Waves, Polarization, and Coherence

Transcription

1 Waves, Polarization, and Coherence Lecture 6 Biophotonics Jae Gwan Kim jaekim@gist.ac.kr, X 0 School of nformation and Communication ngineering Gwangju nstitute of Sciences and Technolog Outline Models of Light Light as an lectromagnetic Wave Polarization of Light Reflection and Refraction Wave Propagation through Anisotropic Media nterference and Coherence of Light

2 Light as an lectromagnetic Wave lectromagnetic wave varies in space and time lectric field can be written as a : scalar z ( z, t Acos ( t cos( A kz t δ is the phase constant or a vector ( z, t Acos( kz t The direction of the electric field vector (which is not the same as the direction of light propagation! is called the polarization direction. Please remember: cos( Re[ e i ] Polarization of Monochromatic Plane Waves Consider a plane M wave propagating in the z direction will lie in the (, plane where the comple envelope : ( z, t Re Ae j( tkz A ˆ ˆ A e j A e j For some z=constant the components of the field will var as: A cos t A cos t where

3 Linear Polarized Light. 0, n phase A cost A cos( t A A cost A A cost A Linear equation A A A 0., 90 degree out of phase A cost A cos( t For particular A case of Circular Polarized Light A cost A sint A A A Standard elliptical equation A A A Left hand polarization Right hand polarization 3

4 General cases lliptical Polarized Light A cost cos sin A cos( t A A A A General elliptical equation A A A A A A A A Linear Polarization in 3D Movies Two snchronized projectors project two images on the screen, each with a different polarization (the images are projected through linear polarizers The glasses allow onl one of the images into each ee. The two images are separated for each ee creating depth 4

5 mportance of Polarization Polarization plas an important role in the interaction of light with matter: The amount of light reflected at the boundar between two materials depends on the polarization of the incident wave. The amount of light absorbed b certain materials is polarization dependent Light scattering from matter is generall polarization dependent The refractive inde of anisotropic materials depends on the polarization Opticall active materials have the natural abilit to rotate the polarization plane of linearl polarized light. These polarization phenomena are used for building important polarization devices. Polarizing Filter A polarizing filter cuts down the reflections (top and made it possible to see the photographer through the glass at roughl Brewster's angle although reflections off the back window of the car are not cut because the are less strongl polarized, according to the Fresnel equations 5

6 Fresnel quations deduced b Augustin Jean Fresnel, describe the behavior of light when moving between media of differing refractive indices. The reflection of light that the equations predict is known as Fresnel reflection. s vs p polarization Plane of ncidence n n Reflected wave k 3 k θ r θ θ t i k mirror perpendicular polarization (or T or s polarization, s easier to remember if we think of the arrow slapping the mirror mirror parallel polarization (or TM or p polarization, p easier to remember if we think of the arrow poking the mirror B solving a boundar value problem for the electromagnetic wave at the interface one can derive the Fresnel equations. This set of 4 equations gives the amounts of perpendicular and parallel polarized that reflected and transmitted at the interface. 6

7 s vs p polarization perpendicular ( component of polarization (transverse electric (T or s polarization from German senkrecht parallel ( // component of polarization (transverse magnetic (TM or p polarization Fresnel quations Reflection coefficient f incident light is s polarized, f incident light is p polarized Transmission coefficient T s = R s, T p = R p f the incident light is unpolarized, R = (R s + R p / 7

8 Fresnel quations The amplitudes of reflection coefficient R and transmission coefficient T are R = and where r and t are the ratio of the reflected/transmitted wave s comple electric field amplitude to that of the incident wave Brewster s Angle An angle of incidence at which light with a particular polarization is perfectl transmitted through a transparent dielectric surface, with no reflection. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectl polarized polarizer 8

9 Brewster s Angle, Critical Angle n < n eternal reflection (e: reflection from air to glass Brewster s angle the incidence angle at which the parallel polarized wave is not reflected n B tan n n > n internal reflection (e: reflection from glass to air Critical angle the incidence angle for which the refraction angle is 90 0 (for θ>θ c all the incident light is totall reflected n c sin n Brewster, Critical Angle Applic. For θ>θc total internal reflection used for light propagation in optical fibers s polarized θ B Partiall p polarized A Brewster window transmits TM (parallel polarized light with no reflection loss (used in lasers cavities Polarizer-a device which converts an unpolarized beam into a beam with single polarization state f unpolarized light is incident on a surface at Brewster angle, the reflected light is linearl polarized with the electric vector perpendicular to the plane of incidence (the parallel component is not reflected polarization b selective reflection 9

10 Polarizer Liner polarizer Absorptive polarizer: the unwanted polarization states are absorbed b the device Crstals: tourmaline, herapathite PVA plastic with an iodine doping is stretched during the manufacturing process Wire grid polarizer: Parallel to the wire is reflected while the perpendicular to the wire is transmitted The separation distance between the wires must be less than the wavelength of the radiation, and the wire width should be a small fraction of this distance. This means that wire grid polarizers are generall onl used for microwaves and for far and mid infrared light. Polarizer Liner polarizer Beam splitting polarizer: the unpolarized beam is split into two beams with opposite polarization states Polarization b reflection Birefringent polarizer Thin film polarizer: glass substrates on which a special optical coating is applied causing an interference effects 0

11 Birefringence An anisotropic crstal ehibits different refractive indices for different polarization components of the light when light refracts at the surface of an anisotropic crstal (quartz or calcite, the two polarizations refracts at different angles, being spatiall separated (birefringence or double refraction. Usuall, two cemented prisms made of anisotropic (uniaial crstals in different orientations are used to obtain polarized from unpolarized light. Optical Ais An optical ais is a line along which there is some degree of rotational smmetr in an optical sstem such as a camera lens or microscope. For an optical fiber, the optical ais is along the center of the fiber core, and is also known as the fiber ais.

12 Optic Ais of a Crstal t is the direction in which a ra of transmitted light suffers no birefringence Uniaial crstals: the heagonal, tetragonal, and trigonal crstal sstems have one optic ais Biaial crstals: orthorhombic, monoclinic, and triclinic have two optic aes f the light beam is not parallel to the optic ais, then the beam is split into two ras (the ordinar and etraordinar when passing through the crstal. These ras will be mutuall orthogonall polarized. Crstal Structures

13 Ordinar vs traordinar f unpolarized light enters the birefringent material at some angle of incidence, the component of the incident radiation whose polarization is perpendicular to the crstal ais (ordinar ra will be refracted according to the standard law of refraction for a material of refractive inde n o, the other polarization component, the so called etraordinar ra will refract at a different angle determined b the angle of incidence, the orientation of the optic ais, and the birefringence Birefringent Polarizer Nicole prism Glan Thomson prism Glan Foucault prism Glan Talor prism 3

14 Birefringent Polarizer Wollaston Prism Senarmont Prism Crstal ais traordinar ra or e ra 5~45 o Ordinar ra or o ra Rochon Prism Malus law When a perfect polarizer is placed in a polarized beam of light, the intensit,, of the light that passes through is given b Where o is the initial intensit θ i is the angle between θ 0 andθ 4

15 Polarizer Circular polarizer (polarizing filter to create circularl polarized light or alternativel to selectivel absorb or pass clockwise and counter clockwise circularl polarized light Polarizing filters in photograph 3D Glasses Wave Ratarder (Wave plate A tpical wave plate is made of anisotropic materials (birefringent crstal. There is a phase dela between the two polarization components which see different refractive indices of the anisotropic material The phase difference is given b: (n n L where L is the length of the wave plate; n, n -the refractive indices corresponding to the two polarization components a half wavelength, Half wave plate a quarter wavelength, Quarter wave plate Wave plate (retarder 5

16 Half Wave Plate A cos( t The light remains linear polarized, but the polarization plane will be rotated at θ. The polarization plane can be rotated b different angles if the half wave plate is rotated A cost origin For linear polarized light (δ origin =0 or π, after passing a half wave plate: total origin 0(or (or 0 linear polarization Half Wave Plate When do we need to use a half wave plate? -in an eperimental set-up when the plane of polarization of a laser beam needs to be rotated - when the laser power needs to be attenuated, a wave plate and a polarizer can be used for this purpose details, principle of operation & videos: 6

17 Possible am Question A quarter waveplate shifts the relative phase b. What is the effect of a quarter waveplate on linear and circular polarized light? Possible am Question A quarter waveplate shifts the relative phase b. What is the effect of a quarter waveplate on linear and circular polarized light? A quarter wave plate converts a linear polarization into a circular polarization and vice-versa. 7

18 Outline Models of Light Light as an lectromagnetic Wave Polarization of Light Reflection and Refraction Wave Propagation through Anisotropic Media nterference and Coherence of Light nterference 8

19 nterference The phenomenon b which (electromagnetic waves interact with one another nterference is the result of the superposition principle: ( r, t ( r, t Consider two fields (most general case: different directions, different frequencies : i i ( r, t 0 cos( k r t ( r, t 0 cos( k r t nterference occurs when the two fields are overlapped spatiall and temporall ntensit (rradiance: energ/area/time We do not actuall measure the electric field ( r, t. We measure the intensit of the light. ( r, t ~ ( r, t Add intensities is not alwas the same as adding the electric fields. ( r, t ( r, t ( r, t ( r, t for coherent light. We onl add intensities when the light is incoherent. 9

20 nterference The measurable quantit is the intensit! where < > stands for time average / / / / / / / /, ( t r nterference term /, (, ( /, ( t r t r t r α cos dot product 0 for No interference for perpendicular polarizations! Maimum interference for parallel polarizations nterference Assume parallel polarizations: where For simplicit, we ll get rid of the time average < > and assume real fields So, ( 0 t z k j e ( 0 t z k j e ( 0 ( 0 ( 0 ( 0 / / / / t z k j t z k j t z k j t z k j e e e e cos cos ( e e t z k k j t z k k j t z k k cos

21 nterference cos k k z t n for n 0,,... in phase ( n for n out of phase 0,,... ( n for n 0,, No interference Constructive interference Destructive interference nterferometer Light source Wave splitting ntroduction of phase difference Wave combination ntroduction of phase difference Ke elements in an interferometer Light source lement for splitting the light into two partial waves Different propagation paths where the partial waves undergo different phase contributions lement for superposing the partial waves Detector for observation of the interference

22 Young s nterferometer Wavefront division Pin hole Screen Pin hole L z D D z D ( 0( cos( z Young s interferometer Michelson nterferometer Amplitude division Mirror Slit Mirror / Detector 4 t

23 Coherence no interference the beams are incoherent cos the beams are coherent f θ=0, π and =, = 4 ( cos the beams are partiall coherent incoherent 0 ( coherent Degree of coherence ncoherence no interference the beams are incoherent This can occur if, for e: k k frequencies drasticall different or man waves of different frequencies ( temporal incoherence k vectors drasticall different or man waves originating from different locations ( spatial incoherence e e j k z ( t k k j( kzt No interference! ncoherent source! 3

24 Coherence spatial vs temporal Spatial coherence means a strong correlation (fied phase relationship between the electric fields arriving at the same time but different locations. For e, within a cross section of a beam from a laser with diffraction limited beam qualit, the electric fields at different positions oscillate in a correlated wa. Spatial coherence tells us how collimated (non-divergent a source is! Temporal coherence means a strong correlation (fied phase relationship between the electric fields at one location but different times. two fields oscillating at the same frequenc but delaed with each other perfectl correlated Temporal coherence tells us how monochromatic a source is! Coherence spatial vs temporal ZeXc 4

25 Spatial and Temporal Coherence Spatial coherence D ( 0( cos( z D z Temporal coherence Slit Mirror 4 ( t 0( cos Mirror Detector mission Spectrum, Temporal Coherence Wavelength (nm Wavelength (nm Fourier Transform. L c Lc L c : coherence length, Temporal coherence is determined b the emission spectrum of the light source 5

26 Quantifing Coherence Coherence time c - the source bandwidth Coherence length 0 Lc c c ~ c the speed of light, Δλ: source spectral width A narrow linewidth means high temporal coherence and long coherence length! Coherence Time 6

27 References B..A. Saleh, M.C. Teich, Fundamentals of Photonics M. Born,. Wolf, Principles of Optics: lectromagnetic Theor of Propagation, nterference and Diffraction of Light 7