EE485 Introduction to Photonics. Introduction

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1 EE485 Introduction to Photonics Introduction

2 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. Geometrical (ray) optics The Strange Story of the Quantum Banesh Hoffmann, 1947 Lih Y. Lin 2

3 History of Optics Quantum optics Geometrical optics (Ray optics) Ray optics Enunciated by Euclid in Catoptrics, 3 B.C. Early 16: First telescope by Galileo Galilei End of the 17 th century: Light as wave to explain reflection and refraction, by Christian Huygens 174: Corpuscular nature of light (light as moving particles) to explain refraction, dispersion, diffraction, and polarization, by Issac Newton Early 18: 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 19: Max Plank quantum theory of light 195: Albert Einstein photoelectric effect experiment, light behaves as particles with energies E = h : de Broglie quantum mechanics explaining the wave-particle duality of light 195s: Communication and information theory 196: First laser E-M wave optics Lih Y. Lin 3

4 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

5 Electromagnetic Spectrum Optical frequencies Lih Y. Lin 5

6 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: Pedrotti 3, Sec , Sec

7 Historic Young s Double-slit Experiment (182) Water waves from two point sources Light is wave Lih Y. Lin 7

8 What does an optical wave look like Water waves Direct measurement of light waves (Goulielmakis, et al., Science, V. 35, p , August 24) Lih Y. Lin 8

9 1-D Wave Equation 1-D traveling wave function: y f( x vt) 2 2 y 1 y They satisfy 1-D differential wave equation: x v t Quiz: Which one(s) of the following wave functions represent traveling waves? What is the magnitude and direction of the wave velocity? yzt A t z 2 (, ) cos [ ( )] y xt Ax xt t 2 2 (, ) ( 4 4 ) y xt ABx t 2 (, ) ( ) Exercise: Consider a pulse propagating in the x direcion with speed v. The shape of the pulse at t t is given by 2 b yxt (, t) 2 2 a ( xx) 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 A snapshot in time y sin sin A [ ( ) ] or [( ) ] cos k x vt A cos kx t 2 k : Propagation constant 2 f : Angular frequency Harmonic waves with different A,, k

10 Harmonic Waves A snapshot in time y sin sin A [ ( ) ] or [( ) ] cos k x vt A cos kx t 2 k : Propagation constant 2 f : Angular frequency Harmonic waves with different A,, 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 1

11 Exercise A red diode laser, with = 65 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 = ) x Note: Light speed in free c = 3 x 1 8 m/s. Assume amplitude A remains constant as the wave enters the medium. Amplitude displacement at the interface = A/ 2 Lih Y. Lin 11

12 Plane Waves and Spherical Waves Plane wave in +x-direction Plane wave in any direction Spherical wave Ae ikr ( cos t) Ae Define k to represent the propagation constant and direction. ( r, t) ( r)exp( it) D wave equation: 2 2 v t 2 2 Helmholtz equation: ( k ) ( r ) A e i t r i( kr t) ( kr ) Intensity (W/m 2 ) A r Energy conservation obeyed 2 Lih Y. Lin 12

13 Useful Formulas in Vector Calculus Lih Y. Lin 13

14 Light as Electromagnetic Waves (Goulielmakis, et al., Science, V. 35, p , August 24) From Maxwell s equation to Wave equation: Maxwell s equations in free space Wave equation E H Necessary condition u t u 2 2 E E c H xxˆeyyˆ Ezzˆ t E,, or,, t : Electric field (V/m) u Exyz Hxyz E HH 1 8 xxˆhyyˆ Hzzˆ c 31 ( m/ s) H : Magnetic field (A/m) 9 (1/ 36 ) 1 ( F/ m) : Electric permittivity : Speed of light in free space 7 41 ( H/ m) : Magnetic permeability Lih Y. Lin 14

15 Maxwell s Equations in a Medium Assume a non-magnetic medium with no free electric charges or currents D H, D: Electric displacement t B E, B: Magnetic flux density t D B Physical meaning of the electric displacement: DEP ( D if the medium has a charge density ) B H Boundary conditions: Tangential components of E and H are continuous. Normal components of D and B are continuous. Power flow per unit area: SRe{ E} Re{ H} (W/m ): Poynting vector 2 - P + D E E B H Lih Y. Lin 15

16 Linear, Nondispersive, Homogeneous, and Isotropic Media P E : Electric susceptibility D E (1 ) / : Dielectric constant Maxwell s equations: D E H H t t B H E E t t D E B H Identical to Maxwell s equations in free space with replaced by. In free space In a medium u 2 1 u Wave equation: u u c t v t 1 c v 1 Speed of light: c n n / 1 : Refractive index 2 Lih Y. Lin 16

17 Monochromatic Electromagnetic Waves Let s relate harmonic waves to electromagnetic waves (,) r t ()exp( r i t) E Maxwell s equations: H t H E t E H Helmholtz equation: 2 2 ( k ) ( r) Optical intensity: 2 2 E E() r e H H() r e it it (E-M wave represented by complex numbers) Hr () ier () Er () i Hr () Er () Hr () ( k ) u( r) u() r E () r or H () r k xyz,, xyz,, nk I 1 S Re E() r H()* r 2 Lih Y. Lin 17

18 Plane Electromagnetic Wave (I) Ae i( kr t) Substituting into Maxwell s equations: Hr () ier () Er () i Hr () i( kr t) it Er (,) t Ee Er () e i( kr t) it Hr (,) t He Hr () e k H E ke H E, H, and k are mutually orthogonal Transverse electromagnetic (TEM) wave. Lih Y. Lin 18

19 Plane Electromagnetic Wave (II) Relationship between the amplitude of the electric field and the magnetic field: E H Optical intensity: I : Impedance of the medium n : Impedance of free space 1 1 * E S Re E() r H()* r EH 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 = 65 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 = Lih Y. Lin 19

20 Exercise Determine the power of a 1-mW laser beam, (a) With = 44 nm, after traveling 1 km of water at various locations. (b) With = 155 nm, after traveling 1 km of optical fiber. Lih Y. Lin 2

21 Absorption Bands of Optical Materials Lih Y. Lin 21

22 Applications of Optical Fibers Optical fiber structure Wires for light Optical fiber transmission system Inter-continental optical fiber network Lih Y. Lin 22

n cladding n cladding Most of the rays entering the top of the

23 Total Internal Reflection (TIR) c 1 1 c : Critical angle 1 n2 c sin n1 Example 1: Diamond Example 2: Optical Fiber 125 m n core n core > n cladding n cladding Most of the rays entering the top of the diamond will exit from the top due to total internal reflection. Lih Y. Lin 23

24 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: NA.. nsin m n n N.A.: Numerical Aperture m : Acceptance angle Calculate NA and m for a silica glass fiber in air with n 1 = and n 2 = Lih Y. Lin 24

Electromagnetic optics!

1 EM theory Electromagnetic optics! EM waves Monochromatic light 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals