Chapter 8 NONLINEAR OPTICS

CHAPTER 8----NONLINEAT OPTICS

23/4/19Fundamentals of Photonics 1

Chapter 8Chapter 8

NONLINEAR OPTICSNONLINEAR OPTICS



Question:

Is it possible to change the color of a monochromatic light?

output

NL

O s

am

ple

input

Answer: Not without a laser light



Nicolaas Bloembergen (born 1920) has carried out pioneering studies in nonlinear

optics since the early 1960s. He shared the 1981 Nobel Prize with Arthur Schawlow.



Part 0 ： Comparison

Linear optics:

★Optical properties, such as the refractive index and the absorption coefficient independent of light intensity.

★ The principle of superposition, a fundamental tenet of classical, holds.

★ The frequency of light cannot be altered by its passage through the medium.

★ Light cannot interact with light; two beams of light in the same region of a linear optical medium can have no effect on each other. Thus light cannot control light.



Part 0 ： Comparison

Nonlinear optics:

★The refractive index, and consequently the speed of light in an optical medium, does changechange with the light intensity.

★ The principle of superposition is violated.

★ Light can alter its frequency as it passes through a nonlinear optical material (e.g., from red to blue!).

★ Light can control light; photons do interact

Light interacts with light via the medium. The presence of an optical field modifies the properties of the medium which, in turn, modify another optical field or even the original field itself.



Part 1 ： phenomena involved

frequency conversionSecond-harmonic generation (SHG)Parametric amplificationParametric oscillation

third-harmonic generationself-phase modulationself-focusingfour-wave mixingStimulated Brillouin ScatteirngStimulated Raman Scatteirng

Optical solitonsOptical bistability

Two-orderTwo-order

Three-orderThree-order



P N p��

19.1 Nonlinear optical media

Origin of Nonlinear

,p ex F eE ��

p x E ��

if Hooke’s law is satisfied

Linear!

if Hooke’s law is not satisfied

p x E ��

Noninear!

the dependence of the number density N on the optical field

the number of atoms occupying the energy levels involved in theabsorption and emission



Figure 19.1-1 The P-E relation for (a) a linear dielectric medium, and (b) a nonlinear medium.

P P

EE



The nonlinearity is usually weak.

The relation between P and E is approximately linear for small E, deviating only slightly from linearity as E increases.

2 31 2 3

1 1

2 6P a E a E a E

2 (3) 30 2 4P E dE E

basic description for a nonlinear optical medium

2

1

4a

3

1

24a

In centrosymmetric media, d vanish, and the lowest order nonlinearity is of third order



In centrosymmetric media: d=0

the lowest order nonlinearity is of third order

Typical values

24 2110 10d (3) 34 2910 10



The Nonlinear Wave Equation

2 22

02 2 20

1 E PE

c t t

0 NLP E P 2 (3) 32 4NLP dE E

22

2 20

1 EE J

c t

2

0 2NLPJt

2

1/ 20 0 0

0

1

1/( )

/

n

c

c c n

nonlinear wave equation



There are two approximate approaches to solving the nonlinear wave equation:

★The first is an iterative approach known as the Born approximation.

★ The second approach is a coupled-wave theory in which the nonlinear wave equation is used to derive linear coupled partial differential equations that govern the interacting waves.

This is the basis of the more advanced study of wave interactions in nonlinear media.



19.2 Second-order Nonlinear Optics

22NLP dE

2)2(

(2) (2) * (2) 2 2( ) 2 ( C.C.)i tP t EE E e



A. Second-Harmonic Generation and Rectification

( ) { ( )exp( )}E t Re E j t

( ) (0) { (2 )exp( 2 )}NL NL NLP t P Re P j t

complex amplitude

*(0) ( ) ( )NLP dE E

(2 ) ( ) ( )NLP dE E

Substitute it into (9.2-l)



This process is illustrated graphically in Fig. 9.2-1.

Figure 9.2-1 A sinusoidal electric field of angular frequency w in a second-order nonlinear optical medium creates a component at 2w (second-harmonic) and a steady (dc) component.

P

E0

E(t)

t

t

t+

t

PNL(t)

dc second-harmonic



Second-Harmonic Generation

2 2(2 ) 2P dE E

1 1 2 2

1 1 2 1 2

1 1 2 1 2

1 2 1 2 2

2 2 * * 21 2 1 1 2 2

2 ( ) ( )2 2 *1 1 1 2 1 2

2 ( ) ( )2 2 * * *1 1 1 2 1 2

( ) ( ) 2* 21 2 1 2 2

1 1( ) [ ( ) ( )]

2 21(

4

i t i t i t i t

i t i t i t

i t i t i t

i t i t i t

E E E Ae A e A e A e

A e A A A e A A e

A A e A A e A A e

A A e A A e A e A

1 2 1 2 2

22

( ) ( ) 2* * * 2 21 2 1 2 2 2 )i t i t i tA A e A A e A A e

SHG SFG

DHG



Component of frequency 2w SHG

complex amplitude 20(2 ) 4 ( ) ( )S dE E

intensity

22 4 2 4 2 ( )

(2 )2

ES d I d

The interaction region should also be as long as possible.

Guided wave structures that confine light for relatively long distances offer a clear advantage.



Figure 9.2-2 Optical second-harmonic generation in (a) a bulk crystal; (b) a glass fiber; (c) within the cavity of a semiconductor laser.



Optical Rectification

The component PNL(0) corresponds to a steady (non-time-varying) polarization density that creates a dc potential difference across the plates of a capacitor within which the nonlinear material is placed.

An optical pulse of several MW peak power, may generate a voltage of several hundred uV.



B. The Electra-Optic Effect

( ) (0) { ( ) exp( )}E t E Re E j t

( ) (0) { ( )exp( )} { (2 )exp( 2 )}NL NL NL NLP t P Re P j t Re P j t

22(0) [2 (0) ( ) ]NLP d E E

( ) 4 (0) ( )NLP dE E

(2 ) ( ) ( )NLP dE E

Substitute it into (9.2-l)

9.2-8



If the optical field is substantially smaller in magnitude than the electric field

2 2( ) (0)E E

(2 )NLP ( )NLP (0)NLP

Can be negleted



22(0) [2 (0) ( ) ]NLP d E E

( ) 4 (0) ( )NLP dE E

(2 ) ( ) ( )NLP dE E

9.2-8

a linear relation between PNL(w) and E(w)

0( ) ( )NLP E 0(4 / ) (0)d E

incremental change of the refractive index

0

2(0)

dn E

n 9.2-9



the nonlinear medium exhibits the linear electro-optic effect

Pockels effect

31(0)

2n n rE

Pockels coefficient

Comparing this formula with (9.2-9)

40

4r d

n



C. Three-Wave Mixing

Frequency Conversion

E(t) comprising two harmonic components at frequencies w1 and w2

22NLP dE

2 2

1 2(0) [ ( ) ( ) ]NLP d E E

1 1 1(2 ) ( ) ( )NLP dE E

2 2 2(2 ) ( ) ( )NLP dE E

1 2( ) 2 ( ) ( )NLP dE E

*1 2( ) 2 ( ) ( )NLP dE E

1 1 2 2( ) Re{ ( )exp( ) ( ) exp( )}E t E j t E j t

Frequency up-conversion

Frequency down-conversion



Figure 9.2-5 An example of frequency conversion in a nonlinear crystal

Although the incident pair of waves at frequencies w1 and w2 produce polarization densities at frequencies 0, 2wl, 2w2, wl+w2, and w1-w2, all of these waves are not necessarily generated, since certain additional conditions (phase matching) must be satisfied, as explained presently.

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Phase Matching

1 1 1( ) exp( )E A jk r ��

2 2 2( ) exp( )E A jk r ��

3 1 2

3 1 2k k k

3 1 2 1 2 3( ) 2 ( ) ( ) 2 exp( )NLP dE E dA A jk r ��

where

Frequency-Matching ConditionFrequency-Matching Condition

Phase-Matching ConditionPhase-Matching Condition

Figure 9.2-6 The phase-matching condition



★same direction: nw3/c0=nw1/c0+ nw2/c0, w3=w1+w2

frequency matching ensures phase matching.

★different refractive indices, nl, n2, and n3: n3w3/c0=n1w1/c0+n2w2/c0

n3w3=n1w1+n2w2

The phase-matching condition is then independent of the frequency-matching condition w3=w1+w2; both conditions must be simultaneously satisfied.

Precise control of the refractive indices at the three frequencies is often achieved by appropriate selection of the polarization and in some cases by control of the temperature.



Three- Wave Mixing

We assume that only the component at the sum frequency w3=w1+w2 satisfies the phase-matching condition. Other frequencies cannot be sustained by the medium since they are assumed not to satisfy the phase-matching condition.

Once wave 3 is generated, it interacts with wave 1 and generates a wave at the difference frequency w2=w3-w1. Waves 3 and 2 similarly combine and radiate at w1. The three waves therefore undergo mutual coupling in which each pair of waves interacts and contributes to the third wave.

three-wave mixingparametric interaction



parametric interaction

◆Waves 1 and 2 are mixed in an up-converter, generating a wave

at a higher frequency w3=w1+w2. A down-converter is realized by an interaction between waves 3 and 1 to generate wave 2, at the difference frequency w2=w3-w1.

◆ Waves 1, 2, and 3 interact so that wave 1 grows. The device

operates as an amplifier and is known as a parametric amplifier. Wave 3, called the pump, provides the required energy,

whereas wave 2 is an auxiliary wave known as the idler wave. The

amplified wave is called the signal.

◆ With proper feedback, the parametric amplifier can operate as a

parametric oscillator, in which only a pump wave is supplied.



Figure 9.2-7 Optical parametric devices: (a) frequency up-converter; (b) parametric amplifier; (c) parametric oscillator.

Crystal

Crystal

Pump

w3 w1w1

w2

w1

w3w3

w1 Amplified signal

w2

Pump signal

Crystalsignal w1

Pump w2

Up-converted signal

w3=w1+w2w1, w2

Filter

Filter

(a)

(b)

(c)



Two-wave mixing can occur only in the degenerate case, w2=2w1, in which the second-harmonic of wave 1 contributes to wave 2;

and the subharmonic w2/2 of wave 2, which is at the frequency difference w2-w1, contributes to wave 1.

Parametric devices are used for coherent light amplification,

for the generation of coherent light at frequencies where no lasers are available (e.g., in the UV band),

and for the detection of weak light at wavelengths for which sensitive detectors do not exist.





Wave Mixing as a Photon Interaction Process

conservation of energy and momentum require

3 1 2

3 1 2k k k

Figure 9.2-8 Mixing of three photons in a second-order nonlinear medium: (a) photon combining; (b) photon splitting.



Photon-Number Conservation

Manley-Rowe Relation

31 2 dd d

dz dz dz

31 2

1 2 3

( ) ( ) ( )II Id d d

dz dz dz



19.3 Coupled-wave theory of three-wave mixing

Coupled- Wave Equations

22

2 2

1 EE J

c t

2

0 2NLPJt

22NLP dE

*

1,2,3 1,2,3

1( ) Re[ exp( )] [ exp( ) exp( )]

2q q q q q qq q

E t E j t E j t E j t

1, 2, 3

1( ) exp( )

2 q qq

E t E j t

, 1, 2, 3

1( ) exp[ ( ) ]

2NL q r q rq r

P t d E E j t

20

, 1, 2, 3

1( ) exp[ ( ) ]

2 q r q r q rq r

J d E E j t

Rewrite in the compact form q q *q qE E



22

2 2

1 EE J

c t

1, 2, 3

1( ) exp( )

2 q qq

E t E j t

20

, 1, 2, 3

1( ) exp[ ( ) ]

2 q r q r q rq r

J d E E j t

2 21 1 1( )k E S

2 22 2 2( )k E S

2 23 3 3( )k E S

3 1 2

2 *1 0 1 3 22S dE E

2 *2 0 2 3 12S dE E

2 *3 0 3 1 22S dE E

2 2 2 *1 1 0 1 3 2( ) 2k E dE E

2 2 2 *2 2 0 2 3 1( ) 2k E dE E

2 2 23 3 0 3 1 2( ) 2k E dE E

Frequency-MatchingCondition

Three-wave Mixing Coupled Equations



Mixing of Three Collinear Uniform Plane Waves

1/ 2(2 ) exp( ), 1, 2,3q q q qE a jk z q 2qq q

q

Ia

2 2( )[ exp( )] 2 exp( )qq q q q q

dak a jk z j k jk z

dz

*13 2 exp( )

dajga a j kz

dz

*23 1 exp( )

dajga a j kz

dz

31 2 exp( )

dajga a j kz

dz

2 3 21 2 32g d 3 2 1k k k k

exp( )q q qE A jk z /q qk c 1/ 2 1/ 20 0 0 0/(2 ) , / , ( / )q q qa A n

slowly varying envelope

approximation

2 2 2 *1 1 0 1 3 2( ) 2k E dE E

2 2 2 *2 2 0 2 3 1( ) 2k E dE E

2 2 23 3 0 3 1 2( ) 2k E dE E

Three-wave Mixing Coupled Equations



A. Second-Harmonic Generation

a degenerate case of three-wave mixing w1=w2=w and w3=2w

Two forms of interaction occur:

☆ Two photons of frequency o combine to form a photon of frequency 2w (second harmonic).

☆ One photon of frequency 2w splits into two photons, each of frequency w.

☆ The interaction of the two waves is described by the Helmholtz with equations sources.

k3=2k1



Coupled- Wave Equations for Second-Harmonic Generation.

*13 1 exp( )

dajga a j kz

dz

31 1 exp( )

2

da gj a a j kz

dz

2 3 3 24g d 3 12k k k where

0k perfect phase matching

*13 1

dajga a

dz

31 12

da gj a a

dz

Coupled Equations(Second-Harmonic Generation)



the solution

11 1

(0)( ) (0)sec

2

ga za z a h

13 1

(0)( ) (0) tan

2 2

ga zja z a h

Consequently, the photon flux densities

21 1( ) (0)sec

2

zz h

23 1

1( ) (0) tan

2 2

zz h

2 2 2 2 2 3 3 2 3 21 1 1 12 (0) 2 (0) 8 (0) 8 (0)g a g d d I



Figure 9.4-1 Second-harmonic generation. (a) A wave of frequency w incident on a nonlinear crystal generates a wave of frequency 2w. (b) Two photons of frequency w combine to make one photon of frequency 2w. (c) As the photon flux density (z) of the fundamental wave decreases, the photon flux density 3(z) of the second-harmonic wave increases. Since photon numbers are conserved, the sum 1(z)+23(z)= 1(0) is a constant.



The efficiency of second-harmonic generation for an interaction region of length L is

23 3 3 3

1 1 1 1

( ) ( ) 2 ( )tanh

(0) ( ) (0) 2

I L L L L

I L

For large L (long cell, large input intensity, or large nonlinear parameter), the efficiency approaches one. This signifies that all the input power (at frequency w) has been transformed into power at frequency 2w; all input photons of frequency w are converted into half as many photons of frequency 2w.

For small L (small device length L, small nonlinear parameter d, or small input photon flux density (0)), the argument of the tanh function is small and therefore the approximation tanhx=x may be used. The efficiency of second-harmonic generation is then

2 23 230 3

1

( )2

(0)

I L d LP

I n A



Effect of Phase Mismatch

*13 1 exp( )

dajga a j kz

dz

31 1 exp( )

2

da gj a a j kz

dz

2 23 1 10( ) (0) exp( ') ' ( ) (0)[exp( ) 1]

2 2

Lg ga L j a j kz dz a j kL

k

0k

Efficiency

Solution

2 2 23 31

1 1

( ) 2 ( ) 1(0)sin

(0) (0) 2 2

I L L kLg L c

I



-2/L -1/L 0 1/L 2/L0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 9.4-2 The factor by which the efficiency of second-harmonic generation is reduced as a result of a phase mismatch kL△ between waves interacting within a distance L.

2sin ( )2

kLc

2

k



B. Frequency Conversion

*13 2 exp( )

dajga a j kz

dz

*23 1 exp( )

dajga a j kz

dz

31 2 exp( )

dajga a j kz

dz

A frequency up-converter converts a wave of frequency w1 into a wave of higher frequency w3 by use of an auxiliary wave at frequency w2, called the “pump.” A photon from the pump is added to a photon from the input signal to form a photon of the output signal at an up-converted frequency w3=w1+w2.

2 1

3

The conversion process is governed by the three coupled equations. Forsimplicity, assume that the three waves are phase matched (△k = 0) and that the pump is sufficiently strong so that its amplitude does not change appreciably within the interaction distance of interest.

*122

daj a

dz

*212

daj a

dz

1 1( ) (0)cosh2

za z a

2 1( ) (0)sinh2

za z ja

21 1( ) (0)cos

2

zz

23 1( ) (0)sin

2

zz



Figure 9.4-3 The frequency up-converter: (a) wave mixing; (b) photon interactions; (c) evolution of the photon flux densities of the input w1-wave and the up-converted w3-wave. The pump w2-wave is assumed constant

23 3

1 1

( )sin

(0) 2

I L z

I

Efficiency2 2

3 230 3 23

1

( )2

(0)

I L d LP

I n A



C. Parametric Amplification and Oscillation

Parametric Amplifiers

The parametric amplifier uses three-wave mixing in a nonlinear crystal to provide optical gain. The process is governed by the same three coupledequations with the waves identified as follows:

★ Wave 1 is the “signal” to be amplified. It is incident on the crystal with a small intensity I(0).

★ Wave 3, called the “pump,” is an intense wave that provides power to the amplifier.

★ Wave 2, called the “idler,” is an auxiliary wave created by the interaction process

*122

daj a

dz

*212

daj a

dz

1 1( ) (0)cosh2

za z a

2 1( ) (0)sinh2

za z ja

21 1( ) (0)cosh

2

zz

23 1( ) (0)sinh

2

zz



Figure 9.4-4 The parametric amplifier: (a) wave mixing; (b) photon mixing; (c) photon flux densities of the signal and the idler; the pump photon flux density is assumed constant.

Parametric Amplifier Gain Coefficient

23 1/ 230 1 2 3

[8 ]Pd

n A



Parametric Oscillators

A parametric oscillator is constructed by providing feedback at both the signal and the idler frequencies of a parametric amplifier. Energy is supplied by the pump.

Figure 9.4-5 The parametric oscillator generates light at frequencies w1 and w2. A pump of frequency w3=w1+w2 serves as the source of energy.


Frequency Upconversion

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Chapter 8 NONLINEAR OPTICS