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 III12 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,

EEEHHH1  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 )]

EEE12 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 2 exp[ 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 

I2 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)

 II20  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   2d 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 m1 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 (1RR )  (1  ) Transmittance T tr 12 22 Ei (1RRRR1 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  2nd ()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  /2 0 < V < 30V transmit o 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 vvgp1  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 d2 → 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