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CHAPTER 8----NONLINEAT OPTICS
Chapter 8
NONLINEAR OPTICS
Fundamentals of Photonics
2017/5/24
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CHAPTER 8----NONLINEAT OPTICS
Question:
Is it possible to change the color of a
monochromatic light?
NLO sample
input
output
Answer:
Not without a laser light
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CHAPTER 8----NONLINEAT OPTICS
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.
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CHAPTER 8----NONLINEAT OPTICS
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.
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CHAPTER 8----NONLINEAT OPTICS
Part 0： Comparison
Nonlinear optics:
★The refractive index, and consequently the speed of light in an optical
medium, does change 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.
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CHAPTER 8----NONLINEAT OPTICS
Part 1：phenomena involved
frequency conversion
Second-harmonic generation (SHG)
Parametric amplification
Parametric oscillation
third-harmonic generation
self-phase modulation
self-focusing
four-wave mixing
Stimulated Brillouin Scatteirng
Stimulated Raman Scatteirng
Two-order
Three-order
Optical solitons
Optical bistability
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CHAPTER 8----NONLINEAT OPTICS
19.1 Nonlinear optical media
Origin of Nonlinear
PNp
p  ex, F  eE
the dependence of
the number density N
on the optical field
if Hooke’s law is satisfied
pxE
the number of atoms
occupying the energy
levels involved in the
absorption and
emission
Fundamentals of Photonics
Linear!
if Hooke’s law is not satisfied
pxE
Noninear!
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CHAPTER 8----NONLINEAT OPTICS
P
P
E
E
Figure 19.1-1 The P-E relation for (a) a linear
dielectric medium, and (b) a nonlinear medium.
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CHAPTER 8----NONLINEAT OPTICS
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.
1
1
P  a1 E  a2 E 2  a3 E 3  
2
6
P   0  E  2dE  4  E  
2
1
a2
4
(3)
3
1
a3
24
In centrosymmetric media, d vanish, and the
lowest order nonlinearity is of third order
basic description for a nonlinear optical medium
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CHAPTER 8----NONLINEAT OPTICS
In centrosymmetric media: d=0
the lowest order nonlinearity is of third order
Typical values
d  1024
1021
 (3)  1034 1029
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CHAPTER 8----NONLINEAT OPTICS
The Nonlinear Wave Equation
1 2 E
2 P
 E  2 2  0 2
c0 t
t
2
n2  1  
P   0  E  PNL
c0  1/( 0 0 )1/ 2
PNL  2dE 2  4  (3) E 3  
c  c0 / n
2
1

E
2 E  2 2   J
c0 t
 2 PNL
J   0
t 2
nonlinear wave equation
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CHAPTER 8----NONLINEAT OPTICS
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.
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CHAPTER 8----NONLINEAT OPTICS
19.2 Second-order Nonlinear Optics
P(2) (t )  2 (2) EE*  (  (2) E 2e2it  C.C.)
2


( 2)

PNL  2dE 2
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CHAPTER 8----NONLINEAT OPTICS
A. Second-Harmonic Generation and Rectification
complex amplitude
E (t )  Re{E ( ) exp( jt )}
Substitute it into (9.2-l)
PNL (t )  PNL (0)  Re{PNL (2 ) exp( j 2t )}
PNL (0)  dE ( ) E ( )
*
PNL (2 )  dE ( ) E ( )
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CHAPTER 8----NONLINEAT OPTICS
This process is illustrated graphically in Fig. 9.2-1.
P
PNL(t)
0
E
t
t
dc
+
t
second-harmonic
E(t)
t
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.
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CHAPTER 8----NONLINEAT OPTICS
Second-Harmonic Generation
P(2 )  2dE 2  E 2
1
1
i1t
*  i1t
E  ( E1  E2 )  [ ( A1e  A1 e )  ( A2ei2t  A2*e i2t )]2
2
SHG 2
SFG
1 2 i 21t
 ( A1 e
 A12  A1 A2ei (1 2 )t  A1 A2*ei (1 2 )t
4
DHG
 i (1  2 ) t
2
2  i 21t
*
* *  i (1  2 ) t
 A1  A1 e
 A1 A2e
 A1 A2 e
2
2
 A1 A2 ei (1 2 )t  A1* A2 ei (1 2 )t  A22ei 22t  A22
 A1 A2*ei (1 2 )t  A1* A2*e i (1 2 )t  A22  A22e i 22t )
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CHAPTER 8----NONLINEAT OPTICS
SHG
Component of frequency 2w
complex amplitude
intensity
S (2 )  4 0 2 dE ( ) E ( )
S (2 )   d I   d 
2
4
2
4
2
E ( )
2
2
The interaction region should also be as long as possible.
Guided wave structures that confine light for relatively long
distances offer a clear advantage.
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CHAPTER 8----NONLINEAT OPTICS
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.
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CHAPTER 8----NONLINEAT OPTICS
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.
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CHAPTER 8----NONLINEAT OPTICS
B. The Electra-Optic Effect
E (t )  E (0)  Re{E ( ) exp( jt )}
Substitute it into (9.2-l)
PNL (t )  PNL (0)  Re{PNL ( ) exp( jt )}  Re{PNL (2 ) exp( j 2t )}
PNL (0)  d [2 E (0)  E ( ) ]
2
2
PNL ( )  4dE (0) E ( )
9.2-8
PNL (2 )  dE ( ) E ( )
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CHAPTER 8----NONLINEAT OPTICS
If the optical field is substantially smaller in magnitude than the electric field
E ( )
PNL (2 )
2
E (0)
PNL (0)
2
PNL ( )
Can be negleted
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CHAPTER 8----NONLINEAT OPTICS
PNL (0)  d [2 E (0)  E ( ) ]
2
2
PNL ( )  4dE (0) E ( )
9.2-8
PNL (2 )  dE ( ) E ( )
a linear relation between PNL(w) and E(w)
PNL ( )   0  E ( )
  (4d /  0 ) E (0)
incremental change of the refractive index
n 
Fundamentals of Photonics
2d
E (0) 9.2-9
n 0
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CHAPTER 8----NONLINEAT OPTICS
the nonlinear medium exhibits the linear electro-optic effect
Pockels effect
1 3
n   n rE (0)
2
Pockels coefficient
Comparing this formula with (9.2-9)
r
Fundamentals of Photonics
4
d
4
 0n
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CHAPTER 8----NONLINEAT OPTICS
C. Three-Wave Mixing
Frequency Conversion
E(t) comprising two harmonic components at frequencies w1 and w2
E (t )  Re{E (1 ) exp( j1t )  E (2 ) exp( j2t )}
PNL  2dE 2
PNL (0)  d [ E (1 )  E (2 ) ]
2
2
PNL (21 )  dE (1 ) E(1 )
PNL (22 )  dE (2 ) E (2 )
Frequency
up-conversion
PNL ( )  2dE (1 ) E (2 )
PNL ( )  2dE (1 ) E * (2 )
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Frequency
down-conversion
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CHAPTER 8----NONLINEAT OPTICS
Figure 9.2-5 An example of frequency conversion in a nonlinear crystal
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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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CHAPTER 8----NONLINEAT OPTICS
Phase Matching
E(1 )  A1 exp( jk1  r )
E(2 )  A2 exp( jk2  r )
PNL (3 )  2dE(1 ) E(2 )  2dA1 A2 exp( jk3  r )
where
3  1  2
Frequency-Matching Condition
k 3  k1  k 2
Phase-Matching Condition
Figure 9.2-6 The phase-matching condition
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CHAPTER 8----NONLINEAT OPTICS
★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.
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CHAPTER 8----NONLINEAT OPTICS
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 w 2=w3-w1.
Waves 3 and 2 similarly combine and radiate at w 1.
The three waves therefore undergo mutual coupling in which each pair of
waves interacts and contributes to the third wave.
three-wave mixing
parametric interaction
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CHAPTER 8----NONLINEAT OPTICS
parametric interaction
◆Waves 1 and 2 are mixed in an up-converter, generating a wave
at a higher frequency w 3=w1+w2. A down-converter is realized by
an interaction between waves 3 and 1 to generate wave 2, at the
difference frequency w 2=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.
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CHAPTER 8----NONLINEAT OPTICS
signal w1
(a)
Up-converted
signal
w3=w1+w2
Crystal
w1, w2
Pump w2
Filter
(b)
Pump
signal
w3
w3
w1
Crystal
Amplified
signal
w2
w1
Filter
(c)
Pump
w2
w3
w1
Crystal
w1
Figure 9.2-7 Optical parametric devices: (a) frequency upconverter; (b) parametric amplifier; (c) parametric oscillator.
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CHAPTER 8----NONLINEAT OPTICS
Two-wave mixing can occur only in the degenerate case, w 2=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.
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CHAPTER 8----NONLINEAT OPTICS
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CHAPTER 8----NONLINEAT OPTICS
Wave Mixing as a Photon Interaction Process
conservation of energy and momentum require
3  1  2
k 3  k1  k 2
Figure 9.2-8 Mixing of three photons in a second-order nonlinear
medium: (a) photon combining; (b) photon splitting.
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CHAPTER 8----NONLINEAT OPTICS
d3
d1 d2


dz
dz
dz
Photon-Number
Conservation
d I1
d I2
d I3
( ) ( ) ( )
dz 1
dz 2
dz 3
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Manley-Rowe
Relation
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CHAPTER 8----NONLINEAT OPTICS
19.3 Coupled-wave theory of three-wave mixing
Coupled- Wave Equations
E (t ) 

q 1,2,3
Re[ Eq exp( jq t )] 
1
[ Eq exp( jq t )  Eq* exp(  jqt )]

q 1,2,3 2
Rewrite in the compact form
E (t ) 
1 2 E
 E  2 2  J
c t
2
 q  q E q  Eq*
1
Eq exp( jq t )
q 1, 2, 3 2

 2 PNL
J   0
t 2
PNL (t ) 
J
PNL  2dE 2
1
d  Eq Er exp[ j (q  r )t ]
2 q ,r 1,2,3
1
0 d  (q  r ) 2 Eq Er exp[ j (q  r )t ]
2
q , r 1, 2, 3
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CHAPTER 8----NONLINEAT OPTICS
E (t ) 
1
Eq exp( jq t )

2
q 1, 2, 3
J
1
0 d  (q  r ) 2 Eq Er exp[ j (q  r )t ]
2
q , r 1, 2, 3
2
1

E
2 E  2 2   J
c t
Frequency-Matching
Condition
3  1  2
( 2  k12 ) E1   S1
S1  2012 dE3 E2*
( 2  k22 ) E2   S 2
S 2  2 022 dE3 E1*
( 2  k32 ) E3   S3
S3  2032 dE1 E2*
( 2  k12 ) E1  2 012 dE3 E2*
Three-wave Mixing Coupled Equations
( 2  k22 ) E2  2 022 dE3 E1*
( 2  k32 ) E3  2032 dE1 E2
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CHAPTER 8----NONLINEAT OPTICS
Mixing of Three Collinear Uniform Plane Waves
Eq  Aq exp( jkq z )
kq  q / c aq  Aq /(2 q )1/ 2 ,  0 / n,0  (0 /  0 )1/ 2
Eq  (2 q ) aq exp( jkq z), q  1, 2,3 q 
1/ 2
slowly varying envelope
approximation
(2  kq2 )[aq exp( jkq z )]   j 2kq
Three-wave Mixing Coupled Equations
( 2  k12 ) E1  2 012 dE3 E2*
( 2  k22 ) E2  2 022 dE3 E1*
( 2  k32 ) E3  2032 dE1 E2
Fundamentals of Photonics
daq
dz
q
 aq
2
exp( jkq z )
da1
  jga3 a2* exp( j kz )
dz
da2
  jga3a1* exp( j kz )
dz
da3
  jga1a2 exp( j kz )
dz
g 2  2 123 3 d 2
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Iq
k  k3  k2  k1
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CHAPTER 8----NONLINEAT OPTICS
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
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CHAPTER 8----NONLINEAT OPTICS
Coupled- Wave Equations for Second-Harmonic Generation.
da1
  jga3 a1* exp( jkz )
dz
da3
g
  j a1a1 exp( j kz )
dz
2
where k  k3  2k1 g 2  4
perfect phase matching
k  0
da1
  jga3a1*
dz
da3
g
  j a1a1
dz
2
Fundamentals of Photonics
 3 3d 2
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Coupled Equations
(Second-Harmonic Generation)
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CHAPTER 8----NONLINEAT OPTICS
the solution
ga1 (0) z
a1 ( z )  a1 (0)sec h
2
a3 ( z )  
ga (0) z
j
a1 (0) tan h 1
2
2
Consequently, the photon flux densities
1 ( z )  1 (0) sec h 2
1
2
z
2
3 ( z )   1 (0) tan h 2
z
2
 2  2 g 2 a12 (0)  2 g 21 (0)  8d 2 3  31 (0)  8d 2 3 2 I1 (0)
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CHAPTER 8----NONLINEAT OPTICS
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 1(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)+23(z)= 1(0) is a constant.
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CHAPTER 8----NONLINEAT OPTICS
The efficiency of second-harmonic generation for an interaction
region of length L is
I 3 ( L)
33 ( L) 23 ( L)
L


 tanh 2
I1 (0)
11 ( L) 1 (0)
2
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 1(0)), the argument of the tanh function is small and
therefore the approximation tanhx=x may be used. The efficiency of secondharmonic generation is then
2
2
I 3 ( L)
3 2 d L
 20 3
P
I1 (0)
n A
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CHAPTER 8----NONLINEAT OPTICS
Effect of Phase Mismatch
da1
  jga3 a1* exp( jkz )
dz
da3
g
  j a1a1 exp( j kz )
dz
2
k  0
Solution
a3 ( L)   j
L
g 2
g
a1 (0)  exp( jkz ')dz '  (
)a12 (0)[exp( jkL)  1]
0
2
2k
Efficiency
I 3 ( L) 23 ( L) 1 2 2
kL

 g L 1 (0) sin c 2
I1 (0)
1 (0) 2
2
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CHAPTER 8----NONLINEAT OPTICS
sin c 2 (
kL
)
2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-2/L
-1/L
0
1/L
2/L
k
2
Figure 9.4-2 The factor by which the efficiency of secondharmonic generation is reduced as a result of a phase
mismatch △kL between waves interacting within a distance L.
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CHAPTER 8----NONLINEAT OPTICS
B. Frequency Conversion
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 2 is added to a photon 1 from the input
signal to form a photon 3 of the output signal at an up-converted frequency
w3=w1+w2.
The conversion process is governed by the three coupled equations. For
simplicity, 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.
da1
  jga3 a2* exp( j kz )
dz
da2
  jga3a1* exp( j kz )
dz
da3
  jga1a2 exp( j kz )
dz
Fundamentals of Photonics
a1 ( z )  a1 (0) cosh
da1

  j a2*
dz
2
da2

  j a1*
dz
2
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z
a2 ( z )   ja1 (0) sinh
1 ( z )  1 (0) cos 2
2
z
2
z
2
z
3 ( z )  1 (0) sin 2
2
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CHAPTER 8----NONLINEAT OPTICS
Efficiency
I 3 ( L) 3
z

sin 2
I1 (0) 1
2
2
2
I 3 ( L)
3 2 d L
 20 3 3
P2
I1 (0)
n A
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 w3wave. The pump w2-wave is assumed constant
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CHAPTER 8----NONLINEAT OPTICS
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 coupled
equations 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
da1

  j a2*
dz
2
da2

  j a1*
dz
2
a1 ( z )  a1 (0) cosh
z
a2 ( z )   ja1 (0) sinh
Fundamentals of Photonics
2
z
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2
1 ( z )  1 (0) cosh 2
3 ( z )  1 (0) sinh 2
z
2
z
2
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CHAPTER 8----NONLINEAT OPTICS
Parametric Amplifier Gain Coefficient
d 2 P3 1/ 2
  [8 12 3 ]
n A
3
0
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.
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CHAPTER 8----NONLINEAT OPTICS
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.
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CHAPTER 8----NONLINEAT OPTICS
Frequency Upconversion
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