EE485 Introduction to PhotonicsIntroduction

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Nature of Light

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“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.”

The Strange Story of the QuantumBanesh Hoffmann, 1947

Geometrical (ray) optics

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History of Optics• Geometrical optics (Ray optics)

– Enunciated by Euclid in Catoptrics, 300 B.C.• Early 1600: First telescope by Galileo Galilei• End of the 17th century: Light as wave to explain reflection and refraction, by

Christian Huygens• 1704: Corpuscular nature of light (light as moving particles) to explain refraction,

dispersion, diffraction, and polarization, by Issac Newton• Early 1800: 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

– 1900: Max Plank – quantum theory of light– 1905: Albert Einstein – photoelectric effect experiment, light behaves as

particles with energies E = h– 1925-1935: de Broglie –quantum mechanics explaining the wave-particle

duality of light• 1950s: Communication and information theory• 1960: First laser

Quantum optics

E-M wave opticsRay optics

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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

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Electromagnetic Spectrum

Optical frequencies

EE485 Introduction to PhotonicsLight as Electromagnetic Waves1. Wave equations2. Harmonic waves3. Electromagnetic waves4. Energy flow and absorption5. Fiber optics

Reading: Pedrotti3, Sec. 4.1-4.8, Sec. 10.1-10.6

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Historic Young’s Double-slit Experiment (1802)

Light is wave

Water waves from two point sources

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What does an optical wave look like

Direct measurement of light waves

(Goulielmakis, et al., Science, V. 305, p. 1267-1269, August 2004)

Water waves

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1-D Wave Equation

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1-D traveling wave function: ( )y f x vt

They satisfy 1-D differential wave equation: 2 2

2 2 2

1y yx v t

Quiz: Which one(s) of the following wave functions represent traveling waves? What is the magnitude and direction of the wave velocity?

2( , ) cos [ ( )]y z t A t z 2 2( , ) ( 4 4 )y x t A x xt t

2( , ) ( )y x t A Bx t

0t t2

0 2 20

( , )( )by x t t

a x x

Exercise: Consider a pulse propagating in the –x direcion with speed v. The shape of the pulse at is given by

Such a pulse is known as a Lorentzian pulse. Determine the shape of the pulse at an arbitrary time t.

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Harmonic Waves

0 0sin sin

[ ( ) ] or [( ) ]cos cos

y A k x vt A kx t

Harmonic waves with different A, 0, k and vor k and form a complete set of functions. Any periodic wave form can be decomposed into linear combination of harmonic wave functions. → Fourier Optics.

A snapshot in time

2 : Propagation constant k

+

+

+…

=

2 : Angular frequencyf

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ExerciseA red diode laser, with = 650 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 ⁄ .

Note: Light speed in free c = 3 x 108 m/s. Assume amplitude A remains constant as the wave enters the medium. Amplitude displacement at the interface = / 2A

x

y(Snapshot at t = 0)

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Plane Waves and Spherical WavesPlane wave in +x-direction Plane wave in any direction Spherical wave

( cos )i kr tAe Define k to represent the propagation constant and direction.

( )i tAe k r ( )i tA er

k r

Intensity (W/m2)2A

r

→ Energy conservation obeyed3-D wave equation: 2

22 2

1v t

( , ) ( )exp( )t i t r r

Helmholtz equation: 2 2( ) ( ) 0k r

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Useful Formulas in Vector Calculus

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Light as Electromagnetic Waves

(Goulielmakis, et al., Science, V. 305, p. 1267-1269, August 2004)

From Maxwell’s equation to Wave equation:

0

0

90

70

00

(1/36 ) 10 ( / ) : Electric permittivity

4 10 ( / ) : Magnetic permeability

t

t

F m

H m

EH

HE

EH

Maxwell’s equations in free space2

22 2

, , , ,

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0 0

1 0

or

1 3 10 ( / )

: Speed of light in free space

x y z x y z

uuc t

u E H

c m s

Wave equationNecessary condition

ˆ ˆ ˆ

: Electric field (V/m)ˆ ˆ ˆ

: Magnetic field (A/m)

x y z

x y z

E E E

H H H

E x y z

H x y z

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E

Maxwell’s Equations in a Medium, : Electric displacement

, : Magnetic flux density

00

t

t

DH D

BE B

DB

Assume a non-magnetic medium with no free electric charges or currents

Physical meaning of the electric displacement:

- - - --

+ + + ++P

0

( if the mediumhas a charge density )

D E PD

0 B HBoundary conditions:• Tangential components of E and H are continuous.• Normal components of D and B are continuous.

E H

B

DPower flow per unit area:2Re{ } Re{ } (W/m ): Poynting vector S E H

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Linear, Nondispersive, Homogeneous, and Isotropic Media

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2 2

In free space

1 0uuc t

0

: Electric susceptibility

P E0

0

(1 )/ : Dielectric constant

D E

Maxwell’s equations:

0

00

t

t

EH

HE

EH

00

t

t

DH

BE

DB

Identical to Maxwell’s equations in free space with replaced by 0.

0 0

1c

Wave equation:2

22 2

In a medium

1 0v

uut

Speed of light: 0

0

1v

/ 1 : Refractive index

cn

n

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Monochromatic Electromagnetic Waves

( )

( )

i t

i t

e

e

E E r

H H r

Let’s relate harmonic waves to electromagnetic waves

( , ) ( )exp( )t i t r r

Maxwell’s equations:

0

00

t

t

EH

HE

EH

0

( ) ( )( ) ( )

( ) 0( ) 0

ii

H r E rE r H r

E rH r

2 2( ) ( ) 0k r

Helmholtz equation:2 2

, , , ,

0 0

( ) ( ) 0( ) ( ) or ( ) x y z x y z

k uu E H

k nk

rr r r

(E-M wave represented by complex numbers)

Optical intensity:

1| | Re ( ) ( )*2

I

S E r H r

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Plane Electromagnetic Wave (I)( )i tAe k r

( )0

( )0

( , ) ( )

( , ) ( )

i t i t

i t i t

t e e

t e e

k r

k r

E r E E r

H r H H r

0

( ) ( )( ) ( )

ii

H r E rE r H r

Substituting into Maxwell’s equations:

0 0

0 0 0

k H Ek E H

→ E, H, and k are mutually orthogonal ― Transverse electromagnetic (TEM) wave.

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Plane Electromagnetic Wave (II)Relationship between the amplitude of the electric field and the magnetic field:

0

0

0

00

0

: Impedance of the medium

120 377 : Impedance of free space

EH

n

Optical intensity:2

0*0 0

1 1| | Re ( ) ( )*2 2 2

EI E H

S E r H r

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 = 650 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 = 1.5.

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ExerciseDetermine the power of a 10-mW laser beam,

(a) With = 440 nm, after traveling 1 km of water at various locations.

(b) With = 1550 nm, after traveling 100 km of optical fiber.

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Absorption Bands of Optical Materials

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Applications of Optical Fibers

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“Wires” for light

Optical fiber structure

Optical fiber transmission system

Inter-continental optical fiber network

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Total Internal Reflection (TIR)

1 2

1sinc

nn

c 1 1

c: Critical angle

Example 1: Diamond

Most of the rays entering the top of the diamond will exit from the top due to total internal reflection.

Example 2: Optical Fiber

ncore

125 m

ncladding

ncore > ncladding

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Numerical Aperture of an Optical FiberLet’s do an exercise:

Show that the maximum incident angle m for TIR in the optical fiber is related to the refractive indices by:

2 20 1 2. . sin mN A n n n

N.A.: Numerical Aperture m: Acceptance angle

Calculate NA and m for a silica glass fiber in air with n1 = 1.475 and n2 = 1.460.