EE485 Introduction to Photonics Introduction Nature of Light “They could but make the best of it and went around with woebegone faces, sadly complaining that on Mondays, Wednesdays, and Fridays, they must look on light as a wave; on Tuesdays, Thursdays, and Saturdays, as a particle. On Sundays they simply prayed.” The Strange Story of the Quantum Banesh Hoffmann, 1947 Geometrical (ray) optics Lih Y. Lin 2 History of Optics Quantum optics • Geometrical optics (Ray optics) E-M wave optics Ray optics – Enunciated by Euclid in Catoptrics, 300 B.C. • Early 1600: First telescope by Galileo Galilei • End of the 17th century: Light as wave to explain reflection and refraction, by Christian Huygens • 1704: Corpuscular nature of light (light as moving particles) to explain refraction, dispersion, diffraction, and polarization, by Issac Newton • Early 1800: Interference experiment by Thomas Young – light is wave • Maxwell equation (1864) ― Light as electromagnetic waves, by James Clerk Maxwell • How about emission and absorption? • Quantum theory ― Light as photons – 1900: Max Plank – quantum theory of light – 1905: Albert Einstein – photoelectric effect experiment, light behaves as particles with energies E = h – 1925-1935: de Broglie –quantum mechanics explaining the wave-particle duality of light • 1950s: Communication and information theory • 1960: First laser Lih Y. Lin 3 Topics we plan to cover • Light as electromagnetic waves • Polarized light • Superposition of waves and interference • Diffraction • Photon and laser basics • Laser operation • Nonlinear optics and light modulation Lih Y. Lin 4 Electromagnetic Spectrum Optical frequencies Lih Y. Lin 5 EE485 Introduction to Photonics Light as Electromagnetic Waves 1. Wave equations 2. Harmonic waves 3. Electromagnetic waves 4. Energy flow and absorption 5. Fiber optics Reading: Pedrotti3, Sec. 4.1-4.8, Sec. 10.1-10.6 Historic Young’s Double-slit Experiment (1802) Water waves from two point sources Light is wave Lih Y. Lin 7 What does an optical wave look like Water waves Direct measurement of light waves (Goulielmakis, et al., Science, V. 305, p. 1267-1269, August 2004) Lih Y. Lin 8 1-D Wave Equation 1-D traveling wave function: yf ()xvt 22yy1 They satisfy 1-D differential wave equation: xvt222 Quiz: Which one(s) of the following wave functions represent traveling waves? What is the magnitude and direction of the wave velocity? yzt(,) A cos[(2 t z )] y(,)xt Ax (22 4 xt 4 t ) y(,)xt ABx (2 t ) Exercise: Consider a pulse propagating in the –x direcion with speed v. The shape of the pulse at tt 0 is given by b2 yxt(, t0 ) 22 axx()0 Such a pulse is known as a Lorentzian pulse. Determine the shape of the pulse at an arbitrary time t. Lih Y. Lin 9 Harmonic Waves sin sin A snapshot in time yA [( kxvt ) ] or A[( kxt ) ] cos00 cos 2 k : Propagation constant 2f : Angular frequency Harmonic waves with different A, 0, k and v or k and form a complete set of functions. Any periodic wave form can be decomposed into linear combination of harmonic wave functions. → Fourier Optics. + = + +… Lih Y. Lin 10 Exercise A red diode laser, with = 650 nm in free space, incidents from air to a medium with refractive index n equal to 1.5, as shown below. Derive its harmonic wave functions in the air and in the medium. The speed of light in the medium is ⁄ . y (Snapshot at t = 0) x Note: Light speed in free c = 3 x 108 m/s. Assume amplitude A remains constant as the wave enters the medium. Amplitude displacement at the interface = A/2 Lih Y. Lin 11 Plane Waves and Spherical Waves Plane wave in +x-direction Plane wave in any direction Spherical wave A Aeikr(cos t ) Aeit()kr eit()kr r Define k to represent the propagation constant and direction. 2 A (rr ,tit ) ( )exp( ) Intensity (W/m2) 1 2 r 3-D wave equation: 2 vt22 → Energy conservation obeyed Helmholtz equation: ()()022kr Lih Y. Lin 12 Useful Formulas in Vector Calculus Lih Y. Lin 13 Light as Electromagnetic Waves (Goulielmakis, et al., Science, V. 305, p. 1267-1269, August 2004) From Maxwell’s equation to Wave equation: Maxwell’s equations in free space Wave equation E Necessary condition 2 H 2 1 u 0 t u 0 Exyz EEEˆˆˆ ct22 H xyz E uE or H 0 t : Electric field (V/m) xyz,, xyz ,, 1 8 E 0 HxyzHHHxyzˆˆˆ cms310(/) H 0 : Magnetic field (A/m) 00 9 : Speed of light in free space 0 (1/36 ) 10 (Fm / ) : Electric permittivity 7 0 4 10 (Hm / ) : Magnetic permeability Lih Y. Lin 14 Maxwell’s Equations in a Medium D HD , : Electric displacement Assume a non-magnetic t B medium with no free electric EB , : Magnetic flux density charges or currents t D 0 B 0 Physical meaning of the electric displacement: DEP 0 - - - - - (D if the medium P E has a charge density ) + + + + + BH0 Boundary conditions: • Tangential components of E and H are continuous. B • Normal components of D and B are continuous. Power flow per unit area: D 2 SERe{ } Re{ H } (W/m ): Poynting vector E H Lih Y. Lin 15 Linear, Nondispersive, Homogeneous, and Isotropic Media DE PE 0 (1 ) : Electric susceptibility 0 /0 : Dielectric constant Maxwell’s equations: D E H H t t B H E E 0 t t D 0 E 0 B 0 H 0 Identical to Maxwell’s equations in free space with replaced by 0. In free space In a medium 1 2u 1 2u Wave equation: 2u 0 2u 0 ct22 v22t 1 c v 1 n Speed of light: c 0 00 n /10 : Refractive index Lih Y. Lin 16 Monochromatic Electromagnetic Waves Let’s relate harmonic waves to electromagnetic waves EEr ()eit (,)rrtit ()exp( ) HHr ()eit (E-M wave represented by complex numbers) E Maxwell’s equations: H t Hr()i Er () H Er() i 0 Hr () E 0 t Er() 0 E 0 Hr() 0 H 0 Helmholtz equation: ()()022kur 22 ()()0k r uE()rr xyz,, () or H xyz ,,() r Optical intensity: knk 00 1 I ||SErHrRe () ()* 2 Lih Y. Lin 17 Plane Electromagnetic Wave (I) Er(,)te Eit()kr Er () e it Aeit()kr 0 it()kr it Hr(,)te H0 Hr () e Substituting into Maxwell’s equations: Hr()i Er () kH 00 E Er() i 0 Hr () kE000 H → E, H, and k are mutually orthogonal ― Transverse electromagnetic (TEM) wave. Lih Y. Lin 18 Plane Electromagnetic Wave (II) Relationship between the amplitude of the electric field and the magnetic field: E 0 H0 0 : Impedance of the medium n 0 0 120 377 : Impedance of free space 0 Optical intensity: 2 11* E0 IEH|SErHr | Re () ()* 00 222 Example: Let’s describe an optical wave mathematically. A laser beam of radius 1 mm carries a power of 5 mW. (a) Determine its average intensity and the amplitude of its electric and magnetic fields. (b) Assume the laser beam is a TEM wave (actually not a completely correct assumption) with = 650 nm, propagating in x-direction, and the electric field is along y- direction (Slide 11). Determine the complex wave functions for the electric and magnetic fields. (c) Determine the wave functions after entering a medium with n = 1.5. Lih Y. Lin 19 Exercise Determine the power of a 10-mW laser beam, (a) With = 440 nm, after traveling 1 km of water at various locations. (b) With = 1550 nm, after traveling 100 km of optical fiber. Lih Y. Lin 20 Absorption Bands of Optical Materials Lih Y. Lin 21 Applications of Optical Fibers Optical fiber structure “Wires” for light Optical fiber transmission system Inter-continental optical fiber network Lih Y. Lin 22 Total Internal Reflection (TIR) c: Critical angle c 1 1 1 n2 c sin n1 Example 1: Diamond Example 2: Optical Fiber 125 m ncore ncore > ncladding ncladding Most of the rays entering the top of the diamond will exit from the top due to total internal reflection. Lih Y. Lin 23 Numerical Aperture of an Optical Fiber Let’s do an exercise: Show that the maximum incident angle m for TIR in the optical fiber is related to the refractive indices by: 22 NA.. n012 sinm n n N.A.: Numerical Aperture m: Acceptance angle Calculate NA and m for a silica glass fiber in air with n1 = 1.475 and n2 = 1.460. Lih Y. Lin 24.