Class overview: Brief review of physical optics, wave propagation, interference, diffraction, and polarization Introduction to Integrated optics and integrated electrical circuits Guide-wave optics: 2D and 3D optical waveguide, optical fiber, mode dispersion, group velocity and group velocity dispersion. 4. Mode-coupling theory, Mach-zehnder interferometer, Directional coupler, taps and WDM coupler. 5. Electro-optics, index tensor, electro-optic effect in crystal, electro- optic coefficient 6. Electro-optical modulators 7. Passive and active optical waveguide devices, Fiber Optical amplifiers and semiconductor optical amplifiers, Photonic switches and all optical switches 8. Opto-electronic integrated circuits (OEIC)

Complex numbers or C: amplitude : angle b C  a Real numbers do not have phases Complex numbers have phases  is the phase of the complex number

Physical meaning of multiplication of two complex numbers times Phase: Phase delay Why 90 i 180 -1 1 -i 270

1 90 180 270 -1 Optical wave, frequency domain Vertical polarization, k  polarization, transwave k,  x k vector, to x-direction Initial phase Horizontal polarization Phase delay by position Amplitude, I=|A|2, Optical intensity

Optical wave propagation in air (free-space) k,  Time and frequency domain of optical signal Phase delay

Optical wave propagation in air (free-space) k ∆x Wave front Phase is the same on the plane --- plane wave Phase distortion in atmosphere k ∆x Wave front distortion Additional phase delay,  non ideal plane wave

Optical wave to different directions Phase delay Refractive index n In vacuum, c: speed of light In media, like glass,

Optical wave to different directions Phase delay In media with refractive index n n Phase delay nx1, nx2, optical path

Interference of two optical waves Upper arm : a1 a3 50% A Aout b1 b2 Lower arm : Aout = + Phase delay between the two arms: = If phase delay is 2m, then: If phase delay is 2m +, then: = -

March-Zehnder interferometer Upper arm : a3 a1 50% A Aout n2 b3 b1 b2 Lower arm : Phase delay between the two arms: = If phase delay is 2m, then: If phase delay is 2m +, then: = - Electro-optic effect, n2 changes with E -field

Electro-optic modulator based on March-Zehnder interferometer Electro-optic effect, n2 changes with E -field 50% A Aout n2 b3 b1 b2 Phase delay between the two arms: If phase delay is 2m, then: = If phase delay is 2m +, then: = -

Reflection, refraction of light 1 1 n1 n2 > n1 2 < 1 n2 2 Refraction reflection 1 1 n1 n2 < n1 2 > 1 n2 2 Refraction

Total internal reflection (TIR) 1 1 n1 n2 < n1 2 > 1 n2 2 Refraction When 2 = 900, 1 is critical angle

Total internal reflection (TIR) Total reflection When 2 = 900, 1 is critical angle 1 1 n1 n2 2 = 900 No refraction Application – optical fiber n1 Low loss, TIR n2 Flexible n1 Communication system Electrical signal Electrical signal Laser EO modulator Detector Optical fiber

Reflection percentage Intensity reflection = [(n1-n2)/(n1+n2)]2 Example 1: n1 Refractive index of glass n = 1.5; n2 Refractive index of air n = 1; The reflection from glass surface is 4%; Example 2: detector is usually made from Gallium Arsenide (GaAs), n = 3.5; Intensity reflection = [(n1-n2)/(n1+n2)]2 n1 The reflection from GaAs surface is 31%; N2 = 3.5 Detector, GaAs

Interference of two optical beams, applications 1, Mach-Zehnder interferometer n a2 a3 a1 Electro-optic effect, n2 changes with E -field 50% A Aout n2 b3 b1 b2

2, Michelson interferometer Phase difference: M1 Beam splitter n1 d1 n2 d2 Measuring small moving or displacement Mirror 2 If the detected light changes from bright to dark, Then the distance moved is half wavelength/2 Detector Michelson interferometer measuring speed of light in ether, speed of light not depends on direction Phase difference: d1 d2 Beam splitter Detector n2 n1 M1 Mirror 2

3, Measuring surface flatness Accuracy: 0.1 wavelength = 0.1*500nm = 50nm n2 n2 4, Measuring refractive index of liquid or gas Gas in

Wave optics Maxwell equations: Dielectric materials Maxwell equations in dielectric materials: phasor

Wave optics approach Helmholtz Equation: Free-space solutions k ∆x Wave front Phase is the same on the plane --- plane wave

Wave optics Helmholtz Equation: 2-D Optical waveguide x Core, refractive index n1 Cladding, n2 z y n2 < n1 d TM mode: TE mode:

Normal reflection at material interface Helmholtz Equation: Er Ein n1 x n2 z Et

Normal reflection at material interface Helmholtz Equation: Er Ein n1 x n2 z Et Continuous @ z=0 Continuous @ z=0 Why?

Normal reflection at material interface Helmholtz Equation: Er Ein n1 x n2 Z = 0 Z = 0

Normal reflection at material interface Helmholtz Equation: Er Ein n1 x n2

Reflection, refraction of light Ein 1 1 y n1 Er x n2 z Et 2

Reflection, refraction of light Ein 1 1 y n1 Er x n2 z Et 2 Continuous @ z=0 Continuous @ z=0 Why?

Reflection, refraction of light Ein 1 1 y n1 Er x n2 z Et 2 Continuous @ z=0

Reflection, refraction of light Ein 1 1 y n1 Er x n2 z Et 2 Continuous @ z=0

Reflection, refraction of light Ein 1 1 y n1 Er x n2 z Et 2 r t 1 1

Ein 1 1 1 1 y n1 Er x n2 2 z Et 2

Ein 1 1 1 1 y n1 Er x n2 2 z Et 2 continuous

Ein 1 1 1 1 y n1 Er x n2 2 z Et 2 continuous =

Ein 1 1 1 1 y n1 Er x n2 2 z Et 2

Ein 1 1 1 1 y n1 Er x n2 2 z Et 2

Ein 1 1 1 1 y n1 Er x n2 2 z Et 2 Ein 1 1 Brewster’s angle 1 n1 Er n2 Et 2 http://buphy.bu.edu/~duffy/semester2/c27_brewster.html

Ein 1 1 1 1 y n1 Er x n2 2 z Et 2

Normal reflection at material interface Helmholtz Equation: Er Ein n1 x n2 n2 n1 r t r2t r4t rt r3t r5t Er Ein n1 x n2

Equal difference series + = + =

Power series … x - =

Power series When |b|<1 b can be anything, number, variable, complex number or function Optical cavity, multiple reflections n2 n1 r t r2t r4t rt r3t r5t

Optical cavity, wavelength dependence, resonant R, intensity reflection

Optical cavity, wavelength dependence, resonant Example: n2 = 1.5, R=0.04, L = 0.05mm, =0.55µm R, intensity reflection Intensity Wavelength (um) Wavelength (um)

Optical cavity, wavelength dependence, resonant Resonant condition Destructively Interference Constructively Interference Resonant condition, m = 1, 2, 3, … m=1, half- cavity m=2,  cavity

Optical cavity, Free-spectral range (FSR) Resonant condition, m = 1, 2, 3, … - Example: n2 = 1.5, R=0.04, L = 1mm, =0.55µm

Optical cavity, Free-spectral range (FSR) L increase, FSR decrease, FSR not dependent on R Example: n2 = 1.5, R=0.04, L = 0.05mm, =0.55µm L = 0.5mm FSR FSR

Loss, and photon lifetime of an resonant cavity Intensity left after two mirror reflection: r1 r1r2

Quality factor Q of an resonant cavity Intensity left after two mirror reflection: r1 r1r2 Time domain Frequency domain

Transmission matrix analysis of resonant cavity B1 B2 At this interface

Transmission matrix analysis of resonant cavity B1 B2 In Matrix form:

Transmission matrix analysis of resonant cavity B3 B1 B2 In Matrix form:

Transmission matrix analysis of resonant cavity B3 B4 B1 B2 In Matrix form:

Transmission matrix analysis of resonant cavity B3 B4 B1 B2 In Matrix form:

Transmission matrix analysis of resonant cavity B3 B4 B1 B2 In Matrix form:

E-field profile inside the resonant cavity B3 B4 B1 B2

E-field profile inside the resonant cavity B3 B4 B1 B2

E-field profile inside the resonant cavity B3 B4 B7 B8 B1 B2 B5 B6

Bragg gratings /4 n1 n2 d1 d2

Ring cavity, Ring resonator L = 2R0 95%, r R0

Ring cavity, Ring resonator filter L = 2R0

Lens and optical path Focal length Focal plane Same optical path f f focus lens

Lens and optical path Focal length f Focal plane Same optical path f f focus lens Same optical path f f focus lens

Lens and optical path Focal length f Focal plane Same optical path

Interference of multiple Waves, gratings Near field pattern x1 x2 Screen

Diffraction of Waves Far field pattern When ∆ = 2m, bright When ∆ = 2m+ , dark 2m = (2/) d*sin() x1  m  = d*sin() ∆L m  = d* ∆L /f   d ∆L = f*sin() ∆L= m *f/d, bright spots x2 f ∆x Focal plane ∆x = d*sin() Screen ∆ = k*∆x =(2/) d*sin()

Multiple slots, grating Far field pattern When ∆ = 2m, bright Focal plane  When ∆ = 2m+ , dark  d 2m = (2/) d*sin() ∆x1 m  = d*sin() d ∆x2 tan() = ∆L /f ∆L   ~ tan() ~ sin(), when  is small ∆L = f*sin() m  = d* ∆L /f f ∆L= m *f/d, bright spots Bright spot still at small position ∆xm ∆xm = md*sin() ∆m = k*∆xm =(2/) md*sin() Screen

Multiple slots, grating Screen f Focal plane ∆xm  d ∆xm = md*sin() ∆m = k*∆xm =(2/) md*sin() ∆L ∆L = f*sin() ∆x1 ∆x2

Optical wave, photon Photon energy:  is the frequency of the photon, momentum of the photon, For photon:

More on grating, Grating period, Phase matching condition k vector, k = (2/), along the beam propagation direction   d is called grating period, often use symbol: , d ∆x1 Grating vector = 2 /, Phase-matching condition d ∆x2 kout = 2/ kgrating = 2/  ∆xm = md*sin() kout *sin() = kgrating = 2/ ∆xm ∆m = k*∆xm =(2/) md*sin() d*sin() = 

Distributed feedback grating (DFB grating) Effective reflection   kin kin k kout k kout kout = kin – k kout = kin – k = - kin Phase-matching condition Phase-matching condition 2/ = 2*2/ 2nout / out = 2nin / in – 2/  = /2

Diffraction: Multiple beam interference

Diffraction:

Single slot diffraction:  

Single slot diffraction:   When bright When dark

Single slot diffraction:   When bright When dark

Angular width :   first dark: Half angular width

Diffraction of a lens d Focus size d Focus size

Resolution of optical instrument d Focus size ∆’ d = 16mm. eye

Polarization of light: Linear polarization

Polarization of light: linear polarization Any linear polarization can be decomposed into two primary polarization with the same phase Why decompose into two primary polarization directions? In crystal, the refractive index is different along different polarizations z no z y x y x no ne ne Refractive index ellipsoid

Polarization of light: circular polarization Lag /2

Polarization of light: circular polarization Clockwise

Polarization of light: circular polarization Count-clockwise elliptical polarization

Delays along different directions: z no z d y x y x no ne ne d Refractive index ellipsoid no no lag Phase difference: ne ne when Right (clockwise) circular polarization Quarter-wavelength /4 plate

Right (clockwise) circular polarization /4 plate 45º d no no lag Phase difference: ne ne when Change the direction of the polarization half-wavelength /2 plate

Change the direction of the polarization by 2 /2 plate   EO modulator polarizer analyzer dark no no Bright ne

EO modulator dark analyzer polarizer ne no Bright  

Circular polarizations and linear polarizations

Circular polarizations and linear polarizations + Linear polarization

Circular polarizations and linear polarizations +  Linear polarization

Faraday rotation Apply B field generate delay for left or right polarization

Jones Matrix

Jones Matrix

Jones Matrix Linear +

Jones Matrix /4 plate /2 phase delay /4 plate /2 phase delay

Jones Matrix y’ y x’ x

Jones Matrix y’ y x’ x

Jones Matrix

Jones Matrix

Jones Matrix Linear +

Faraday rotation Apply B field generate delay for left or right polarization