Lecture 8: March, Diffraction, gratings and spectroscopy

Transcription

1 Photographic Technology Lecture 8: March, 14 7 Diffraction, gratings and spectroscopy Rom Clement Diffraction 1

2 Maxwell equation What s light? Coupling between magnetic field H [A/m] and electric r r r r r r field E [V/m] { } i ω t E(, t) = R E ( ) e { } i ω H (, t) = R H e t ( ) Intensity of light is time average of Poynting vector For a plane wave Intensity of light proportional to E Contrast c Π = R E H 8π I C = I max max I + I min min c I = 4π ε r E µ c = 4π ε r H µ 3/14/7 PhotoTech Rom Clement 3 Assumptions Vacuum Media (simplified Maxwell s equations) Linear Homogenous div ( B r ) = Isotropic div ( E r ) = r Plane wave r B rot ( E ) = Far enough from the t r source so that ~ plane r E wave rot ( B) = µ ε t 3/14/7 PhotoTech Rom Clement 4

3 E = E 1 + E E = E1 + E + E1 E = I1 + I + I = I1 + I + I1I cos( δ ) π nxd δ = n integer a J Coherence Criteria Same wavelength Same source Max difference of optical path (coherence s length of the source) Young s experiment 1 I + max = I1 + I I1I I min = I1 + I I1I s s d 1 = S1P = a + y + x d = SP = a + y + x + s s xd 1 = 3/14/7 PhotoTech Rom Clement 5 What s diffraction Diffraction occurs when the wavelength of a wave (visible light, IR, UV, sound, radio waves ) is about the same size as an object Example1: a needle s hole in a sheet of paper generates diffraction for light Example: an opened door can represent a hole in a wall for which diffraction will occur θ a a ~.1mm 3/14/7 PhotoTech Rom Clement 6 3

4 Fraunhofer Diffraction Aperture A in a sheet Po is source creates a plane wave P(p,q) is point on a screen located at (p,q) U amplitude of the wavefront ( ξ, η) coordinates of a point Q in the aperture G( ξ, η) pupil function (depends on transmission characteristics) In our case we will consider General diffracted amplitude iπ ( pξ + qη ) 3/14/7 PhotoTech Rom Clement 7 U ( p, q) = G( ξ, η) e dξdη G( ξ, η) = constant G( ξ, η) = G( ξ, η) = 1 G( ξ, η) = at point of the opening at points outside the opening at point of the opening at points outside the opening Diffraction of a rectangular aperture Diffraction of rectangular aperture: a width, b height U ( P) = C a a b a b ik ( pξ + qη ) ik ( pξ + qη ) ik( pξ + qη ) e b dξdη = C a sin( kpa) sin( kqb) = I( P) = U ( P) I kpa kqb e dξ e π k = b dη p,q direction (angle) from which the aperture is seen 3/14/7 PhotoTech Rom Clement 8 4

5 D plots, Rectangular and Circular Aperture Rectangular Aperture Circular aperture sin( kpa) sin( kqb) J1( kaw) = U ( P) = I ( P) = U ( P) = I I( P) I kpa kqb π k = p,q angle from which the aperture is seen kaw w direction angle from which the aperture is seen 3/14/7 PhotoTech Rom Clement 9 Comparison rectangular/circular aperture Rectangular opening Circular opening /14/7 PhotoTech Rom Clement 1 5

6 Comparison rectangular/circular aperture 3/14/7 PhotoTech Rom Clement 11 How to define the spot? Total Energy Integral 84% 91% 94% 3/14/7 PhotoTech Rom Clement 1 6

7 The problem of Resolution 3/14/7 PhotoTech Rom Clement 13 How to determine when objects are distinguishable? Abbe s criterion (1873) When maximum of first PSF coincides with 1st minimum of nd PSF θ Abbe = D Rayleigh s criterion (1879) Airy disk contains 84% of energy points can be resolved if centers of Airy disks are separated by ½ the diameter of the Airy disk θ Rayleigh = 1. D Abbe s/rayleigh Criterion 3/14/7 PhotoTech Rom Clement 14 7

8 How to determine when objects are distinguishable? Sparrow s criterion (1916) Based on summed line profiles Where the saddle point first develops θ Sparrow =. 94 D Houston s criterion (196) FWHM (Full Width at Half Maximum) The two diffraction patterns have to be separated by the FWHM Sparrow criterion Houston s criterion Sparrow s is the least stringent, then Houston s then Rayleigh s ~= Abbe 3/14/7 PhotoTech Rom Clement 15 Which pixel s size to chose? The optimum: Nyquist criteria i.e. pixels inside the airy disk 3/14/7 PhotoTech Rom Clement 16 8

9 Another example: diffraction of an straight edge C( ) + is( ) = 1 1 I = C( w) + e i( π /) τ dτ S( w) I 3/14/7 PhotoTech Rom Clement 17 Gratings and Spectroscopy 9

10 Spectroscopy s Principles Continous spectrum Emission spectrum Absorption spectrum Spectrum In order to have a spectrum, we need to scatter the light, prism is a way but not the only one 3/14/7 PhotoTech Rom Clement 19 Line emission: theory Absorption hν =E1-E Emission Ritz s Law 1/ = RH (1/n²-1/m²) RH Rydberg s constant /m n,m integers and m>n Lyman s serie (n=1), Balmer (n=), Paschen (n=3) Balmer m=3 Hα (=6563 Å) m=4 Hβ (=4861 Å) m=5 Hγ (=434 Å) m=6 Hδ (=411 Å). 3/14/7 PhotoTech Rom Clement 1

11 Lines The wavelength of the lines allow to characterize precisely the components of the light source (or absorbing source) but the study of the distortion of these lines allows to learn more Line spreading Natural width (/1 Å) Thermal shaking (Doppler-Fizeau s effect) (1/1 Å) Spreading due to high pressure Stark effect, electric field (several Angströms) Star rotating speed (Doppler-Fizeau effect) (several Angströms) Lines split Zeeman effect (triple lines, 1/4Å), magnetic field 3/14/7 PhotoTech Rom Clement 1 Gratings Principles Repetition of a pattern introduces a slight difference of optical path which introduces a difference of phase and generates interferences Everyday life s examples CD s/dvd iridescence Credit cards hologram 3/14/7 PhotoTech Rom Clement 11

12 Reflective and transmission: gratings 3/14/7 PhotoTech Rom Clement 3 Gratings: examples Reflective gratings Stiffer Aluminized epoxy layer glued on a thick glass Higher efficiency (blazed),5 lines/mm Price $13 for 3mm*3mm (visible light) $5 for 5mm*5mm (visible light) Transmission gratings Polyester film 1, lines/mm Lower efficiency Cheaper $5 3/14/7 PhotoTech Rom Clement 4 1

13 Transmission gratings, the math Assume a transmission grating made of N slits (D case) of aperture s size s and distant of distance d The difference of optical path between successive beams is π δ = d( sin( θ ) sin( θ )) If we sum the amplitude over all N slits we obtain I( P) = U ( P) N 1 ikndp 1 e U ( p) = U ( p) e = U ( p) 1 e 1 cos( Nkdp) = I 1 cos( kdp) where I ( p) = U ( p) and U ( p) n= inkdp ikdp sin( Nkdp / ) ( p) = sin( kdp / ) diffracted amplitude of a single slit I ( p) Diffracted light π sin( Nkdp / ) k = I( P) = sinc ( ksp / ) sin( kdp / ) p = sin θ sin θ ( ) ( ) Incident light 3/14/7 PhotoTech Rom Clement 5 Transmission gratings, the maths Normalized intensity 1 sin( Nkdp / ) I( P) = sinc ( ksp / ) N N sin( kdp / ) Modulation function due to N slits Intensity of one single slit For a given wavelength, pics appear at specific angular positions Nd sin θ sin Resolving Power = Angular dispersion ( ) ( θ ) ( θ ) 1 sin = cos ( θ ) sin( θ ) θ 3/14/7 PhotoTech Rom Clement 6 I(P) Order 1 Order 13

14 Overlapping of orders Transmission grating Reflective grating θ θ θ θ θ θ m sin( θ ) sin( θ ) = ( m =, ± 1, ±,...) d m is the order of interference d is the distance between groves (or lines, or step ) But [ 4nm, nm] visible 8 Assume 1 groves/mm, i.e. d = 1µm and θ =1 sin( θ ) + sin( θ) = 1 = 4nm m d = 8nm 19.5 Pb!! 4.4 3/14/7 PhotoTech Rom Clement 7 m=1 m= m= Overlapping of grating spectra 3/14/7 PhotoTech Rom Clement 8 14

15 Amateur spectroscope Adjustable slot Collimator (Zoom 75-mm) To telescope Reflective grating 1 groves/mm on VHS magnetoscope head Adaptator 31,75mm Camera optics (5mm f:1.4) Digital Camera (Modified Webcam, cooled by Peltier module) 3/14/7 PhotoTech Rom Clement 9 Post processing steps (Procyon) Crude image (~ 67Å) Alignment Binning Spectrum Normalization Comparison with reference spectrum Polynom of interpolation 5 =,63x 1 P( i) +,85P( i) + cste Instrument response 3/14/7 PhotoTech Rom Clement 3 15

16 Entire spectrum of Procyon (type F5 IV) Stiching of 9 calibrations Reference spectrum of star F5 IV Ca I 46,7 Hγ 434,5 Fe I 57 Hβ 4861,5 Mg I Fe I 538 Na Hα 656,5 Bande O 6884 H stripe O 761 3/14/7 PhotoTech Rom Clement 31 Conclusion Diffraction s phenomena comes from the wave approach of light A perfect point source is always diffracted by an opening Its diffracted pattern is called the airy disk Diffraction usually size the aperture of optical components Diffraction size the optimum sensor spatial resolution Spectroscopy How diffraction can be used to analyze light: gratings, a pattern of.1µm allows to extract information at the Angström s scale = 1 e-1 m (factor of 1 3!) θ a a ~.1mm 3/14/7 PhotoTech Rom Clement 3 16

17 References Books Principles of Optics, 7th edition, Max Born and Emil Wolf (hardcore but very very good!) Websites Wolfram: Definition of focal length: Edmund Optics for gratings Astronomical spectroscopy for amateur 6 different spectrographs for different budgets and objectives Visual Spec: spectrum analyzer software IRIS: Astronomical images processing software All for free! 3/14/7 PhotoTech Rom Clement 33 Appendixes 17

18 Diffraction of a circular aperture: the maths Change for Cartesian reference to cylindrical reference ρ cos( θ ) = ξw cos(ψ ) = p ρ sin( θ ) = ξ w sin(ψ ) = q Leads to the following integral U( P) = C a π ikρw cos( θ ψ ) e ρdρd θ Bessel function n i π π ix cos( ) inα e [ ] α n + 1 n+ 1 x J ( x) = x J ( x ) e dα = J ( x) n and recurrence relation d dx n+ 1 n So the integral can be expressed as a function of J U( P) = πc a J ( kρw ρdρ ) Or J ( kaw) U P) = CD kaw D = πa 1 ( with And ultimately we obtain the intensity s expression: J1( kaw) I( P) = U ( P) = I kaw a aperture radius [m] k wave vector [1/m] w angle from which the aperture is seen 3/14/7 PhotoTech Rom Clement 35 Stars Classification Type Color External layer s temperature Principle characteristics Examples O Blue > 5 K Ionized helium Fort strong UV radiation 1 Lacerta B Blue 11 à 5 K Helium absorption lines Rigel Spica A Blue 75 à 11 K Hydrogen emission lines Sirius Vega F Blue to white 6 à 75 K Metals lines Canopus Procyon G White to yellow 5 à 6 K Absorption lines of metal atoms Soleil Capella K Orange to red 35 à 5 K Several metals lines Arcturus Aldebaran M Red < 35 K Stripes of Titanum oxydes molecules Betelgueuse Antares There is a much, much more well defined classification inside each category 3/14/7 PhotoTech Rom Clement 36 18

19 Type of information extracted from light Analysis of lines Chemical composition of the star Analysis of lines width/distortion Characterization of emission milieu: pressure, magnetic field (Zeeman effect) Doppler s effect : Extraction of star s rotational velocity Extraction of expansion speed of star s envelope Need high resolution of spectroscopes 3/14/7 PhotoTech Rom Clement 37 19