DOCSLIB.ORG

Fraunhofer Diffraction Effects on Total Power for a Planckian Source

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fraunhofer Diffraction Effects on Total Power for a Planckian Source Volume 106, Number 5, September–October 2001 Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 106, 775–779 (2001)] Fraunhofer Diffraction Effects on Total Power for a Planckian Source Volume 106 Number 5 September–October 2001 Eric L. Shirley An algorithm for computing diffraction ef- Key words: diffraction; Fraunhofer; fects on total power in the case of Fraun- Planckian; power; radiometry. National Institute of Standards and hofer diffraction by a circular lens or aper- Technology, ture is derived. The result for Fraunhofer diffraction of monochromatic radiation is Gaithersburg, MD 20899-8441 Accepted: August 28, 2001 well known, and this work reports the re- sult for radiation from a Planckian source. [email protected] The result obtained is valid at all temper- atures. Available online: http://www.nist.gov/jres 1. Introduction Fraunhofer diffraction by a circular lens or aperture is tion, because the detector pupil is overfilled. However, a ubiquitous phenomenon in optics in general and ra- diffraction leads to losses in the total power reaching the diometry in particular. Figure 1 illustrates two practical detector. situations in which Fraunhofer diffraction occurs. In the All of the above diffraction losses have been a subject first example, diffraction limits the ability of a lens or of considerable interest, and they have been considered other focusing optic to focus light. According to geo- by Blevin [2], Boivin [3], Steel, De, and Bell [4], and metrical optics, it is possible to focus rays incident on a Shirley [5]. The formula for the relative diffraction loss lens to converge at a focal point. In practice, even with in spectral power is already well known, and it is noted the focal point at the center of a finite-sized circular below. However, a formula for the diffraction loss in the detector, some of the light incident on the lens fails to total power for the case of a Planckian source such as a be collected, giving rise to a “diffraction loss.” star or blackbody appears to have been given only in the The second example can occur when using a source, high-temperature limit. This article reports a formula such as a blackbody cavity source viewed through a for the diffraction loss in the power for such a source at small “pinhole” aperture, to calibrate optical systems all temperatures. The formula is useful for predicting intended for observing celestial objects or for use in the loss in cases such as the examples discussed, and it remote-sensing applications. In this case, the angle sub- can be used as an independent check for numerically tended by the cavity opening at the aperture is larger calculated diffraction losses. than the angle subtended by the detector pupil on the For the radiation present at a given wavelength ␭, the dark side of the aperture [1]. Because the former angle diffraction loss can be described in terms of a unit-less is larger than the latter one, geometrical optics suggests parameter, v =2␲⌿R/␭. R is the radius of the lens or that the total power detected depends only on the black- aperture. The angle ⌿ is defined in either of two ways: body temperature and a geometrical factor related to the either 2⌿ is the full angle subtended by the detection pinhole aperture, detector optics, and relative separa- pupil at the focusing optic, or 2⌿ is the full angle 775 Volume 106, Number 5, September–October 2001 Journal of Research of the National Institute of Standards and Technology Fig. 1. Two optical systems are shown, for which the (spectral) power detected is subject to Fraunhofer diffraction. In the first system (a), a lens concentrates radiation on the detector. In the second system (b), radiation from a blackbody cavity may pass through the blackbody cavity opening, pass through the pinhole aperture, and reach the detector pupil. subtended by the blackbody cavity opening at the 2. Derivation of Formula pinhole aperture. For a source in thermal equilibrium at temperature T, the effects of temperature on the In the Fraunhofer case, the ratio of the true spectral distribution of spectral power enter relevant equations power to the spectral power predicted by geometrical through the ratio, c2/(␭T) ≡ Av . Here, c2 = hc/k = optics is 1.438 7752(25) ϫ 10Ϫ2 m K is the second radiation con- 2 2 stant, where the number in parenthesis is the one-stan- F(v)=1Ϫ J0(v) Ϫ J1(v). (1) dard-deviation uncertainty in the last two digits. This implicitly defines another unit-less parameter, Here, Jm (v) is a cylindrical Bessel function. It is neces- A = c2/(2␲⌿RT). The relative effects of diffraction on sary to incorporate this wavelength-dependent factor the total power only depend on this one parameter, A. when evaluating the ratio of the total power detected to As a final note, if the detection pupil subtends a the total power predicted by geometrical optics. For a considerable angle, the diffraction losses can be approx- source whose spectral output obeys the Planck distribu- imately given as the weighted average of losses arising, tion, this unit-less ratio is given by in the case of an infinitesimal detector, for values of ⌿ ⌿ ϱ varying between minus the half-angle subtended by 3 Ϫ1 2 2 ͐dvv [exp(Av ) Ϫ 1] [1 Ϫ J0(v) Ϫ J1(v)] the detector and ⌿ plus the same half-angle. Analogous 0 F(A)= ϱ . (2) weighting can be used, in the context of the first exam- Ϫ ͐dvv 3[exp(Av ) Ϫ 1] 1 ple, to account for the finite angular diameter of a source 0 (smaller than 2⌿). Such averaging is discussed else- where [5] but is not discussed further in this work, The value of the denominator is familiar, being given which henceforth assumes a single effective value of ⌿. by 776 Volume 106, Number 5, September–October 2001 Journal of Research of the National Institute of Standards and Technology ϱ ϱ ϱ ͐dvv 3[exp(Av ) Ϫ 1]Ϫ1 = ͚ ͐dvv 3 exp(ϪnAv) High-Temperature Case 0 n=1 0 ϱ =6͚ (nA)Ϫ4 =6AϪ4␨(4). (3) To evaluate the numerator in the high-temperature n=1 case, the relation in Eq. (2) is rewritten to render Here, ␨(n) is the Riemann zeta function. When evaluat- ing the numerator, two techniques have been found to be Q(A)=6AϪ4␨(4)[1 Ϫ F(A)] helpful: one for the “low-temperature” case (defined as ϱ ϱ 3 2 2 A > 4) and one for the “high-temperature” case (defined = ͚ ͐dvv exp(ϪnAv)[J0(v)+J1(v)] as A < 4). By such definitions, “high” and “low” temper- n=1 0 ature cases arise depending on the average diffraction ϱ = ͚ Q1(nA), (8) loss for a source at temperature T. This depends on T n=1 and the geometry through A. In either case, one first with makes use of the relation, ϱ ϱ ͐ 3 Ϫ 2 2 2 2 Q1(A)= dvv exp( Av )[J0(v)+J1(v)]. (9) J0[2v sin(␪/2)] = J0(v)+2͚ Jm (v)cos(m␪), (4) 0 m=1 Use of Eq. (5) and the relation, from which follows, ϱ 1 1 2␲ ͐dvexp(Ϫ␣v)J (␤v)= , (10) 2 2 ͐ ␪ ␪ ␪ 0 2 2 J0(v)+J1(v)= d (1 + cos )J0[2vsin( /2)]. (5) 0 ͙␣ + ␤ 2␲ 0 one has Low-Temperature Case 3 2␲ ␪ ͩϪ d ͪ 1 ͐ ␪ 1 + cos To evaluate the numerator in the low-temperature Q1(A)= d . (11) dA 2␲ 0 ͙ 2 2 ␪ case, series expansion of the latter Bessel function in Eq. A + 4sin ( /2) (5) yields Making the abbreviation, w = 4/(A 2 + 4), one has 1 ϱ (Ϫ1)s v 2s 2␲ 2 v 2 v ͚ ͐ ␪ Ϫ 2 ␪ 2s ␪ 3 ␲/2 2␾ J0( )+J1( )=␲ 2 d [1 sin ( /2)]sin ( /2) ͩϪ d ͪͫ 4 ͐ ␾ cos ͬ s=0 (s!) 0 Q1(A)= d dA ␲͙A 2 +4 0 ͙1 Ϫ wcos2␾ ϱ (Ϫ1)s v 2s (2s Ϫ 1)!! (2s + 1)!! ͚ ͫ Ϫ ͬ 3 K w Ϫ E w =2 2 ͩϪ d ͪͭ 4 ͫ ( ) ( )ͬͮ s=0 (s!) (2s)!! (2s + 2)!! = dA ␲͙A 2 +4 w ϱ (Ϫ1)s v 2s (2s Ϫ 1)!! ͚ ͫ ͬ 3 ͙A 2 =2 2 (6) ͩϪ d ͪͭ +4 Ϫ ͮ s=0 (s!) (2s + 2)!! = [K(w) E(w)] , (12) dA ␲ This permits one to rewrite Eq. (2) as follows: where E(w)andK(w) are respectively complete elliptic Ϫ integrals of the first and second kind, defined according Q(A) ≡ 6A 4␨(4)[1 Ϫ F(A)] to the convention,7 ϱ ϱ ␲/2 ͸͵ 3 Ϫ 2 2 ͐ ␾͙ Ϫ 2␾ = dvv exp( nAv)[J0(v)+J1(v)] E(w)= d 1 wsin , n=1 0 0 ␲ ϱ /2 d␾ (Ϫ1)s(2s + 3)!␨(2s +4) (2s Ϫ 1)!! K(w)= ͐ . (13) ͸ ͫ ͬ 2 =2 2 2s+4 0 ͙1 Ϫ wsin ␾ s=0 (s!) A (2s + 2)!! ϱ (In a different convention, w is replaced by w 2 in the (2␲)2s+4(2s + 3)!(2s Ϫ 1)!!B = Ϫ ͸ 2s+4 , (7) integrand but nowhere else.) From the properties of (s!)2(2s + 4)!(2s + 2)!!A 2s+4 s=0 elliptic integrals [8], one may obtain In this expression, B is a Bernoulli number. The sum i 4 ͫ4(2 + A 2)E(w) Ϫ A 2K(w)ͬ converges for all A > 2, and one obtains an error in F(A) Q1(A)= 3 2 3/2 , (14) Ϫ ␲A (A +4) estimated to be as small as 10 14 for A > 4 if one sums up to s = 28.6 and 777 Volume 106, Number 5, September–October 2001 Journal of Research of the National Institute of Standards and Technology Ϫ Ϫ d ͫ 4E(w) ͬ One should include only the first m 1 terms in the Q1(A)= .
Load more

Recommended publications
  • Phase Retrieval Without Prior Knowledge Via Single-Shot Fraunhofer Diffraction Pattern of Complex Object
    Phase retrieval without prior knowledge via single-shot Fraunhofer diffraction pattern of complex object An-Dong Xiong1 , Xiao-Peng Jin1 , Wen-Kai Yu1 and Qing Zhao1* Fraunhofer diffraction is a well-known phenomenon achieved with most wavelength even without lens. A single-shot intensity measurement of diffraction is generally considered inadequate to reconstruct the original light field, because the lost phase part is indispensable for reverse transformation. Phase retrieval is usually conducted in two means: priori knowledge or multiple different measurements. However, priori knowledge works for certain type of object while multiple measurements are difficult for short wavelength. Here, by introducing non-orthogonal measurement via high density sampling scheme, we demonstrate that one single-shot Fraunhofer diffraction pattern of complex object is sufficient for phase retrieval. Both simulation and experimental results have demonstrated the feasibility of our scheme. Reconstruction of complex object reveals depth information or refraction index; and single-shot measurement can be achieved under most scenario. Their combination will broaden the application field of coherent diffraction imaging. Fraunhofer diffraction, also known as far-field diffraction, problem12-14. These matrix complement methods require 4N- does not necessarily need any extra optical devices except a 4 (N stands for the dimension) generic measurements such as beam source. The diffraction field is the Fourier transform of Gaussian random measurements for complex field15. the original light field. However, for visible light or X-ray Ptychography is another reliable method to achieve image diffraction, the phase part is hardly directly measurable1. with good resolution if the sample can endure scanning16-19. Therefore, phase retrieval from intensity measurement To achieve higher spatial resolution around the size of atom, becomes necessary for reconstructing the original field.
    [Show full text]
  • Fraunhofer Diffraction with a Laser Source
    Second Year Laboratory Fraunhofer Diffraction with a Laser Source L3 ______________________________________________________________ Health and Safety Instructions. There is no significant risk to this experiment when performed under controlled laboratory conditions. However: • This experiment involves the use of a class 2 diode laser. This has a low power, visible, red beam. Eye damage can occur if the laser beam is shone directly into the eye. Thus, the following safety precautions MUST be taken:- • Never look directly into the beam. The laser key, which is used to switch the laser on and off, preventing any accidents, must be obtained from and returned to the lab technician before and after each lab session. • Do not remove the laser from its mounting on the optical bench and do not attempt to use the laser in any other circumstances, except for the recording of the diffraction patterna. • Be aware that the laser beam may be reflected off items such as jewellery etc ______________________________________________________________ 26-09-08 Department of Physics and Astronomy, University of Sheffield Second Year Laboratory 1. Aims The aim of this experiment is to study the phenomenon of diffraction using monochromatic light from a laser source. The diffraction patterns from single and multiple slits will be measured and compared to the patterns predicted by Fourier methods. Finally, Babinet’s principle will be applied to determine the width of a human hair from the diffraction pattern it produces. 2. Apparatus • Diode Laser – see Laser Safety Guidelines before use, • Photodiode, • Optics Bench, • Traverse, • Box of apertures, • PicoScope Analogue to Digital Converter, • Voltmeter, • Personal Computer. 3. Background (a) General The phenomenon of diffraction occurs when waves pass through an aperture whose width is comparable to their wavelength.
    [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]
  • Monte Carlo Ray-Trace Diffraction Based on the Huygens-Fresnel Principle
    Monte Carlo Ray-Trace Diffraction Based On the Huygens-Fresnel Principle J. R. MAHAN,1 N. Q. VINH,2,* V. X. HO,2 N. B. MUNIR1 1Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, USA 2Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, VA 24061, USA *Corresponding author: [email protected] The goal of this effort is to establish the conditions and limits under which the Huygens-Fresnel principle accurately describes diffraction in the Monte Carlo ray-trace environment. This goal is achieved by systematic intercomparison of dedicated experimental, theoretical, and numerical results. We evaluate the success of the Huygens-Fresnel principle by predicting and carefully measuring the diffraction fringes produced by both single slit and circular apertures. We then compare the results from the analytical and numerical approaches with each other and with dedicated experimental results. We conclude that use of the MCRT method to accurately describe diffraction requires that careful attention be paid to the interplay among the number of aperture points, the number of rays traced per aperture point, and the number of bins on the screen. This conclusion is supported by standard statistical analysis, including the adjusted coefficient of determination, adj, the root-mean-square deviation, RMSD, and the reduced chi-square statistics, . OCIS codes: (050.1940) Diffraction; (260.0260) Physical optics; (050.1755) Computational electromagnetic methods; Monte Carlo Ray-Trace, Huygens-Fresnel principle, obliquity factor 1. INTRODUCTION here. We compare the results from the analytical and numerical approaches with each other and with the dedicated experimental The Monte Carlo ray-trace (MCRT) method has long been utilized to results.
    [Show full text]
  • Chapter 14 Interference and Diffraction
    Chapter 14 Interference and Diffraction 14.1 Superposition of Waves.................................................................................... 14-2 14.2 Young’s Double-Slit Experiment ..................................................................... 14-4 Example 14.1: Double-Slit Experiment................................................................ 14-7 14.3 Intensity Distribution ........................................................................................ 14-8 Example 14.2: Intensity of Three-Slit Interference ............................................ 14-11 14.4 Diffraction....................................................................................................... 14-13 14.5 Single-Slit Diffraction..................................................................................... 14-13 Example 14.3: Single-Slit Diffraction ................................................................ 14-15 14.6 Intensity of Single-Slit Diffraction ................................................................. 14-16 14.7 Intensity of Double-Slit Diffraction Patterns.................................................. 14-19 14.8 Diffraction Grating ......................................................................................... 14-20 14.9 Summary......................................................................................................... 14-22 14.10 Appendix: Computing the Total Electric Field............................................. 14-23 14.11 Solved Problems ..........................................................................................
    [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]
  • Diffraction I Single-Slit Diffraction
    Lecture 22 (Diffraction I Single-Slit Diffraction) Physics 2310-01 Spring 2020 Douglas Fields Single-Slit Diffraction • As we have already hinted at, and seen, waves don’t behave as we might have expected from our study of geometric optics. • We can see interference fringes even when we only have a single slit! • To understand this phenomena, we have to go back to Huygens’ Principle and phasor diagrams. Doesn’t have to be a slit… • Any obstruction can cause an interference pattern. Single-Slit Diffraction • Because real slits have finite width, then there is not just a single source of Huygens wavelets, but many (infinite?). • As we saw with the two-slit problem, the geometry of the problem is much easier when we go to the limit that the distance to the screen (where the interference pattern is viewed) is much larger than the width of the slit. – Fraunhofer or far-field diffraction. Fraunhofer diffraction of circular • When this isn’t the case, you still aperture get diffraction, it just looks different. – Fresnel or near-field diffraction. "Circular Aperture Fresnel Diffraction high res" by Gisling - "Airy-pattern2" by Epzcaw - Own work. Licensed under CC Own work. Licensed under CC BY-SA 3.0 via Commons - BY-SA 3.0 via Commons - https://commons.wikimedia.org/wiki/File:Circular_Aperture_F https://commons.wikimedia.org/wiki/File:Airy-pattern2.jpg#/m resnel_Diffraction_high_res.jpg#/media/File:Circular_Apertur edia/File:Airy-pattern2.jpg e_Fresnel_Diffraction_high_res.jpg Fresnel and Fraunhofer Diffraction • The only distinction is the distance between the slit and the screen. • In Fraunhofer diffraction the distance is large enough to consider the rays to be parallel.
    [Show full text]
  • Chapter 11. Fraunhofer Diffraction
    ChapterChapter 11.11. FraunhoferFraunhofer Diffraction Diffraction Last lecture • Numerical aperture of optical fiber • Allowed modes in fibers • Attenuation • Modal distortion, Material dispersion, Waveguide dispersion This lecture • Diffraction from a single slit • Diffraction from apertures : rectangular, circular • Resolution : diffraction limit • Diffraction from multiple-slits DiffractionDiffraction regimesregimes Fraunhofer diffraction • Specific sort of diffraction – far-field diffraction – plane wavefront – Simpler maths FraunhoferFraunhofer DiffractionDiffraction Fresnel Diffraction • This is most general form of diffraction – No restrictions on optical layout • near-field diffraction • curved wavefront – Analysis difficult Obstruction Screen FresnelFresnel DiffractionDiffraction 11-1.11-1. FraunhoferFraunhofer Diffraction Diffraction fromfrom aa SingleSingle SlitSlit • Consider the geometry shown below. Assume that the slit is very long in the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction. Δ r0 The contribution to the electric field amplitude Δ at point P due to the wavelet emanating from the element ds in the slit is given by dE ⎛⎞0 r dEP =−⎜⎟exp ⎣⎦⎡⎤ i() krω t 0 ⎝⎠r Let r== r0 for the source element ds at s 0. Then for any element ⎛ dE ⎞ dE = 0 exp ikr⎡⎤+Δ −ω t P ⎜⎟⎜ ⎟ {}⎣⎦()0 ⎝ ()r0 +Δ ⎠ why? We can neglect the path differenceΔ in the amplitude term,. but not in the phase term We letdE0 = EL d s,, where EL is the electric field amplitude assumed uniform over
    [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]