EE485 Introduction to Photonics Superposition of Waves and Interference 1. Two-beam interference and interferometry 2. Multi-wave interference 3. Fabry-Perot interferometer 4. Group/phase velocity and dispersion Reading: Pedrotti3, Sec. 5.1-5.2, 5.5-5.6, 7.1-7.2, 7.4, 7.9, 8.1-8.7, 8-9, 10.7-10.8 Reference: J. S. Verdeyen, “Laser Electronics,” 3rd ed., Sec. 6.1-6.3, Prentice Hall Superposition Principle 11(r ,tI ), 22(r ,tI ), + √ (,)(,)(,)rt 12 r t r t or X III12 or X Both? Formally speaking, If 1 and 2 are independently solutions of the wave equation, 1 2 2 vt22 then the linear combination, ab 12 ab, : constants is also a solution. For electromagnetic waves, EEEHHH1 2 1 2 (Orientation of the fields must be taken into account.) Lih Y. Lin 2 Two-Beam Interference E1 E 01exp[i ( k 1 r t 1 )] E 01 exp[ i ( 1 t )] Same frequency, different phase E2 E 02exp[i ( k 2 r t 2 )] E 02 exp[ i ( 2 t )] EEE12 Recall from Light as Electromagnetic Waves, optical intensity = 1 2 I ||S Re E () r H ()* r E () r 2 For the combined wave, ** I (()E1 r E 2 ())( r E 1 () r E 2 ()) r If E01 and E02 are parallel to each other, I I1 I 2 I 1 I 2exp[ i ( 1 2 )] I1 I 2exp[ i ( 2 1 )] IIIII 1 2 2 1 2 cos II 12 Fringe contrast max min IImax min Lih Y. Lin 3 Young’s Double-slit Experiment kasin I2 I0 [1 cos( ka sin )] Discussion If a thin plate of glass is placed over one 2 asin 4I0 cos of the slits, what will happen to the fringe pattern? For points P near the optical axis, 2 ay II 40 cos L Lih Y. Lin 4 Michelson Interferometer (I) II20 1 cos 2 : Optical path length difference Dark fringes: 2dm cos m || λ = 632.8 nm λ = 420 nm 2d sin Lih Y. Lin 5 Michelson Interferometer (II) Sensing small refractive index change Start with dark fringe in the center, (nx 1)2 I( n ) 2 I0 1 cos 2 Input n Example: x 632.8 nm x 0.5 105 3.164 mm →n of 10-5 results in 2 phase change. Resolving small wavelength difference (Spectrometer) Start with coincident bright fringe in the center Input for both 1 and 2. Move the mirror by d until the next coincidence occurs. 2 d 2d Lih Y. Lin 6 Mach-Zehnder Interferometer Free-space Fiber-optic Example: Design a fiber-optic Mach-Zehnder interferometer that can demultiplex two light signals of free-space wavelength 1 = 1551 nm and 2 = 1550 nm. Through which output port would light of 3 = 1549 nm and 4 = 1548 nm exit? Lih Y. Lin Interference in Dielectric Films High-reflection coating (dielectric mirror) high/4 low/4 high/4 low/4 Additional phase shift occurs when high/4 reflected by higher index material. low/4 In normal incidence (from n1 material to n2 material), the reflection coefficient 1 n r 1 n n n21/ n : Relative index Anti-reflection (AR) coating (0-th order): nfs n0 n, t f / 4 Lih Y. Lin 8 Multi-Wave Interference ― Equal Amplitude and Equal Phase Difference Umo Iexp[ i ( m 1) ], m 1,2,..., M M 1 exp(iM ) UUI mo m1 1 exp(i ) 2 2 sin (M / 2) IUI 0 sin2 ( / 2) Interesting features: M = 5 • Mean intensity 1 2 I Id MI0 2 0 2 • Peak intensity = M I0 • Intensity drops to zero at = 2/M from peak intensity. • Sensitivity to increases with M. • (M - 2) minor peaks between major peaks. Lih Y. Lin 9 Multi-Wave Interference Example: Bragg Reflection Intensity of the reflected light maximizes at n sin 2d One of the applications: Characterizing the lattice constant of a crystal. = 1.16 Å For (200) direction, 1.16 d 2.48 Å 2sin(27 / 2) Lih Y. Lin 10 Multi-Wave Interference ― Progressively Smaller Amplitude and Equal Phase Difference Parallel plate Fabry-Perot Interferometer Two high-reflection plates separated by a distance d. d is often tunable. r, r’: Reflection coefficient t, t’: Transmission coefficient Lih Y. Lin 11 Fabry-Perot Interferometer ― Principle r1, r2: Reflection coefficients t1, t2: Transmission coefficients (These all work on fields.) Consider just to the right of mirror M1, E0 En i2 0 1 r12 r e 2 nd 0 Furthermore, E01 t Ei i Etr t2 e E n Lih Y. Lin 12 Fabry-Perot Interferometer ― Spectra 22 Define power reflectivity of the mirror R r, t 1 R 2 E (1RR ) (1 ) Transmittance T tr 12 22 Ei (1RRRR1 2 ) 4 1 2 sin 2 Er Reflectance RTnet 1 Ei Quiz: For sharper transmission spectrum, do we want higher R1,2 or lower R1,2? Lih Y. Lin 13 Fabry-Perot Interferometer ― Parameters Free spectral range (FSR) c 2 , 0 FSR2nd FSR c FSR FWHM c 1 RR12 1/2 1/4 2nd ()RR12 Cavity Q-factor 2nd ()RR 1/ 4 Q 0 12 1/ 2 0 1 RR12 Example Finesse A HeNe laser ( = 632.8 nm) cavity is defined by d ~ 1 m, ()RR 1/4 FSR 12 R1 = R2 = 0.99 F 14 8 1/2 1 RR12 → 0 = 4.74 x 10 , 1/2 = 480 kHz, Q = 9.88 x 10 , and F = 313. Lih Y. Lin 14 Fabry-Perot Interferometer ― Tunable Filter and Spectrometer T Resonant condition 2 nd dm c 0 2nd Resonant wavelength 0 m But the Fabry-Perot interferometer has finite passband width … T 2 1 d ()min Resolving criterion: Spacing less than what’s defined by T = 0.5 Tmax 2 1 RR 0 12 Exercise: min 2nd ()RR 1/4 12 Design a tunable filter that can separate 1 = 1550.918 nm and = 1552.524 nm (Two standard Resolving power R 2 min wavelengths in optical fiber communication systems). Lih Y. Lin 15 MEMS Tunable Filters and Modulators Filter Tuning plot Output fiber 0 21 V 18 V 15 V 12 V 8 V -10 Ground Top curved mirror -20 V Bottom mirror -30 -40 -50 100 mm Transmission(dB) -60 1500 1520 1540 1560 1580 Wavelength (nm) 22 mm Via-hole silicon Input fiber 100 mm 0.7 input 0.6 reflect gap 3 o /4 V drive 0.5 /4 SiNx 0.4 PSG 0.3 Silicon 0.2 Reflectivity 0.1 o gap o /2 0 < V < 30V transmit drive 0.0 3/4 < gap < /2 1300 1400 1500 1600 Wavelength (nm) Ford, Walker, Greywall & Goossen, IEEE J. Lightwave Tech. 16, 1998 Lih Y. Lin 16 Two-Wave Interference ― Different f and constructive interference destructive interference E1 E 0cos( k 1 x 1 t ) E2 E 0cos( k 2 x 2 t ) E E1 E 2 2 E 0 cos( kp x p t )cos( k g x g t ) kk k 1 2, 1 2 pp22 |kk | | | k 1 2, 1 2 gg22 cos(kpp x t ) : Average cos(kgg x t ) : Modulation, beating Beat frequency bg 2 12 Lih Y. Lin 17 Phase Velocity, Group Velocity, and Dispersion Assume the two waves are close in and k. 1 1 p 12 c Phase velocity: v E(x) p k k k k n p 12 Envp1(x) 0 t0 Envp2(x) g 12 d Group velocity: vg 1 1 kg k12 k dk 0 2 4 6 8 10 0 x 10 1 dn 1 vvgp1 nd E(x) Envp1(x) 0 t1 Envp2(x) Dispersion. n = n(). Light with different wavelength travels with different velocity in a medium. 1 1 0 2 4 6 8 10 0 x 10 Normal dispersion: dn/d < 0, vg < vp. 1 1 v determines the speed with which energy is g E(x) transmitted. It is the directly measurable speed ofEnvp1 (x) t2 0 the wave. Envp2(x) 1 1 0 2 4 6 8 10 0 x 10 Lih Y. Lin 18 Dispersion in an Optical Fiber c Group index: v g N dn Nn d Dispersion coefficient: dn2 D (s/m-nm) c d2 → A measure of time delay per wavelength (nm) after certain distance. Bandwidth limited by dispersion: 0.5 L max (/)L If the original pulse width cannot be neglected compared to the broadening, 2 2 2 f 0 () max 0.5 / f Lih Y. Lin 19 Pulse Broadening in an Optical Fiber √ X or, √ X Lih Y. Lin 20 Multi-Wave Interference ― Different Frequencies |Uf| 1, 2, …, M At a given position r, (M 1)/ 2 U( t ) I00 exp[ i 2 ( f q f ) t ] qM( 1)/ 2 2 2 sin (M t / TF ) f I( t ) | U ( t ) | I0 sin2 (tT / ) f0 f F 1 TF M = 5 f (Compare this with multi-wave interference of same frequencies but different phases.) 1 Mf Total frequency bandwidth E.g., 1 ps pulse can be generated by combining 1000 waves separated by 1 GHz from each other. (Note: The exact relation between pulse width and bandwidth depends on the pulse shape. This is a subject of Fourier Optics.) Lih Y. Lin 21 .