ( ) ( ) ( )[ ]{ }( ) ( ) ( ) ( )[ ]{ } ( ) ( ) ( )[ ]{ } ( )[ ] ( ) ( ) ( )[ ] ( )[ ]SFSFSFS SFzSFySFxSzyx SFzSFySFxzyxS SFzSFySFxSSSFS rrkjkrrkjkjkjrrkjkj zzkyykxxkjzjkyjkxjk zzkyykxxkjzjkyjkxjk zzkyykxxkjrrkj   −⋅−−=−⋅−−⋅−=−⋅−∇⋅−= −+−+−−∇⋅      −−−=       −+−+−−      −−−⋅∇= −+−+−−∇⋅∇=−⋅−∇ →→→ →→→ expexpexp exp111 exp111 expexp 2 2 Therefore: ( ) ( )[ ] ( ) ( )[ ]∫∫ −⋅−−=−⋅−      +− SFSF rrkjdkrrkj v kkgdk  exp 2 4 exp, 3 32 2 23 π πω ω Because this is true for all k and ω, we obtain ( )             − = 2 2 2 2 1 2 1 , v k kg ωπ ω 
    80. 80. 80 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 3) ( ) ( ) ( )[ ] ( )[ ]∫∫ −⋅−             − =−⋅−= SFSFFS rrkj v k d dkrrkjkgdkrrG  exp 2 1 exp,; 2 2 2 3 2 3 ω ω π ω We can see that the integral in k has to singular points for v k ω ±= To find the integral let change ω by ω + jδ where δ is a small negative number ( ) [ ] ( )∫             + − ⋅− = 2 2 2 3 2 exp 2 1 ; v j k rkj dkrrG FS δωπ   where .SF rrr  −=
    81. 81. 81 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 4) In the plane ω we close the integration path by the semi-circle with and the singular points on the upper side, for τ > 0 (for t > t’) ( ) ( ) ( )[ ]∫ ∫ ∞ ∞− ⋅−             + − = rkj v j k d dktrtrG FS  ωτ δω ω π exp 4 1 ',;, 2 2 2 3 3 ∞→r ( ) ( ) ( )'00exp ttdjf UPC >>=∫ τωωτω ( ) ( ) ( )'00exp ttdjf DOWNC <<=−∫ τωωτω ( ) ( ) ( ) ( ) ( ) ( )        <− >− =− ∫ ∫ ∫ ∞≤≤∞− + ∞≤≤∞− +∞ ∞− 0exp 0exp exp τωωτω τωωτω ωωτω ω ω DOWN UP C C djf djf djf δjvk −− δjvk − ωRe ωIm
    82. 82. 82 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 5) ( ) ( )[ ] ( ) ( )[ ] ( )[ ] ( )[ ] ( )[ ]∫∫∫         −+−++ ⋅− =⋅−             + − =⋅−             + − = ∞ ∞− CC jvkjvk rkjv drkj v j k d rkj v j k d I δωδω ωτ ωωτ δω ω ωτ δω ω   exp expexp 2 2 2 2 2 2 2 Let use the Cauchy Integral for a complex function f (z) continuous on a closed path C, in the complex z plane: ( ) ( ) ( )0 0 2lim2 0 zfjzfjdz zz zf zz C ππ == − →∫ We have: ( )[ ] ( )[ ] ( )[ ] ( )[ ] ( ) ( ) ( ) ( ) ( ) k kvrkj v vk jkv vk jkv rkjvj jvk rkjv j jvk rkjv jI vkvk τ π ττ π δω ωτ π δω ωτ π δωδω sinexp 2 2 exp 2 exp exp2 exp 2lim exp 2lim 2 2 0, 2 0,    ⋅− =      + − − ⋅−−= ++ ⋅−− + −+− ⋅−− = →→→−→ Therefore, we can write: ( ) ( )[ ] ( ) ( ) ∫∫ ∫ ⋅− −=             − ⋅− = k vkrkj dk v v k rkj ddktrtrG FS τ πω ωτ ω π sinexp 2 exp 4 1 ',;, 3 2 2 2 2 3 3  
    83. 83. 83 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 6) Let use spherical coordinates relative to vector: ( ) ( ) ∫ − ⋅− = 2 2 2 3 2 exp 2 1 ; v k rkj dkrrG FS ωπ        = = =      = = = rr r r kk kk kk z y x z y x 0 0 cos sinsin cossin θ ϕθ ϕθ ϕθ dk sin θdk dk ( )( ) ϕθθ dddkkdk sin23 = θ ϕ r x y z
    84. 84. 84 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 7) ( )kvd v r jkv v r jkv v r jkv v r jkv r ∫ ∞ ∞−             −−+            +−−            +−            − = 4 expexpexpexp 1 ττττ π r v rr tt v rr tt SFSF         − +−−        − −− =  '' δδ r v r v r dx v r jx v r jx r       +−      − =                   +−            −= ∫ ∞ ∞− τδτδ ττ π expexp 2 11 ( ) ( )kvd v r jkv v r jkv r v kvd v r jkv v r jkv r ∫∫ ∞ ∞− ∞ ∞−             +−−            −− +             +−            − = 4 expexp 4 expexp 1 ττ π ττ π ( ) ( ) ( ) ( ) ∫ ∞ ∞−       −−       −− = dk j jvkjvk j jkrjkr r v 2 expexp 2 expexp ττ π ( ) ( )∫∫ ∞ ∞− ∞ − −= − −= dkkr v k k r dkkr v k k r sin 1 sin 2 2 2 2 2 0 2 2 2 2 ωπωπ ( ) ( ) ( ) ∫∫ ∞∞ = =       −− − −= − − − = 0 2 2 2 2 0 0 2 2 2 2 2 expexp2cosexp1 dk j jkrjkr v k k r dk jkr jkr v k k ωπ θ ωπ πθ θ ( ) ∫∫ ∫ ∞ − − = 0 0 2 0 2 2 2 2 2 sin cosexp 2 1 π π θϕθ ω θ π dkddk v k jkr ( ) ( ) ∫ − − = 2 2 2 3 2 cosexp 2 1 ; v k jkr dkrrG FS ω θ π  ( ) ∫∫ ∞ − − = 0 0 2 2 2 2 sin cosexp1 π θθ ω θ π dkdk v k jkr
    85. 85. 85 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 8) We can see that represents a progressive wave and represents a regressive wave:         − +−=        − −− v rr tt v rr tt SFSF  '' δδ         − −−=        − +− v rr tt v rr tt SFSF  '' δδ Hence ( ) SF SFSF FS rr v rr tt v rr tt trtrG    −         − +−−        − −− = '' ',;, δδ We shall consider only the progressive wave and use: ( ) SF SF FS rr v rr tt trtrG    −         − +− = ' ',;, δ Retarded Green Function The other solution is: ( ) SF SF FS rr v rr tt trtrG    −         − −− = ' ',;, δ Advanced Green Function  n i iSS 1= = iS nS dV dSn → 1 V Fr  Sr  F 0r SF rrr  −= PositionSourcerS  PositionFieldrF  Table of Content
    86. 86. 86 SOLO ( ) ( ) ( )[ ] { }gFTydxdyfxfjyxgffG yxyx =+−= ∫∫Σ π2exp,:, The two dimensional Fourier Transform F of the function f (x, y) The Inverse Fourier Transform is ( ) ( ) ( )[ ] { }GFTfdfdyfxfjyxgyxg F yxyx 1 2exp,, − =+= ∫∫ π ( ) ( ) ( ) ( )[ ] { }Σ Σ =+−= ∫∫ fFTddkkjfkkF yxyx ηξηξηξ π exp, 2 1 :, 2 Two Dimensional Fourier Transform Two Dimensional Fourier Transform (FT) Fraunhofer Diffraction and the Fourier Transform In Fraunhofer Diffraction we arrived two dimensional Fourier Transform of the field within the aperture P 0P Q 1x 0x 1y 0y η ξ Sr'  Sr  ρ  r  O Sθ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn θcos11 =⋅ θcos11 =⋅ rnS z Sn1 'r  Fr  F rPP  =0 SrQP  =0 rQP  = SrOP '0  = '1 rOO  = Using kx = 2 π fx and ky = 2 π fy we obtain: Diffractions
    87. 87. 87 SOLO ( ) ( ){ } ( ){ } ( ){ }yxhFTyxgFTyxhyxgFT ,,,, βαβα +=+ 1. Linearity Theorem Two Dimensional Fourier Transform (FT) Fourier Transform Theorems ( ){ } ( )yx ffGyxgFT ,, = 2. Similarity Theorem ( ){ }       = b f a f G ba ybxagFT yx , 1 ,If then ( ){ } ( )yx ffGyxgFT ,, = 3. Shift Theorem ( ){ } ( ) ( )[ ]bfafjffGbyaxgFT yxyx +−=−− π2exp,, If then Diffractions
    88. 88. 88 SOLO Two Dimensional Fourier Transform (FT) Fourier Transform Theorems (continue – 1) ( ){ } ( )yx ffGyxgFT ,, = 4. Parseval’s Theorem ( ) ( )∫∫∫∫ = yxyx fdfdffGydxdyxg 22 ,, If then ( ){ } ( )yx ffGyxgFT ,, = 5. Convolution Theorem ( ) ( ){ } ( ) ( )yxyx ffHffGddyxhgFT ,,,, =−−∫∫ ηξηξηξ If then ( ){ } ( )yx ffHyxhFT ,, =and ( ){ } ( )yx ffGyxgFT ,, = 6. Autocorrelation Theorem ( ) ( ){ } ( )2* ,,, yx ffGddyxggFT =−−∫∫ ηξηξηξ If then similarly ( ){ } ( ) ( )∫∫ −−= ηξηξηξηξ ddffGGgFT yx ,,, *2 Diffractions
    89. 89. 89 SOLO Two Dimensional Fourier Transform (FT) Fourier Transform Theorems (continue – 2) ( ){ } ( ){ } ( )yxgyxgFTFTyxgFTFT ,,, 11 == −− 7. Fourier Integral Theorem Diffractions
    90. 90. 90 SOLO Two Dimensional Fourier Transform (FT) Fourier Transform for a Circular Symmetric Optical Aperture To exploit the circular symmetry of g (g (r,θ) = g (r) ) let make the following transformation ( ) ( ) φρφ φρρ θθ θ sin/tan cos sin/tan cos 1 22 1 22 == =+= == =+= − − yxy xyx fff fff ryxy rxyxr { } ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]∫ ∫ ∫ ∫∫∫ −−= +−=+−= = = Σ a o rgrg a o drdrydxd yx drjrdrrg drjrgrdrydxdyfxfjyxggFT πθ πθ θφθρπ θθφθφρπθπ 2 0 , 2 0 cos2exp sinsincoscos2exp,2exp, Use Bessel Function Identity ( ) ( )[ ]∫ −−= π θφθ 2 0 0 cosexp dajaJ ( ) ( ){ } ( ) ( )∫== a o rdrrgrJrgFTG ρπρ 2: 00 to obtain J0 is a Bessel Function of the first kind, order zero. Diffractions
    91. 91. 91 SOLO Two Dimensional Fourier Transform (FT) Fourier Transform for a Circular Symmetric Optical Aperture For a Circular Pupil of radius a we have ( )    > ≤ = ar ar rg 0 1 Use Bessel Function Identity J1 is a Bessel Function of the first kind, order one. ( ) ( ){ } ( )∫== a o rdrrJrgFTG ρπρ 2: 00 ( ) ( )xJxdJ x o 10 =∫ ςςς ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( )ρπ ρπ ςςς ρπ ρπρπρπ ρπ ρ ρπ aJ a dJ rdrrJrgFTG a o a o 2 22 1 222 2 1 : 1 2 02 020 == == ∫ ∫ Bessel Functions of the first kind Diffractions
    92. 92. 92 SOLO E. Hecht, “Optics” Circular Aperture Two Dimensional Fourier Transform (FT) Fourier Transform for a Circular Symmetric Optical Aperture ( ) ( ){ } ( ) ( )ρπ ρπ ρ a aJ argFTG 2 2 : 12 0 == Table of Content Diffractions
    93. 93. 93 SOLO References Diffractions 2. 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, 1976 8. M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986 6. E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979 3. 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 9. J. Meyer-Arendt, “Introduction to Classical & Modern Optics”, 3th Ed., Prentince Hall, 1989
    94. 94. 94 SOLO References [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
    95. 95. 95 SOLO References [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, 1986 Table of Content OPTICS
    96. 96. January 4, 2015 96 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
    97. 97. 97 SOLO Diffraction Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Circular Aperture Hecht p.491 Hecht p.492
    98. 98. 98 SOLO Diffraction Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Circular Obstacles
    99. 99. 99 SOLO Diffraction Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Fresnel Zone Plate
    100. 100. 100 SOLO Diffraction Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Fresnel Diffraction by a Slit Hecht p.504 a Fresnel Diffraction Hecht p.504 b
    101. 101. 101 SOLO Diffraction Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Semi-Infinite Opaque Screen Hecht p.506 a Hecht p.506
    102. 102. 102 SOLO Diffraction Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Semi-Infinite Opaque Screen Hecht p.506 a Hecht p.507
    103. 103. 103 SOLO Diffraction Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Diffraction by a Narrow Obstacle
    104. 104. 104 Optics - DiffractionSOLO Elmore p.409
    105. 105. 105 Optics - DiffractionSOLO Hecht p.487 a Hecht p.487 b Hecht p.489 Hecht p.490 a Hecht p.490 b
    106. 106. 106 Optics - DiffractionSOLO Hecht p.500
    107. 107. 107 Optics - DiffractionSOLO Jenkins p.320 Jenkins p.323 Jenkins p.343 Klein Furata p.362
    108. 108. 108 Optics - DiffractionSOLO Reynolds 19 Reynolds 20 a Reynolds 20 b
    109. 109. 109 Optics - DiffractionSOLO Reynolds 62 a Reynolds 62 b Reynolds 62 c
    110. 110. 110 Optics - DiffractionSOLO Reynolds 63 a Reynolds 63 b

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