Light matter interaction Hydrogen atom Ground state spherical electron cloud Excited state : 4 quantum numbers n principal (energy) L angular momentum, 2,3... L L z projection of angular momentum S z projection of spin (S=1/2) 1 SZ. 2 n 1 L L, ( L1),...0 Z
Electron transition between levels emission or absorption of a photon Absorption Spontaneous emission Selection rules L 1, L 0, Z Photon carries angular momentum Stimulated emission Relation between angular momentum of a photon and its polarization
Atomic configurations Atoms have hydrogen like orbitals. Electrons are fermions, they follow Pauli exclusion principle: No more than one fermion can be placed in the same quantum state.
Atoms come together, forms molecules and solids Energy Energy
Metals Band Empty states Occupied states Energy gap (no states allowed) Filled band C C B Ed ds t A E Bd oj 0 ds t A We cannot neglect free current J E Response to EM wave Attenuation BUT In UV range many metals becomes transparent Plasma frequency 2 2 ne P m 0
Insulators Completely empty band Energy gap Completely filled band Excited electron hole Insulators are transparent for photons that have energy smaller than the band gap (long wavelength photon) Completely filled band no current States in the gap (impurity, vacancy) Behave very much like isolated atoms; Example: States of Cr in ruby Al Often gives color to crystals 2O 3:Cr
Geometrical optics Ideal optical system : all rays emitted from a point of an object arrive in the same point of the image at the same time. (We proved it for a spherical interface for small angles!) v c/ n Speed of light in media with index of refraction n n depends on wavelength of the light; be aware. Snell s law: Total internal reflection Refraction, Spherical surface n1 n2 n2 n1 1 s s R f 0 i
Thin lens nm n 1 1 m 1 nl nm s0 si R1 R2 f Thin lens combinations From left to right: the image of lens#1 is the object for lens #2 and so on. This works always if you use formulas; the ray tracing can give wrong result. Two thin lenses in contact 1 1 1 f f f 1 2
Mirrors 1 1 2 1 s s R f 0 i Spherical mirrors and lens give sharp image only in paraxial (small angle) approximation. Aspherical systems (ellipsoidal, hyperbolic and parabolic are free from these deficiency We can say that they do not have spherical aberrations. We have proved this for parabolic mirror.
Optical aberration
Chromatic aberration Index of refraction depends on wavelength Correction with negative lens We notice that aberration correction requires addition of extra lenses. Recall that only 96% of light transmits via air/glass interface. In multi lens systems lenses must be covered by antireflection interference coating. I I T 0 N
2 2 1 2 2 2 x v t Plane wave ( rt, ) C1 ( kr/ kvt) i( krt) ( rt, ) Ce 1 Wave equation ( x vt) General solution 1 C11( x vt) C22( x vt) Superposition i i( kx t) Acos( kx t ) Re Ae e Harmonic wave f 2 k 2 / vk v f 2 2 2 2 1 2 2 2 2 2 x y z v t 2 1 v 2 t 2 2 Plane wave: wave front is perpendicular to the wave vector
Spherical and cylindrical waves
A A C C Maxwell equations and Electromagnetic waves 1 1 EdS qi dv Gauss law (from Coulomb law) 0 0 V BdS 0 B Ed ds (Faraday law of electromagnetic induction) A t E Bd oj 0 ds (Ampere law with Maxwell displacement term) t A A A C C In vacuum EdS 0 BdS 0 B Ed ds t A E Bd o0 ds t A
EM wave is transverse wave (in vacuum) Poynting vector Energy flux of EM wave energy crossing unit area per unit time 1 2 S EBc 0EB 0 Irradiance Poynting vector averaged over time c 2 0 2 I S Eo T E cb
EM waves in matter A A C C 1 1 EdS qi dv 0 0 V BdS 0 Gauss law (from Coulomb law) B Ed ds (Faraday law of electromagnetic induction) A t E Bd oj 0 ds (Ampere law with Maxwell displacement term) t A These equations are correct both in vacuum and in matter, but E, BJ, must include all charges, free and bound Bound charges and bound currents d dipole moment d P polarization V D 0E P electic displacement Linear media P E o D E E R 0 D represent the effect of free charges - magnetic dipole M magnetization V B M H 0 0 Linear media M H B H H 0 R R H auxiliary magnetic field, represents free charges
Examples Spontaneous magnetization Ferromagnetic materials Spontaneous polarization Ferroelectic materials Piezoelectric materials Applied voltage shrinks/extends a crystal and shrinking the crystal generates voltage across. Quartz
Semiconductors: doping Free carriers electrons Free carriers holes
The state of extra electron on P at low temperatures A 1 EdS qi DdS q 0 A FREE After taking surface integral around point charge we have q D 2 4 R E is the real field, which determines forces between charges D E E In Silicon 11 R 0 R P E Silicon q R 4 R 0 2 Hydrogen atom E q 4 R 0 2 Borh radius of orbit ab 10 nm ab 0.05 nm
2 2 2 E E t Wave equation in matter 0 r 0 r 1 r c c v n 1 r typical case 0 C A o C A B Ed ds t E Bd J ds t EM waves in matter 0 o FREE BOUND C A P E Bd J J ds t t C A B Ed ds t Real current created by bound charges
Fresnel equations
Polarization Linear polarization
Circular polarization Right, clockwise Left, counter clockwise Polarizers The only component that is passes through polarizer is the component of with electrical field along the axis of polarizer. (You need cos() factor.
Birefringent Crystals Dx xx xy xz Ex D ije D y yx yy yz E y D z zx zy zz E z E perpendicular to optical axis ordinary ray E along optical axis extraordinary ray
Retarders
Interference Principle of superposition E E ( x, t) E ( x, t) 1 2 The first slit is needed ensure spatial coherence of the wave front
Photons (electrons) sent one by one Formation of interference pattern is a property of a single photon
Diffraction and Interference with Fullerenes: Wave-particle duality of C60 Markus Arndt, Olaf Nairz, Julian Voss-Andreae, Claudia Keller,Gerbrand van der Zouw, and Anton Zeilinger Nature 401, 680-682, 14.October 1999
Diffraction 1) Superposition principle 2) Huygens Frensel wavelet principle Phase of EM wave is taken into account
Diffraction grating ( kb /2)sin ( ka /2)sin b a width of the slit distance between the slits
Fraunhofer (long distance) diffraction Single slit ( kb/2)sin b width of the slit Circular aperture a radius J 1 Bessel function
Diffraction limit imposed by a circular aperture 1.22 / D Angular spread of the first maxima
Angular resolution of optical instruments Geometrical resolution Distance between photoreceptors d=6 10 micron Focal length of an eye f=2 cm Angular resolution Teta=d/f=3x10 4 Resolution set by diffraction 1.22 / D Wavelength 600 nm Diameter of pupil 4 mm Teta=1.22L/D=1.8x10 4 Eye is well optimized optical instument
Angular resolution of Hubble space telescope Geometrical resolution Distance between photo detectors in CCD camera d=3 microns Effective Focal length f=10 m Angular resolution Teta=d/f=3x10 7 rad Resolution set by diffraction Wavelength 600 nm Diameter of primary mirror 2m Teta=1.22L/D=3.5x10 7 radians The optics of the telescope is also optimazed
Component of the first ruby laser Al2O 3:Cr Elementary processes 1) Excitation 2) Stimulated emission Fabry Perot resonator
Elementary process What is needed; physical principle Stimulated emission 1) We need to create so called inverted population of electrons n n E G E exp kt Equilibrium concentration Excitation using three level process 2) We need external excitation, but we also can not excite electrons directly to lasing level because we have stimulated excitation, which has the same rate as stimulated emission 3) For optical excitation at least three level system must be used
Photon are bosons and like to stay in the same quantum state This is actually wrong way of thinking about lasing
Laser Fabry Perot cavity Quantum state of a photon. From energy in a particular mode we can compute the number of photons in it
The resonance lines are extremely narrow, when reflections at the is high. Stimulated emission, a bit more details Possible emission of stimulated photon
Photons are bosons. They like to be together. PN ( ) P(1) N Probability of emission in a particular quantum state is proportional to the number of photon occupying this state.
Excitation is not optical Semiconductor lasers.