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# wave nature–light is an electromagnetic wave having a very

Chapter 2 Fundamentals of Optics and Waves
! Nature of Light ! Optics Review ! Lightwave Fundamentals ! Optical Waveguides

1

Nature of Light
? Wave nature
– Light is an electromagnetic wave having a very high oscillation frequency

λ=v/f
? Particle nature

v=c=3x108 m/s

in free space

e.g., λ=1.5 ?m" f=2x1014 Hz

– Light is made up of photons – The energy of a single photon is

Wp=hf=hc/λ h=6.626x10-34 J?s Planck’s constant λ (?m) = 1.24/Wp(eV)
e.g., λ=1.55 ?m" Wp=0.8 eV

2

Optics Review
? In a medium, light ray travels at a speed

c v= n

? Rays travel in straight paths unless deflected by some change in the medium ? At a plane boundary between two media, a ray is reflected at an angle equal to the angle of incidence ? If any power crosses the boundary, the transmitted light is determined by Snell’s law

sin θ t n2 = sin θ i n1

Reflected

Transmitted

n1
Incident

θr θi

θt

n2
3

Snell’s Law
Material
Air Water MgF Silica glass Calcite Sapphire LiNbO3 Zinc sulfide InP GaAS Si Ge

n
1.0 1.33 1.38 1.5 1.6 1.8 2.25 2.3 3.21 3.35 3.5 4.0
θ

Air
i

θ

t

GLASS

The transmitted ray is bent toward the normal when traveling from a medium with lower n to a medium with a higher n
Air GLASS

Air

A ray is undeflected after transversing a parallel glass plate
4

Lenses
Convex lens Focusing
Fiber

GRIN rod

Focal length

Collimating

Focal length

5

Numerical Aperture
f f θ d/2 f

d
Photodetector

θ

θ=0

θ = θmax
0.5 0.4 NA 0.3 0.2 0.1 0

θ > θmax

θmax= maximal acceptance angle tan θmax=d/2f NA = n0 sin θmax
θ NA = sin θ

0

10o

20o

30o
6

Acceptance Angle

Lightwave Fundamentals
E=E0 sin(ωt-kz) w=2πf : radian frequency k=ω/c=2πn/λ0 :
propagation constant
Wavelength

1/λ : wavelength number

If the medium is lossy with attenuation coefficient α

E = E0

? z e 2 sin(ωt

α

? kz )

α: power loss coefficient
For a path of length L Loss(dB)=10?log10exp(-αL) Loss(dB/km)=-4.34α(1/km)
7

Resonant Cavity
The left-going and right-going waves can add either constructively and destructively

To produce a stationary standing wave " L=mλ/2 m: integer

0

L
8

Resonant Frequency
? The resonant frequencies
c 2Ln

mc fm = 2nL

fm-1

fm

fm+1

fm+2

? Longitudinal mode spacing

Frequency

c ?f c = 2nL
? mode spacing in wavelength
INTENSITY 2.0 1.5 1.0 0.5 0.0 -3.5 -2.5 -1.5 -0.5

Material gain spectrum

?λc =

λ2 0
c

?f c =

λ2 0
2ng L

0.5 1.5 2.5 3.5

WAVELENGTH DEVIATION(nm)
9

Source Linewidth
? A perfectly coherent source emits light at a single wavelength#Zero linewidth ? Real sources produce radiation over a range of wavelengths

?f ?λ = λ f
Typical Source Spectral Widths
Source Linewidth (? λ )(nm)

1.0

0.5

LED FP Laser diode Nd:YAG laser HeNe laser DFB laser

20-100 1-5 0.1 0.002 0.0002

0 1480 1500 1520 1540 1560

10

Material Dispersion ? Wave velocity v = c n
Refractive index varies with wavelength ? cause velocity variation with wavelength ? dispersion ?pulse spreading

Fiber

=

=
Time

Time

11

Dispersion
Source spectrum L

τ: time to travel through a fiber of length L
Nondispersive medium

Dispersive medium ? (τ /L) ?λ λ1 λ2 Wavelength

τ/L

τ/L

(τ /L)2 (τ /L)1

λ1

λ2

n1 v1

n2 v2

Wavelength

τ1

τ2

Difference in traveling time ? pulse spread per unit length
λ ?τ ? ?τ ? ?τ ? ?? ? = ? ? ? ? ? = ? n′′?λ = ? M?λ c ? L ? ? L ? 2 ? L ?1
12

Dispersion parameter
Refractive index (n)

0
1.45

n'' = d2n/d2λ

n' = dn/dλ

0

λ0 (1.3?m) Wavelength

λ0 Wavelength

λ0 Wavelength
240 200 M [ps/(nm x km)] 160 120 80 40

M=

λ
c

n′′ ps/nm×km

Silica glass

To have small ?τ ? small ?λ and small M

0 40 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Wavelength (?m)
13

Reflection at an interface
? A wave incident on a plane boundary between two dielectrics is partially transmitted and reflected. ? The reflectance varies with the angle and the polarization
For normal incidence n ?n ρ= 2 1 n2 + n1

? n2 ? n1 ? R=? ?n +n ? ? ? 2 1?

2

n1 Ei Er

n2

Et

e.g., n1=1, n2=1.5, ? R=0.04
Er n1 n2

For angled incidence, the wave can be decomposed into two components

?

Er

θr θi

θt

?

Et

n1

n2

Et

θr θi
Ei

θt

? E

Boundary Perpendicular (s) polarization

Boundary Parallel (p) polarization

i

14

Reflectance of Angled Incidence
1.0 Reflectance
Reflectance 1.0

n1=1, n2=1.5
0.5 Rs RP 0o

n1=1.5, n2=1
0.5 Rs 0 0o RP 60o 90o

0

θB 30o Angle of Incidence

90o

θB θC

Angle of Incidence

? R~0.04 for θi < 20o ? R=0 at θB (Brewster angle) for p-wave ? R= 1 (total reflection) for θi ≥ θc (critical angle) if n1 > n2
Brewster angle Critical angle (n1 > n2)

tan θ B =

n2 n1

n2 sin θ c = n1

For air to glass interface θB=56.3o, θc=41.8o for glass(1.48) to glass(1.46) interface θc=80.6o
15

Optical Waveguide
y x z d cladding n3 n1 n2 core

θ

θ

sin θ c = n3 n1 sin θ c = n2 n1

If n1 ≥ n2 and n1 ≥ n3, total internal reflection occurs for the light traveling inside the core

The wave propagates along the z-axis (longitudinal direction) with a propagation constant β and transverse field distribution U(y)
E ( y, z ) = U ( y )sin (ωt ? βz )
vg = ω β n2 ≤ neff ≤ n1
16

Phase velocity:

Effective index: neff = β k0

Wavelength in the waveguide: λ g = λ0 neff

Waveguide Modes
Only certain ray directions that is larger than θc is allowed to propagate along the waveguide without radiation ? modes
Waveguide modes must satisfy the transverse resonance condition
n2 n1 n2 TEo TE1 y

v ?E
TE

v ?H
TM

Round-trip phase shift

= m2π

y

TE2

y

TE3

y

d

n2 n1 n2

Low Order

High Order

17

Waveguide and Modal Dispersion
? Effective index of waveguide varies with d/λ ? waveguide dispersion ? Number of modes existed in a waveguide increases with d/λ and n2-n1 ? Different orders of modes have different effective index ? modal dispersion Waveguide dispersion: neff varies with λ
λ ?τ ? ′′ ?λ = ? M g ?λ ?? ? = ? neff c ?L?
Mg =

λ
c

′′ neff

Modal dispersion: different modes travel different path lengths
NA2 ? τ ? n1 ?? ? = ? = 2cn1 ?L? c

?=

n1 ? n2 n1
18

Bandwidth Limited by Dispersion
The bandwidth of a fiber channel is Written in electrical bandwidth For RZ signals
0.35 BRZ × L = (?τ L )
T 2

f 3dB

1 ≈ 2 ?τ 0.35 ≈ ?τ

f 3dB ( electrical )
Power Spectra

T

1 0 1/T 2/T Frequency

1

1 1 Time

0

1

For NRZ signals
0.7 BRZ × L = (?τ L )

T

Power Spectra

1 0

1 0 1 1 Time

1 0 1/T 1/2T Frequency
19

Optical Waveguide Coupling
? Edge coupling To couple a light into a guided mode ? θ ≥ θc
2 sin θ = n2 n1 ? cosθ = n1 ? n1 2 n1

radiate out θi θ1 θ

n3 n1 n2

Numerical aperture
2 NA = n0 sin θ i = n1 cos θ = n1 ? n1 2 = n1 2 ?

n1 ? n2 ?= n1

Edge coupling using a lens ? Prism coupling: ? Grating coupling
20

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