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23/4/18Fundamentals of Photonics 1
NONLINEAR OPTICS-NONLINEAR OPTICS-IIIIII
23/4/18Fundamentals of Photonics 2
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
23/4/18Fundamentals of Photonics 3
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.
23/4/18Fundamentals of Photonics 4
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.
23/4/18Fundamentals of Photonics 5
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.
23/4/18Fundamentals of Photonics 6
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
Second-Second-orderorder
Third-orderThird-order
23/4/18Fundamentals of Photonics 7
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
23/4/18Fundamentals of Photonics 8
Figure 19.1-1 The P-E relation for (a) a linear dielectric medium, and (b) a nonlinear medium.
P P
EE
23/4/18Fundamentals of Photonics 9
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
23/4/18Fundamentals of Photonics 10
In centrosymmetric media: d=0
the lowest order nonlinearity is of third order
Typical values
24 2110 10d (3) 34 2910 10
23/4/18Fundamentals of Photonics 11
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
23/4/18Fundamentals of Photonics 12
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.
23/4/18Fundamentals of Photonics 13
19.2 Second-order Nonlinear Optics
22NLP dE
2)2(
(2) (2) * (2) 2 2( ) 2 ( C.C.)i tP t EE E e
23/4/18Fundamentals of Photonics 14
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)
23/4/18Fundamentals of Photonics 15
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
23/4/18Fundamentals of Photonics 16
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
23/4/18Fundamentals of Photonics 17
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.
23/4/18Fundamentals of Photonics 18
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.
23/4/18Fundamentals of Photonics 19
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.
23/4/18Fundamentals of Photonics 20
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
23/4/18Fundamentals of Photonics 21
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
23/4/18Fundamentals of Photonics 22
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
23/4/18Fundamentals of Photonics 23
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
23/4/18Fundamentals of Photonics 24
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
23/4/18Fundamentals of Photonics 25
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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23/4/18Fundamentals of Photonics 26
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
23/4/18Fundamentals of Photonics 27
★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.
23/4/18Fundamentals of Photonics 28
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
23/4/18Fundamentals of Photonics 29
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.
23/4/18Fundamentals of Photonics 30
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)
23/4/18Fundamentals of Photonics 31
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.
23/4/18Fundamentals of Photonics 32
23/4/18Fundamentals of Photonics 33
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.
23/4/18Fundamentals of Photonics 34
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
23/4/18Fundamentals of Photonics 35
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
23/4/18Fundamentals of Photonics 36
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
23/4/18Fundamentals of Photonics 37
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
23/4/18Fundamentals of Photonics 38
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
23/4/18Fundamentals of Photonics 39
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)
23/4/18Fundamentals of Photonics 40
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
23/4/18Fundamentals of Photonics 41
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.
23/4/18Fundamentals of Photonics 42
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
23/4/18Fundamentals of Photonics 43
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
23/4/18Fundamentals of Photonics 44
-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
23/4/18Fundamentals of Photonics 45
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
23/4/18Fundamentals of Photonics 46
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
23/4/18Fundamentals of Photonics 47
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
23/4/18Fundamentals of Photonics 48
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
23/4/18Fundamentals of Photonics 49
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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