DOCSLIB.ORG

Fresnel and Fraunhofer Diffraction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fresnel and Fraunhofer Diffraction Fresnel and Fraunhofer Diffraction 4. 1 Background 4.1.1 The intensity of a wave field 4.1.2 The Huygens-Fresnel principle in rectangular coordinates 4.2 The Fresnel approximation 4. 2. 1 Positive vs. negative phases 4.2.2 Accuracy of the Fresnel approximation 4.2.3 The Fresnel approximation and the angular spectrum 4.2.4 Fresnel diffraction between confocal spherical surfaces 4. 3 The Fraunhofer approximation 4.4 Examples of Fraunhofer diffraction patterns 4.4.1 Rectangular aperture 4.4.2 Circular aperture 4. 4. 3 Thin sinusoidal aperture grating 4.4.4 Thin sinusoidal phase grating 4.5 Examples of Fresnel diffraction calculations 4.5.1 Fresnel diffraction by a square aperture 4. 5 2 Fresnel diffraction by a sinusoidal amplitude grating-Talbot images 4.1 Background • In chapter 3 we dealt with most general form of the diffraction theory. • In c hap ter 4 we w ill dea l w ith – Intensity of a wave field – Huygens-Fresnel principle – Certain approximations to reduce the problem to a simpler mathematical form. These approximations are: • Fresnel • Fraunhofer – We consider the wave propagation phenomenon as a system. – The approximations will be valid for certain class of inputs. – Preparation for the calculations related to the approximations Spring 2010 Eradat Physics Dept. SJSU 2 4.1.1 The intensity of a wave field Intensity is the physically measurable attribute of an optical wavefield Intensity and power density are not the same but proportional Intensity of a scalar monochromatic wave at point P IP()|= UP ()|2 For a narrow-band (not perfectly monochromatic) intensity is given by IP( )== | uPt ( ,)|2 and IPt ( ,) | uPt ( ,)|2 An infinite Instantaneous time average Intensity In calculating a diffraction pattern, we are looking for the intensity of the pa ttern. Spring 2010 Eradat Physics Dept. SJSU 3 4.1.2 The Huygens-Fresnel principle in rectangular coordinates According to the first Rayleigh-Somefeld solution the diffracted field on the xy plane due to the aperture on the ξη plane is UI ( P0 ) and the Huygens-Fresnel principle can be written as: 1 e jkr01 z UP( ) = UP() cosθθdsnrwhere cos== cos()( , ) I 01∫∫Σ 01 jλ rr01 01 zejkr01 Uxy( , ) = U()(ξη, ) dd ξ η, ∫∫Σ 2 jrλ 01 η 222 y where rzx01 =+−+−()()ξη y ξ x Two approximations are used in this result: P0 1) inherent approximation in the scalar theory Σ z θ 2) r01 >> λ P Spring 2010 Eradat Physics Dept. SJSU 1 4 4.2 The Fresnel approximation I Goal: to reduce the H-F principle to a simple and usable expression WhithibWe achieve this by approx itifimations for r01 : nn(1)− Binomial expansion: (xy+=+ )nn x nxy n−−122 + x n y ++= ... yn n, 1,2,3,... 2! 11 (1+=±bbb )±1/2 1 ∓ 2 ±−<≤... for 1b 1: the higher order terms are negligable 28 22 2 2 ⎛⎞⎛⎞xy−−ξη⎡⎤11 ⎛⎞⎛⎞ x − ξ y − η rz01 =+11⎜⎟⎜⎟ + ≈+ z⎢⎥ ⎜⎟⎜⎟ + +... ⎝⎠⎝⎠zz⎣⎦⎢⎥22 ⎝⎠⎝⎠ z z Where do we cut the series? We will use r01 in the diffracted field equation zejkr01 Ux( ,(,)y) = Uddξ ηξη ∫∫Σ 2 jrλ 01 jkr01 The term eris very sensitive to the values of 01 specially since it is multiplied blby a very large num ber k = 2/πλλ. IthiiblIn the visible ofthf the or der 107. We keep two 2 terms for the exponent. For rz01 error introduced by dropping all terms but is small. ⎧⎫k jkr01 jkz ∞ ⎡⎤22 zee ⎨⎬jx⎣()()−+−ξη y⎦ UUddUU ( x, yU) ==()(ξξξ,η)( ddUξ η () ξ,η)e⎩⎭2z dd ξη ∫∫Σ 2 ∫ ∫ jrjzλλ01 −∞ The integration limit is let to ∞ using the usual boundary conditions. Spring 2010 Eradat Physics Dept. SJSU 5 4.2 The Fresnel approximation II jkz ∞ ⎧⎫k ⎡ 22⎤ e ⎨⎬jx⎣⎦()()−+−ξη y Uxy(),(,)= ∫∫ Uξη e⎩⎭2z dd ξ η this looks like a convolution jzλ −∞ ∞ jkz ⎧⎫k ⎡⎤22 e ⎨⎬jx⎣⎦()()−+ξη+ y− Uxy( ,(,)(,)(,)) =−−= Uξη hx ξ y η dd ξ η where hxy e⎩⎭2z ∫∫ jzλ −∞ First form of the Fresnel diffraction integral Another form of the Fresnel diffraction integral is expressed as the following k j ξη22+ Fourier transform of the U (ξ ,)η e 2 z( ) which is complex field just to the right of aperture multiplied by a quadratic phase factor jkz kk22∞ 22 2π e jxy()++⎧⎫ j()ξη −+jxy()ξη Uxy(),(,)= e22zz⎨⎬ Uξη e eλz dd ξ η jzλ ∫∫ −∞ ⎩⎭ Seond form of the Fresnel diffraction integral Observation in the near field of the aperture or Fresnel diffraction region xy--ξη Where r >>λ,1,1, < < 01 zz and scalar theory approximation are assumed Spring 2010 Eradat Physics Dept. SJSU 6 4.2.1 Positive vs. negative phases Goal: to understand the meaning of the signs of the y phase exponentials: in the sppqherical wave e jkr01 and its equivalent k jxy()22+ in the quadratic approximation ez2z (0)for > z Sign convention: our phasors rotate in the clockwise direction (the angle becomes more negative as time goes) and their time dependence is e− jt2πν We move in space in such a way that we encounter Wavefront Wavefront emitted emitted portions of the wavefield that were emitted earlier in time. later earlier The phase must become more positive since the y phasor had not have time to rotate as far in clockwise. k We move in space in such a way that we encounter θ z portions of the wavefield that were emitted later in time. The phasor will have advanced in the clockwise direction, therefore the phase must become more negative. Spring 2010 Eradat Physics Dept. SJSU 7 4.2.2 Accuracy of the Fresnel approximation I Fresnel approximation replaced the spherical secondary wavelets with parabolic wavefronts in the Huygens-Fresnel principle jkr jkz ∞ ⎧⎫k 2 2 01 ⎨⎬jx⎡⎤()()−+−ξη y ze e ⎩⎭2z ⎣⎦ Uxy( , ) = U()(ξη, )(2 dd ξ η→ U () ξη, ) e ddξη jrjzλλ∫∫Σ ∫∫ 01 −∞ Parabolic wavelets Spherical wavelets The accuracy of this approximation depends on the size of the higher order terms in binomial expansion. A sufficient condition for accuracy is: ⎡⎤ ⎢⎥ 22 222 ⎢⎥1 ⎛⎞⎛⎞x −ξ 11yxy−−−ηξη⎪⎪⎧⎫ ⎛⎞⎛⎞ rz01 ≈+⎢⎥1 ⎜⎟⎜⎟++⎨⎬ ⎜⎟⎜⎟ + ... ⎢⎥2 ⎝⎠⎝⎠z 28zzz⎩⎭⎪⎪ ⎝⎠⎝⎠ ⎢⎥ maximum phase change due to dropping ⎢⎥2 ⎣⎦b /8 term must be much less than one radian 2 φ ⎧⎫2 2 jkr 2 21πξ⎪⎪⎛⎞x − ⎛⎞y −η eO01 →Δφ() = ⎨⎬⎜⎟+<<⎜⎟ 1 λ 8 ⎝⎠z ⎝⎠z ⎩⎭⎪⎪Max π 222 zxy3 >>()() −ξη + − 4λ {}Max Spring 2010 Eradat Physics Dept. SJSU 8 4.2.2 Accuracy of the Fresnel approximation II Example: calculate the safe distance to use the Fresnel approximation for a circular aperture of size 1cm and a circular observation ragion of 10.525cm with a light of λμ=>> m. (Answere: z cm) π 222 zx3 >>()() −ξη+−yxy hint: − ξη and − should have their 4λ { }Max maximum possible values to evaluate the condition. If the higher order terms do not change the value of the Fresnel integral substantially, we can use the approximation. They do not need to be small in this case. Spring 2010 Eradat Physics Dept. SJSU 9 4.2.3 The Fresnel approximation and the angular spectrum I Goal: understand the implications of the Fresnel approximations from the point of view of angular spectrum method of analysis. We compare the transfer function of propagation through free space, predicted by RS scalar diffraction theory, with the transfer function predicted by the Fresnel analysis General spatial phase dispersion ⎧representing propagation ⎪ ⎪ ⎛⎞2π z 2 2 Hff(,)= ⎜⎟jff1()−−()λλXY ← RS theory RS X Y ⎨ ef⎝⎠λ 22+<1f / λ ⎪ X Y ⎩⎪0 otherwise A constant Qadratic phase phase delay dispersion due to traveling jkz ⎧⎫k ⎡⎤22 ⎨⎬jx()()−+−ξη y jzf22 f e ⎩⎭2z ⎣⎦FT jkz −+πλ ()XY hx(,yyff )=⎯⎯→=eHFXY (ff , ) e e jzλ All plane waves Different plane-wave suffer equally components suffer Fresnel approximation impulse response different phase delays Spring 2010 Eradat Physics Dept. SJSU 10 4.2.3 The Fresnel approximation and the angular spectrum II ⎛⎞2π z 2 2 ⎧ ⎜⎟jff1()−−()λλXY ⎪ ⎝⎠λ 22 eff XY+<1/ λ HffRS(,) X Y =←⎨ RS theory ⎩⎪0 otherwise 22 jkz −+jzfπλ ()XY f HffFXY(,)= ee We can see that HffFXY(,) is an approximation to the H RSXY (,) ff Applying the binomial expansion to the H RS(,)ff X Y we get: 22 222()λλffXY() 1()1−−≈−−()λλf f2 if () λ f 1 and () λ f 1 XY22 X Y ⎛⎞⎡⎤22 2π z ()λλffXY() ⎛⎞2π z 2 2 ⎜⎟j ⎢⎥1−− ⎜⎟jff1()−−()λλXY ⎜⎟λ ⎢⎥22 jzf22 f ⎝⎠⎝⎠λ ⎣⎦jkz −+πλ ()XY ee≈== eeHffFXY(,) Conclusion: 2 2 2 ⎛⎞kx HffRS(,) X Y≈== Hff F (,) X Y When the conditions: ()λα f X ⎜⎟ 1, ⎝⎠||k 2 2 2 ⎛⎞kY ()λβfY ==⎜⎟1 are satisfied. So Fresnel approxilation is equivalent ⎝⎠||k to the paraxial approximation that is limitted to small propagation angles. Spring 2010 Eradat Physics Dept. SJSU 11 4.2.4 Fresnel diffraction between confocal spherical surfaces I Goal: analysis of diffraction between two confocal spherical surfaces Confocal spheres: center of each lies on the surface of the other. We set the spheres tangant to the plance we used before. Located at zzzr==0. and 01 is the distance between two spherical caps. We write the equations for both surfaces and find the distance between them. Make paraxial approximation by using the binomial expansion. Assuming the extend of the spherical caps about the z axis is small compared to the radii of the spheres, i.e. for z>> x − ξ and y −η , we get xyξη η rzx=+−+−→≈222()()ξη y rz -- y 01 01 zz ξ x zejkr01 Uxy(),(,)= Uξη dξηd , ∫∫Σ 2 jλ r01 r01 jkz ∞ ⎧⎫2π e ⎨−+jxy[ ξη]⎬ Uxy,(,)= Uξ ηξη e⎩⎭λz dd ( ) ∫∫ jzλ −∞ Field on the right hand Fourier transform of the Field on the z spherical cap left hand spherical cap Spring 2010 Eradat Physics Dept.
Load more

Recommended publications
  • Fresnel Diffraction.Nb Optics 505 - James C
    Fresnel Diffraction.nb Optics 505 - James C. Wyant 1 13 Fresnel Diffraction In this section we will look at the Fresnel diffraction for both circular apertures and rectangular apertures. To help our physical understanding we will begin our discussion by describing Fresnel zones. 13.1 Fresnel Zones In the study of Fresnel diffraction it is convenient to divide the aperture into regions called Fresnel zones. Figure 1 shows a point source, S, illuminating an aperture a distance z1away. The observation point, P, is a distance to the right of the aperture. Let the line SP be normal to the plane containing the aperture. Then we can write S r1 Q ρ z1 r2 z2 P Fig. 1. Spherical wave illuminating aperture. !!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!! 2 2 2 2 SQP = r1 + r2 = z1 +r + z2 +r 1 2 1 1 = z1 + z2 + þþþþ r J þþþþþþþ + þþþþþþþN + 2 z1 z2 The aperture can be divided into regions bounded by concentric circles r = constant defined such that r1 + r2 differ by l 2 in going from one boundary to the next. These regions are called Fresnel zones or half-period zones. If z1and z2 are sufficiently large compared to the size of the aperture the higher order terms of the expan- sion can be neglected to yield the following result. l 1 2 1 1 n þþþþ = þþþþ rn J þþþþþþþ + þþþþþþþN 2 2 z1 z2 Fresnel Diffraction.nb Optics 505 - James C. Wyant 2 Solving for rn, the radius of the nth Fresnel zone, yields !!!!!!!!!!! !!!!!!!! !!!!!!!!!!! rn = n l Lorr1 = l L, r2 = 2 l L, , where (1) 1 L = þþþþþþþþþþþþþþþþþþþþ þþþþþ1 + þþþþþ1 z1 z2 Figure 2 shows a drawing of Fresnel zones where every other zone is made dark.
    [Show full text]
  • Huygens Principle; Young Interferometer; Fresnel Diffraction
    Today • Interference – inadequacy of a single intensity measurement to determine the optical field – Michelson interferometer • measuring – distance – index of refraction – Mach-Zehnder interferometer • measuring – wavefront MIT 2.71/2.710 03/30/09 wk8-a- 1 A reminder: phase delay in wave propagation z t t z = 2.875λ phasor due In general, to propagation (path delay) real representation phasor representation MIT 2.71/2.710 03/30/09 wk8-a- 2 Phase delay from a plane wave propagating at angle θ towards a vertical screen path delay increases linearly with x x λ vertical screen (plane of observation) θ z z=fixed (not to scale) Phasor representation: may also be written as: MIT 2.71/2.710 03/30/09 wk8-a- 3 Phase delay from a spherical wave propagating from distance z0 towards a vertical screen x z=z0 path delay increases quadratically with x λ vertical screen (plane of observation) z z=fixed (not to scale) Phasor representation: may also be written as: MIT 2.71/2.710 03/30/09 wk8-a- 4 The significance of phase delays • The intensity of plane and spherical waves, observed on a screen, is very similar so they cannot be reliably discriminated – note that the 1/(x2+y2+z2) intensity variation in the case of the spherical wave is too weak in the paraxial case z>>|x|, |y| so in practice it cannot be measured reliably • In many other cases, the phase of the field carries important information, for example – the “history” of where the field has been through • distance traveled • materials in the path • generally, the “optical path length” is inscribed
    [Show full text]
  • Diffraction Effects in Transmitted Optical Beam Difrakční Jevy Ve Vysílaném Optickém Svazku
    BRNO UNIVERSITY OF TECHNOLOGY VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ FACULTY OF ELECTRICAL ENGINEERING AND COMMUNICATION DEPARTMENT OF RADIO ELECTRONICS FAKULTA ELEKTROTECHNIKY A KOMUNIKAČNÍCH TECHNOLOGIÍ ÚSTAV RADIOELEKTRONIKY DIFFRACTION EFFECTS IN TRANSMITTED OPTICAL BEAM DIFRAKČNÍ JEVY VE VYSÍLANÉM OPTICKÉM SVAZKU DOCTORAL THESIS DIZERTAČNI PRÁCE AUTHOR Ing. JURAJ POLIAK AUTOR PRÁCE SUPERVISOR prof. Ing. OTAKAR WILFERT, CSc. VEDOUCÍ PRÁCE BRNO 2014 ABSTRACT The thesis was set out to investigate on the wave and electromagnetic effects occurring during the restriction of an elliptical Gaussian beam by a circular aperture. First, from the Huygens-Fresnel principle, two models of the Fresnel diffraction were derived. These models provided means for defining contrast of the diffraction pattern that can beused to quantitatively assess the influence of the diffraction effects on the optical link perfor- mance. Second, by means of the electromagnetic optics theory, four expressions (two exact and two approximate) of the geometrical attenuation were derived. The study shows also the misalignment analysis for three cases – lateral displacement and angular misalignment of the transmitter and the receiver, respectively. The expression for the misalignment attenuation of the elliptical Gaussian beam in FSO links was also derived. All the aforementioned models were also experimentally proven in laboratory conditions in order to eliminate other influences. Finally, the thesis discussed and demonstrated the design of the all-optical transceiver. First, the design of the optical transmitter was shown followed by the development of the receiver optomechanical assembly. By means of the geometric and the matrix optics, relevant receiver parameters were calculated and alignment tolerances were estimated. KEYWORDS Free-space optical link, Fresnel diffraction, geometrical loss, pointing error, all-optical transceiver design ABSTRAKT Dizertačná práca pojednáva o vlnových a elektromagnetických javoch, ku ktorým dochádza pri zatienení eliptického Gausovského zväzku kruhovou apretúrou.
    [Show full text]
  • Laser Beam Profile & Beam Shaping
    Copyright by Priti Duggal 2006 An Experimental Study of Rim Formation in Single-Shot Femtosecond Laser Ablation of Borosilicate Glass by Priti Duggal, B.Tech. Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering The University of Texas at Austin August 2006 An Experimental Study of Rim Formation in Single-Shot Femtosecond Laser Ablation of Borosilicate Glass Approved by Supervising Committee: Adela Ben-Yakar John R. Howell Acknowledgments My sincere thanks to my advisor Dr. Adela Ben-Yakar for showing trust in me and for introducing me to the exciting world of optics and lasers. Through numerous valuable discussions and arguments, she has helped me develop a scientific approach towards my work. I am ready to apply this attitude to my future projects and in other aspects of my life. I would like to express gratitude towards my reader, Dr. Howell, who gave me very helpful feedback at such short notice. My lab-mates have all positively contributed to my thesis in ways more than one. I especially thank Dan and Frederic for spending time to review the initial drafts of my writing. Thanks to Navdeep for giving me a direction in life. I am excited about the future! And, most importantly, I want to thank my parents. I am the person I am, because of you. Thank you. 11 August 2006 iv Abstract An Experimental Study of Rim Formation in Single-Shot Femtosecond Laser Ablation of Borosilicate Glass Priti Duggal, MSE The University of Texas at Austin, 2006 Supervisor: Adela Ben-Yakar Craters made on a dielectric surface by single femtosecond pulses often exhibit an elevated rim surrounding the crater.
    [Show full text]
  • This Is a Repository Copy of Ptychography. White Rose
    This is a repository copy of Ptychography. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/127795/ Version: Accepted Version Book Section: Rodenburg, J.M. orcid.org/0000-0002-1059-8179 and Maiden, A.M. (2019) Ptychography. In: Hawkes, P.W. and Spence, J.C.H., (eds.) Springer Handbook of Microscopy. Springer Handbooks . Springer . ISBN 9783030000684 https://doi.org/10.1007/978-3-030-00069-1_17 This is a post-peer-review, pre-copyedit version of a chapter published in Hawkes P.W., Spence J.C.H. (eds) Springer Handbook of Microscopy. The final authenticated version is available online at: https://doi.org/10.1007/978-3-030-00069-1_17. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Ptychography John Rodenburg and Andy Maiden Abstract: Ptychography is a computational imaging technique. A detector records an extensive data set consisting of many inference patterns obtained as an object is displaced to various positions relative to an illumination field.
    [Show full text]
  • 4. Fresnel and Fraunhofer Diffraction
    13. Fresnel diffraction Remind! Diffraction regimes Fresnel-Kirchhoff diffraction formula E exp(ikr) E PF= 0 ()θ dA ()0 ∫∫ irλ ∑ z r Obliquity factor : F ()θθ= cos = r zE exp (ikr ) E x,y = 0 ddξ η () ∫∫ 2 irλ ∑ Aperture (ξ,η) Screen (x,y) 22 rzx=+−+−2 ()()ξ yη 22 22 ⎡⎤11⎛⎞⎛⎞xy−−ξη()xy−−ξ ()η ≈+zz⎢⎥1 ⎜⎟⎜⎟ + =+ + ⎣⎦⎢⎥22⎝⎠⎝⎠zz 22 zz ⎛⎞⎛⎞xy22ξη 22⎛⎞ xy ξη =++++−+z ⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠22zz 22 zz⎝⎠ zz E ⎡⎤k Exy(),expexp=+0 ()ikz i() x22 y izλ ⎣⎦⎢⎥2 z ⎡⎤⎡⎤kk22 ×+−+∫∫ exp⎢⎥⎢⎥ii()ξ ηξ exp()xyη ddξη ∑ ⎣⎦⎣⎦2 zz E ⎡⎤k Exy(),expexp=+0 ()ikz i() x22 y r izλ ⎣⎦⎢⎥2 z ⎡⎤⎡⎤kk ×+−+exp iiξ 22ηξexp xyη dξdη ∫∫ ⎢⎥⎢⎥() () Aperture (ξ,η) Screen (x,y) ∑ ⎣⎦⎣⎦2 zz ⎡⎤⎡⎤kk22 =+−+Ci∫∫ exp ⎢⎥⎢⎥()ξ ηξexp ixydd()η ξη ∑ ⎣⎦⎣⎦2 zz Fresnel diffraction ⎡⎤⎡⎤kk E ()xy,(,)=+ C Uξ ηξηξηξηexp i ()22 exp−i() x+ y d d ∫∫ ⎣⎦⎣⎦⎢⎥⎢⎥2 zz k 22 ⎧ j ()ξη+ ⎫ Exy(, )∝F ⎨ U()ξη , e2z ⎬ ⎩⎭ Fraunhofer diffraction ⎡⎤k Exy(),(,)=−+ C Uξη expi() xξ y η dd ξ η ∫∫ ⎣⎦⎢⎥z =−+CU(,)expξ ηξθηθξη⎡⎤ ik sin sin dd ∫∫ ⎣⎦()ξη Exy(, )∝F { U(ξ ,η )} Fresnel (near-field) diffraction This is most general form of diffraction – No restrictions on optical layout • near-field diffraction k 22 ⎧ j ()ξη+ ⎫ • curved wavefront Uxy(, )≈ F ⎨ U()ξη , e2z ⎬ – Analysis somewhat difficult ⎩⎭ Curved wavefront (parabolic wavelets) Screen z Fresnel Diffraction Accuracy of the Fresnel Approximation 3 π 2 2 2 z〉〉[()() x −ξ + y η −] 4λ max • Accuracy can be expected for much shorter distances for U(,)ξ ηξ smooth & slow varing function; 2x−=≤ D 4λ z D2 z ≥ Fresnel approximation 16λ In summary, Fresnel diffraction is … 13-7.
    [Show full text]
  • Numerical Techniques for Fresnel Diffraction in Computational
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2006 Numerical techniques for Fresnel diffraction in computational holography Richard Patrick Muffoletto Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Computer Sciences Commons Recommended Citation Muffoletto, Richard Patrick, "Numerical techniques for Fresnel diffraction in computational holography" (2006). LSU Doctoral Dissertations. 2127. https://digitalcommons.lsu.edu/gradschool_dissertations/2127 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. NUMERICAL TECHNIQUES FOR FRESNEL DIFFRACTION IN COMPUTATIONAL HOLOGRAPHY A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Computer Science by Richard Patrick Muffoletto B.S., Physics, Louisiana State University, 2001 B.S., Computer Science, Louisiana State University, 2001 December 2006 Acknowledgments I would like to thank my major professor, Dr. John Tyler, for his patience and soft-handed guidance in my pursual of a dissertation topic. His support and suggestions since then have been vital, especially in the editing of this work. I thank Dr. Joel Tohline for his unwavering enthusiasm over the past 8 years of my intermittant hologram research; it has always been a source of motivation and inspiration. My appreciation goes out to Wesley Even for providing sample binary star system simulation data from which I was able to demonstrate some of the holographic techniques developed in this research.
    [Show full text]
  • View of Self- Focusing Effects in NGLS; E.G., Spectral Broadening of the Amplified Pulse [25]
    Chapter 8 Beam Shaping and Suppression of Self-focusing in High-Peak-Power Nd:Glass Laser Systems Svetlana G. Lukishova, Yury V. Senatsky, Nikolai E. Bykovsky, and Alexander S. Scheulin Abstract Laser-beam shaping and suppression of beam self-focusing in high-peak-power Nd:glass laser systems are discussed. The role of Fresnel diffraction at apertures at the laser facility as the source of dangerous spatial scales for small-scale self-focusing is illustrated. Methods of the formation of super-Gaussian laser beams for suppressing diffraction ripples on the beam profile and their self-focusing are presented. A brief outline of self-focusing as the primary nonlinear optical process limiting the brightness of Nd:glass laser installations is given. Methods for self-focusing suppression are presented. 8.1 Introduction In 1964, Basov and Krokhin published the concept of using lasers for thermonuclear fusion [1]. After the first fusion neutrons from laser irradia- tion of targets containing deuterium had been detected in 1968 by Basov et al. [2] and were confirmed by Floux et al. in 1970 [3], several laser-fusion centers with high-peak-power Nd:glass laser systems (NGLS) have been established in different countries (USSR, USA, France, UK, Japan, and China). These laser systems operated with light pulses of 10À8 to 10À11 s durations with output powers for a typical laser-amplifier channel from several gigawatts to the terawatt range [4–9]. Modern multi-channel NGLS for fusion applications, e.g., upgraded OMEGA Laser at the University of Rochester’s Laboratory for Laser Energetics (LLE), has 30 kJ at the third harmonic in 60 beamlines [10]; the National Ignition Facility at Lawrence Livermore National Laboratory (LLNL) will have at the same wavelength (351 nm) 1.8 MJ, 500 TW in 192 beamlines [11].
    [Show full text]
  • Introduction to Propagation, Time Reversal and Imaging in Random Media
    An Introduction to Propagation, Time Reversal and Imaging in Random Media Albert C. Fannjiang∗ Department of Mathematics, University of California, Davis, CA 95616-8633 Contents I. Scalar Diffraction Theory 2 A. Introduction 2 B. Kirchhoff's theory of diffraction 5 C. Huygens-Fresnel principle 8 D. Fresnel and Frauhoffer diffraction 9 E. Focal spot size and resolution 11 II. Approximations: weak fluctuation 13 A. Born approximation 13 B. Rytov approximation 13 C. The extended Huygens-Fresnel principle 15 D. Paraxial approximation 15 III. The Wigner Distribution 20 IV. Parabolic Markovian approximation 24 A. White-noise scaling 24 B. Markovian limit 26 V. Two-frequency transport theory 28 A. Parabolic waves 28 1. Two-frequency radiative transfer equations 30 2. The longitudinal and transverse cases 31 B. Spherical waves 32 1. Geometrical radiative transfer 34 2. Spatial (frequency) spread and coherence bandwidth 34 3. Small-scale asymptotics 35 VI. Application: Time reversal 36 ∗Electronic address: [email protected]; I thank the organizers, in particular Professor T. Hou and J. Liu, for the opportunity of visiting Fudan University. A. Anomalous focal spot 38 B. Duality and turbulence-induced aperture 40 C. Coherence length 41 D. Broadband time reversal communications 42 VII. Application: Imaging in random media 43 A. Imaging of phase objects 43 B. Long-exposure imaging 45 C. Short-exposure imaging 47 D. Coherent imaging of multiple point targets in Rician media 49 1. Differential scattered field in clutter 50 2. Imaging functions 51 3. Numerical simulation with a Rician medium 53 E. Coherent imaging in a Rayleigh medium 55 References 58 I.
    [Show full text]
  • Chapter 15S Fresnel-Kirchhoff Diffraction Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 22, 2011)
    Chapter 15S Fresnel-Kirchhoff diffraction Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 22, 2011) Augustin-Jean Fresnel (10 May 1788 – 14 July 1827), was a French physicist who contributed significantly to the establishment of the theory of wave optics. Fresnel studied the behaviour of light both theoretically and experimentally. He is perhaps best known as the inventor of the Fresnel lens, first adopted in lighthouses while he was a French commissioner of lighthouses, and found in many applications today. http://en.wikipedia.org/wiki/Augustin-Jean_Fresnel ______________________________________________________________________________ Christiaan Huygens, FRS (14 April 1629 – 8 July 1695) was a prominent Dutch mathematician, astronomer, physicist, horologist, and writer of early science fiction. His work included early telescopic studies elucidating the nature of the rings of Saturn and the discovery of its moon Titan, the invention of the pendulum clock and other investigations in timekeeping, and studies of both optics and the centrifugal force. Huygens achieved note for his argument that light consists of waves, now known as the Huygens–Fresnel principle, which became instrumental in the understanding of wave-particle duality. He generally receives credit for his discovery of the centrifugal force, the laws for collision of bodies, for his role in the development of modern calculus and his original observations on sound perception (see repetition pitch). Huygens is seen as the first theoretical physicist as he was the first to use formulae in physics. http://en.wikipedia.org/wiki/Christiaan_Huygens ____________________________________________________________________________ Green's theorem Fresnel-Kirchhhoff diffraction Huygens-Fresnel principle Fraunhofer diffraction Fresel diffraction Fresnel's zone Fresnel's number Cornu spiral ________________________________________________________________________ 15S.1 Green's theorem First we will give the proof of the Green's theorem given by (2 2 )d ( ) da .
    [Show full text]
  • Fraunhofer Diffraction
    Fraunhofer Diffraction Equipment Green laser (563.5 nm) on 2-axis translation stage, 1m optical bench, 1 slide holder, slide with four single slits, slide with 3 diffraction gratings, slide with four double slits, slide with multiple slits, pinhole, slide containing an etched Fourier transformed function, screen, tape measure. Purpose To understand and test Fraunhofer diffraction through various apertures. To understand Fraunhofer diffraction in terms of Fourier analysis. To gain experience with laser optics. Theory Suppose, as depicted in Figure 1, that a laser is shone upon a small slit. The light passing through the slit is then allowed to fall on a screen, positioned at some distance, L, from the slit, with its plane perpendicular to the path of the light's propagation. If light traveled in straight lines one would expect the screen to display a single image of diffraction pattern the slit with the rest of the screen in shadow. For a sufficiently small slit, however, it is found that this is not the case. Instead, a diffraction pattern is observed, consisting of a central bright fringe along the slit- screen axis with alternating dark and L screen bright fringes on either side of the central fringe. It is evident, based slit on this observation, that light does laser not travel in straight lines since Figure 1 bright fringes are seen where shadow would be expected. It follows then, that light has the property of being able to "bend" around corners, a property called diffraction. In order to examine diffraction we will assume that light behaves like a wave.
    [Show full text]
  • Fresnel and Fraunhofer Diffraction
    Diffraction • Interference with more than 2 beams – 3, 4, 5 beams – Large number of beams • Diffraction gratings – Equation –Uses • Diffraction by an aperture – Huygen’s principle again, Fresnel zones, Arago’s spot – Qualitative effects, changes with propagation distance – Fresnel number again – Imaging with an optical system, near and far field – Fraunhofer diffraction of slits and circular apertures – Resolution of optical systems • Diffraction of a laser beam LASERS 51 April 03 Interference from multiple apertures L Bright fringes when OPD=nλ 40 x nLλ d x = d Intensity source OPD two slits position on screen screen Complete destructive interference halfway between OPD 1 OPD 1=nλ, OPD 2=2nλ OPD 2 all three waves40 interfere constructively d Intensity source three position on screen equally spaced slits screen OPD 2=nλ, n odd outer slits constructively interfere middle slit gives secondary maxima LASERS 51 April 03 Diffraction from multiple apertures • Fringes not sinusoidal for more than two slits 2 slits • Main peak gets narrower – Center location obeys same 3 slits equation • Secondary maxima appear 4 slits between main peaks – The more slits, the more 5 slits secondary maxima – The more slits, the weaker the secondary maxima become • Diffraction grating – many slits, very narrow spacing – Main peaks become narrow and widely spaced – Secondary peaks are too small to observe LASERS 51 April 03 Reflection and transmission gratings • Transmission grating – many closely spaced slits • Reflection grating – many closely spaced reflecting
    [Show full text]