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Fundamentals of Optics Jiun-You Lin Department of Mechatronics Engineering, National Changhua University of Education • Historically, optical theory developed roughly in the following : sequence: (1) ray optics (2) wave optics (3) electromagnetic optics (4) quantum optics • The theory of quantum optics provides an explanation of virtually all optical phenomena. The theory of electromagnetic optics provides the most complete treatment of light within the confines of classical optics (electromagnetic optics, wave optics, ray optics) • Wave optics is a scalar approximation of electromagnetic optics. Ray optics is the limit of wave optics when the wavelength is very short. Classical Optics Outline • Wave Nature of Light • Polarization and Modulation of Light Wave Nature of Light Light Waves in a Homogeneous Medium A. Plane Electromagnetic Wave B. Maxwell’s Wave Equation and Diverging Waves Refractive Index Magnetic Field, Irradiance, and Poynting Vector Snell’s Law and Total Internal Reflection Fresnel’s Equations A. Amplitude Reflection and Transmission Coefficients B. Reflectance and Transmittance Interference Principles A. Interference of Two Waves B. Interferometer Diffraction Principles A. Diffraction B. Fraunhofer Diffraction C. Diffraction grating Light Waves in a Homogeneous Medium •Ultraviolet 10nm~390nm •Visible 390nm~760nm •Infrared 760nm~1mm Optical frequencies and wavelengths A. Plane Electromagnetic Wave Ex Direction of Propagation k x z z y By An electromagnetic wave is a travelling wave which has time varying electric and magnetic fields which are perpendicular to each other and the direction of propagation, z. A sinusoidal wave Ex Eo cost kz o Ex : the electric field at position z at time t Eo : the amplitude of the wave : the angular frequency k=2/ : the propagation constant o : phase constant (t-kz+o) : the phase of the wave (1) A monochromatic plane wave E and B have constant phase in this xy plane; a wavefront z E E B k Propagation Ex z A plane EM wave travelling along z, has the same E x (or By) at any point in a given xy plane. All electric field vectors in a given xy plane are therefore in phase. The xy planes are of infinite extent in the x and y directions. Eq. (1) can be rewritten as Ex(z,t)=Re[Eoexp(jo)expj(t-kz)] =Re [Ecexpj(t-kz)] (2) where Ec= Eoexp(jo) complex number that represents the amplitude of the wave and includes the constant phase information o y Direction of propagation k E(r, t ) r q O r z A travelling plane EM wave along a direction k . E(r,t)=Eocos(t-kr+o) (3) where kr=kr =kxx+kyy+kzz The relationship between time and space for a given phase, for example, that corresponding to a maximum field, according to Eq. (1): =t-kz+o=constant (4) The phase velocity v=dz/dt=/k= (5) B. Maxell’s Wave Equation and Diverging Waves Wave fronts (constant phase surfaces) Wave fronts k P P k Wave fronts E r O z A perfect plane wave (a) A perfect spherical wave (b) Examples of possible EM waves A divergent beam (c) There are many types of possible EM waves: • Plane wave: the plane wave has no divergence • Spherical wave: wavefronts are spheres and k vectors diverge out E=(A/r)cos(t-kz) (6) where A is a constant • Divergent beam: the wavefront are slowly bent away thereby spreading the wave Ex. The Output from a laser (Gaussian beam) y Wave fronts (b) x 2wo O z Beam axis Intensity q Gaussian (c) r (a) 2w (a) Wavefronts of a Gaussian light beam. (b) Light intensity across beam cross section. (c) Light irradiance (intensity) vs. radial distance r from beam axis ( z ). • 2w: the beam diameter at any point z • w2: the cross sectional area at z point contains 85% of the beam power • 2wo: the beam diameter at point O = the waist of the beam =the spot size • wo: the waist radius • q : divergence angle (7) 2q=4/(2o) In an isotropic and linear dielectric medium, these fields must obey Maxwell’s EM wave equation, 2E 2E 2E 2E 2 2 o r o 2 2 x y z t o : the absolute permeability o : the absolute permittivity r : the relative permittivity of the medium (8) Refractive Index For an EM wave traveling in a nonmagnetic dielectric medium of relative permittivity r, the phase velocity v is given by v= 1/(roo)1/2 (9) For an EM wave traveling in free space, r=1, the phase velocity v is given by (10) v = 1/( )1/2=c=3108ms-1 vacuum o o The refractive index n of the medium: n=c/v= (r )1/2 (11) In free space, the wavenumber: k=2/ (12) In an isotropic medium, the wavenumber: kmedium=nk (13a) medium=/n (13b) Light propagates more slowly in a denser medium that has a higher refractive index, and the frequency remains the same. Magnetic Field, Irradiance and Poynting Vector v Dt z E Area A k Propagation direction B A plane EM wave travelling alongk crosses an area A at right angles to the direction of propagation. In timeDt, the energy in the cylindrical volumeAvDt (shown dashed) flows throughA . Fields in an EM wave Ex=vBy=(c/n)By Energy densities in an EM wave (1/2) ro Ex2= (1/2 o) By2 (15) The EM power flow per unit area (14) S= Energy flow per unit time per unit area =(AvDt) (ro Ex2)/(ADt)=v ro Ex2 = v 2ro ExBy (16) Poynting Vector S= v 2ro EB (17) The energy flow per unit time per unit area in a direction determined by EB Irradiance |S|= v 2ro |EB| (18) Snell’s Law and Total Internal Reflection At A t kt Refracted Light Bt A y O qt z qt B A n1 qi qr qi qr A B ki kr Ai Incident Light n2 Br Bi Ar Reflected Light A light wave travelling in a medium with a greater refractive index ( n1 > n2) suffers reflection and refraction at the boundary. • Ai and Bi are in phase Ar and Br must still be be in phase BB=AA=v1t AB=v1t/sin qi =v1t/sinqr • Ai and Bi are in phase A and B must still be be in phase BB=v1t=ct/n1 ; AA=v2t =ct/n2 AB= BB /sin qi = AA /sin qt qiqr AB= BB /sin qi = AA /sin qt = v1t/sin qi = v2t/sin qt Snell’s law : sin qi /sinqt=v1/v2=n2/n1 (19) qt ki qi q i Incident light Transmitted (refracted) light kt n2 n 1 > n2 kr Evanescent wave qc qc TIR Reflected light (a) qi >qc (b) (c) Light wave travelling in a more dense medium strikes a less dense medium. Depending on the incidence angle with respect toq c, which is determined by the ratio of the refractive indices, the wave may be transmitted (refracted) or reflected. (a)qi < qc (b) q i = qc (c) q i > q c and total internal reflection (TIR). When n1>n2 and the refraction angle qt reaches 90 Critical angle: sin qc =n2/ n1 (20) Fresnel’s Equations A. Amplitude Reflection and Transmission Coefficients Et,// y qt z x into paper Ei,// ki qi qr Ei,^ Incident wave Er,// Transmitted wave kt Et,^ Evanescent wave Et,^ Er,^ (a) qi < qc then some of the wave is transmitted into the less dense medium. Some of the wave is reflected. qi qr Ei,// Ei,^ kr Reflected wave n2 q t =90 Incident wave Er,^ n1 > n 2 Er,// Reflected wave (b) qi > q c then the incident wave suffers total internal reflection. However, there is an evanescent wave at the surface of the medium. Light wave travelling in a more dense medium strikes a less dense medium. The plane of incidence is the plane of the paper and is perpendicular to the flat interface between the two media. The electric field is normal to the direction of propagation . It can be resolved into perpendicular (^ ) and parallel (//) components • Ei^, Er^ ,and Et^: transverse electric field (TE) Ei//, Er// ,and Et// : transverse magnetic field (TM) • Incident wave, reflected wave, and transmitted wave Ei=Eioexpj(t-kir) (21a) Er=Eroexpj(t-krr) (21b) Et=Etoexpj(t-ktr) (21c) According to Maxwell’s EM wave equations and Boundary conditions Reflection and transmission coefficients for E ^ and E// r^=Ero,^ /Eio,^= (22a) t^=Eto,^ /Eio,^=1+ r^ (22b) r//=Ero,// /Eio,//= (22c) t//=Eto,// /Eio,//=(1/n)(1+ r//) (22d) where n=n2/n1 Magnitude of reflection coefficients Phase changes in degrees qc 1 0.9 0.8 (a) (b) 120 0.7 0.6 0.5 0.4 0.3 0.2 0.1 TIR 180 60 qp 0 | r^ | 60 qp | r // | 0 Incidence angle, q i // qc 1 20 1 80 0 10 20 30 40 50 60 70 80 90 ^ 0 10 20 30 40 50 60 70 80 90 Incidence angle, q i Internal reflection: (a) Magnitude of the reflection coefficients r// and r^ vs. angle of incidence q i for n1 = 1.44 and n2 = 1.00. The critical angle is 44? (b) The corresponding phase changes // and ^ vs. incidence angle. • Normal incidence: r//=r^ = (n1 n2)/( n1 +n2) (23) • Polarization angle =Brewser’s angle: r//=0 tan qp =n2/ n1 (24) Reflectance and Trasmittance Reflectance: R^=Ero,^ 2/ Eio,^ 2= r^2 (25) R//=Ero,// 2/ Eio,// 2= r//2 (26) Transmittance: T^= (n2Eto,^ 2)/(n1 Eio,^ 2)= (n2/n1)t^2 (27) T//= (n2Eto,// 2)/(n1Eio,// 2=(n2/n1) t//2) (28) Interference Principles superposed (29) where and (30) If 1. 2. 3. (31) B. Interferometer Delay by a distance d (32) 1. d= m, m=0, 1, 2,…., 2. d= m/2, m=1, 3,…., 3. = 2I0 Diffraction Principles A. Diffraction Light intensity pattern Diffracted beam Incident light wave Circular aperture A light beam incident on a small circular aperture becomes diffracted and its light intensity pattern after passing through the aperture is a diffraction pattern with circular bright rings (called Airy rings). If the screen is far away from the aperture, this would be a Fraunhofer diffraction pattern. Fresnel region Fraunhofer region (near field) (far field) aperture D Zi Fresnel and Fraunhofer diffraction region Zi D2 Zi D2 Fraunhofer region Fresnel region B. Fraunhofer Diffraction Incident plane wave A secondary wave source Another new wavefront (diffracted) New wavefront (a) q z (b) (a) Huygens-Fresnel principles states that each point in the aperture becomes a source of secondary waves (spherical waves). The spherical wavefronts are separated by . The new wavefront is the envelope of the all these spherical wavefronts. (b) Another possible wavefront occurs at an angle q to the z-direction which is a diffracted wave. Huygens-Fresnel principle: every unobstructed point of a wavefront, at a given instant time, serves as a source of spherical secondary waves. The amplitude of the optical field at any point beyond is the superposition of all these wavelets. Incident light wave Y dy y Incident light wave Screen R = Large q y b a q A c y z dy q ysinq z q0 q Light intensity (a) (b) (a) The aperture is divided intoN number of point sources each occupying d y with amplitude dy. (b) The intensity distribution in the received light at the screen far away from the aperture: the diffraction pattern The wave emitted from point source at y: dEdyexp(-jkysinq) All of these waves from point source from y=0 to y=a interfere at the screen and the field at the screen is their sum The resultant field E(q ): y a E (q ) C dy exp( jky sin q ) y 0 Ce 1 j ka sinq 2 1 a sin ka sin q 2 1 ka sin q 2 (33) The intensity I at the screen E(q ) 2 2 1 C a sin 2 ka sin q 1 2 I (0) sin c ( ); ka sin q I (q ) 1 2 ka sin q (34) 2 • The pattern has bright and dark regions, corresponding to constructive and destructive interference of waves from the aperture. • The zero intensity points: sinq=m/a ; m=1, 2,…… (35a) For a circular aperture (35b) sinq=1.22/D angular radius of Airy disk b a The rectangular aperture of dimensionsa b on the left gives the diffraction pattern on the right. C. Diffraction Grating y One possible diffracted beam y Incident light wave Single slit diffraction envelope a m=2 Second-order m=1 First-order m=0 Zero-order m = -1 First-order d q m = -2 Second-order z dsin q Diffraction grating (a) Intensity (b ) (a) A diffraction grating with N slits in an opaque scree. (b) The diffracted light pattern. There are distinct beams in certain directions (schematic) All waves from pairs of slit will interfere constructively when this a multiple of the whole wavelength, Grating Equation: dsinqm=m ; m=0, 1, 2,…… (36) When the incident beam is not normal to the diffraction grating, the diffraction angle qm for the m-th mode: Grating Equation: d(sinqm-sinqi)=m ; m=0, 1, 2,…… (37) First-order Zero-order Incident light wave (a) Transmission grating Incident light wave m = 1 First-order m = 0 Zero-order First-order m = -1 First-order (b) Reflection grating (a) Ruled periodic parallel scratches on a glass serve as a transmission grating. (b) A reflection grating. An incident light beam results in various "diffracted" beams. The zero-order diffracted beam is the normal reflected beam with an angle of reflection equal to the angle of incidence. Polarization and Modulation of Light Polarization A. State of Polarization Birefringent Optical Devices A. Birefringence B. Retarding Plates C. Compensator D. Birefringent Prisms Electro-Optic Effects A. Definitions B. Pockels Effect C. Kerr Effect Acousto-Optic Effects A. Definitions Polarization A. State of Polarization Plane of polarization y E x E ^ y y ^ x E E z (a ) (b) ^ xE x x ^ yE y E (c ) (a) A linearly polarized wave has its electric field oscillations defined along a line perpendicular to the direction of propagation, z . The field vector E and z define a plane of polarization . (b) The E -field oscillations are contained in the plane of polarization. (c) A linearly polarized light at any instant can be represented by the superposition of two fields E x and E y with the right magnitude and phase. Ex Exo cost kz E y E yo cost kz (38a) (38b) where is the phase difference between Ey and Ex By choosing Eyo = Exo and = , the field in the wave: E xˆE x yˆE y xˆE xo cost kz yˆE yo cost kz E o cost kz where E o xˆE xo yˆ E yo (39) the vector Eo at –45 to the x-axis: linear polarization Dz q = k Dz E z Ey E z Ex A right circularly polarized light. The field vector E is always at right angles to z , rotates clockwise around z with time, and traces out a full circle over one wavelength of distance propagated. By choosing Eyo = Exo =A and =/2 , the field in the wave: E xˆE x yˆE y xˆA cost kz yˆA sin t kz (40) E E A : circularly polarized 2 x 2 y 2 • =/2: right circularly polarized (clockwise) • =-/2: left circularly polarized (counterclockwise) y y (a) y (b) (c) x x Exo = 0 Eyo = 1 =0 y Exo = 1 Eyo = 1 =0 (d) E Exo = 1 Eyo = 1 = /2 x E Exo = 1 Eyo = 1 = /2 Examples of linearly, (a) and (b), and circularly polarized light (c) and (d); (c) is right circularly and (d) is left circularly polarized light (as seen when the wave directly approaches a viewer) x y y (a) (b) y E x x Exo = 1 Eyo = 2 =0 (c) E =1 =2 = /4 Exo Eyo x =1 =2 = /2 Exo Eyo (a) Linearly polarized light with Eyo = 2Exo and = 0. (b) When = /4 (45), the light is right elliptically polarized with a tilted major axis. (c) When = /2 (90), the light is right elliptically polarized. If Exo and Eyo were equal, this would be right circularly polarized light. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Birefringent Optical Devices A. Birefringence • Opically anisotropic crystal : the refractive index n of a crystal depends on the direction of the electric field in the propagating light beam. • Birefringence : optically anisotropic crystals are called birefringence because an incident light beam may be doubly refracted B. Retarding Plates z = Slow axis Optic axis E // E // E ne = n3 y E^ E^ no LL x = Fast axis A retarder plate. The optic axis is parallel to the plate face. The o- and e-waves travel in the same direction but at different speeds. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Uniaxial crystal ne>no For E^ : a phase change k^L=(2/)no For E// : a phase change k//L= (2/)ne Relative phase through retarder plate 2 (ne no ) L (41) • Half-wave plate retarder: =, corresponding to a half of wavelength (/2). • Quarter-wave plate retarder: =/2, corresponding to a half of wavelength (/4) Half wavelength plate: = Optic axis Quarter wavelength plate: = /2 z z E x Input = arbitrary x x 0 < 45 = 45 z z E 45陣 z Output x (a) E E x (b) Input and output polarizations of light through (a) a half-wavelength plate and (b) through a quarter-wavelength plate. x C. Compensator: is adjustable E1 E2 Wedges can slide Optic axis d D Plate Optic axis Soleil-Babinet Compensator © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) 1 2 (ne d no D ) 2 2 (no d ne D) (42a) (42b) The phase difference : 2 1 2 (ne no )( D d ) (43) A Soleil-Babinet compensator D. Birefingent Prisms: e-ray Optic axis A B o-ray Optic axis A E1 e-ray E1 E1 E2 E2 E2 q B Optic axis Optic axis o-ray The Wollaston prism is a beam polarization splitter. E1 is orthogonal to the plane of the paper and also to the optic axis of the first prism. E2 is in the plane of the paper and orthogonal to E1. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Commercial Wollaston Prisms Electro-Optic Effects A. Definition Electro-Optic Effects : changes in the refractive index of a material induced by the application of an external electric field, which therefore modulates the optical properties. Field induced refractive index: n=n+a1E+a2E2+… (a) Pockel effect: Dn=a1E (b) Kerr effect : Dn=a2E2 (44) B. Pockels Effect Ey V 45 Input light y Ey d Ea x D Ex z z Output light Ex Tranverse Pockels cell phase modulator. A linearly polarized input light into an electro-optic crystal emerges as a cir cularly polarized light. For a LiNbO3 crystal Ea=0 : the refractive indices are equal no Ea0 : Pockel effect 1 3 n1 no no r22 Ea 2 1 3 n2 no no r22 Ea 2 (45a) (45b) 2n1L 2L 1 3 V E x : 1 (no no r22 ) (46a) 2 d 2n2 L 2L 1 3 V E y : 2 (no no r22 ) (46b) 2 d Transverse Pockels Effect 2 L D 1 2 nr V d 3 o 22 (47) Intensity modulator QWP Transmission intensity V y P Io A Detec tor Input light 45 Crystal Q x z 0 V / V Left: A tranverse Pockels cell intensity modulator. The polarizer P and analyzer A have their transmission axis at right angles and P polarizes at an angle 45 to y-axis. Right: Transmission intensity vs. applied voltage characteristics. If a quarter-wave plate ( QWP) is inserted after P, the characteristic is shifted to the dashed curve. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) The total field at the analyzer (transmission axis=45°): Eo Eo E xˆ cos t yˆ cost D 2 2 (48) The intensity I of the detected beam: V 1 2 I I o sin D I osin 2 2 V / 2 2 where V/2: half-wave voltage (49) Integrated optical modulator V(t) Coplanar strip electrodes Thin buffer layer d Polarized input light L LiNbO 3 Ea EO Subs trate Waveguide x z y LiNbO 3 Cross-section Integrated tranverse Pockels cell phase modulator in which a waveguide is diffused into an electro-optic (EO) substrate. Coplanar strip electrodes apply a transverse field E a through the waveguide. The substrate is an x-cut LiNbO 3 and typically there is a thin dielectric buffer layer ( e.g. ~200 nm thick SiO 2) between the surface electrodes and the substrate to separate the electrodes away from the waveguide. 2 © 1999 S.O. Kasap,Optoelectronics (Prentice Hall) L D nr V d 3 o 22 0.5-0.7 (50) V(t) Electrode C In B B A A Out D Waveguide LiNbO 3 EO Substrate An integrated Mach-Zender optical intensity modulator. The input light is split into two coherent waves A and B, which are phase shifted by the applied voltage, and then the two are combined again at the output. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Ti diffused lithium noibate electro-optic (Pockels effect) modulator C. Kerr Effect: Phase modulatorz Ez E D Input light Ea y Output light Ex x An applied electric field, via the Kerr effect, induces birefringences in an otherwise optically istropic material. . Dn=a2E2=KEa K: Kerr coefficient (51) 2 Acousto-Optic Effects A. Definition Acousto-Optic Effects : changes in the refractive index of a material induced by a strain (S), which therefore modulates the optical properties Photoelastic effect Photoelastic effect: 1 D 2 pS n (52) Acoustic absorber Induced diffraction grating Incident light Acoustic wave fronts q Diffracted light q Through light Piezoelectric crystal Modulating RF voltage Interdigitally electroded transducer Traveling acoustic waves create a harmonic variation in the refractive index and thereby create a diffraction grating that diffracts the incident beam through an angle 2 q. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Incident optical beam A Diffracted optical beam A' B nmin nmax nmin nmax Acoustic wave x x B' q O q Q P si nq si nq Acoustic wave fronts O' vacoustic nmin nma x Simplified n nmin nma x n Actual Consider two coherent optical waves A and B being "reflected" (strictly, scattered) from two adjacent acoustic wavefronts to become A' and B'. These reflected waves can only constitute the diffracted beam if they are in phase. The angle q is exaggerated (typically this is a few degrees). © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) The condition that gives the angle q for a diffracted beam to exist is, Bragg condition : 2sinq=/n (53) If is the frequency of the acoustic wave, the diffracted beam has a Dopper shifted frequency: Doppler shift : (54) where is the angular frequency of the incident optical wave 參考書目 1. S. O. Kasap, Optoelectronics and Photonics: Principles and Practices, Prentice-Hall, Inc., 2001. 2. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, John Wiley & Sons, Inc. 1991.