p19 nonlinear optics
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19. Nonlinear Optics
19. Nonlinear Optics
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0
21
2 31 2 3 0 1 0 2 0 3
2 3
"
"
"
Polarization :
Susceptibil ity :
P E
E E
P P P P E E E
== + + +
= + + + = + + +
Nonlinear optics
Nonlinear optics
+===+=+== 1)1(
0
000cvnEEED
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Second-order Nonlinear optics
Second-order Nonlinear optics
Second-harmonic generation (SHG) and rectif ication
(0)
),2()(
2
22
P
PP =
Electro-optic (EO) effect (Pockells effect)
= )()( 22 EPEE optical
{ } { } { }{ }
DCelectricEnEEPP
EEPEEPEP
EP
,22
22
2
2
2
2
)0()()0()((0),
)()()(2,)()0()(,)0((0)
{ })()0(but,)()0(,
EEEEEopticalDCelectrical
>>+=
Three-wave mixing
22 0 2P E
opticaloptical EEE )()( 21 +=
{ } { }
{ }{ })()()(,)()()(
,)()(2,)()(2
21212
21212
2
2
221
2
12
2
2
EEPEEP
EPEP
EP
+
SHG
Frequency up-converter Parametric amplif ier, parametric oscillator
Index modulation by DC E-field
Frequency doubling
Rectification
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Third-order Nonlinear opticsThird-order Nonlinear optics
Third-harmonic generation (THG)
{ })()3(,)()()( 332
3 EPEEP
Optical Kerr effect
= )()( 33 EPEE optical
33 0 3P E
Self-phase modulation
Frequency tr ipling
)()()()()()(2
3 InEIEEP Index modulation by optical Intensity
)()( 000 nLkInnn =+=+=
{ } { } 00 )()( nxInxInnn >+= Self-focusing, Self-guiding (Spatial soli tons)
{ } { } 00 )()( nxInxInnn >+=
2
DC,
2
DC,3 )0()()0()( electricelectric EnEEP Index modulation by DC E2
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Third-order Nonlinear opticsThird-order Nonlinear optics
Four-wave mixing
33 0 3P E
opticalopticalopticalEEEE )()()( 321 ++=
( ) terms2166,, 333213
3 = EP
Frequency up-converter
Degenerate four-wave mixing
)()()()(: 32143213 EEEPexampleOne ++
)()()()-(: 3*
2143213 EEEPexampleAnother +
4321 ===
If
34321 === If THG
4321 +=+
directionsoppositeintraveling
arethemamongwavestwo
splane wave
Assume
)()()()( *43 EEEP = Optical phase conjugation
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22 0 2P E
1 2
2 20 1 0 2
2 20 1 0 2 0 2
cos
cos cos
1 1cos cos 22 2
o
o o
o o o
E E tP P P
E t E t
E t E E t
== += += + +
Constant (DC) term
Optical rectificationSecond harmonic term
2
: Only for non-centro-symmetry crystals
24-2. Second harmonic generation (SHG)24-2. Second harmonic generation (SHG)
( )2 1cos 1 cos 22
= +
2 22 0 2 0 0 2 0 2 2
1 1( ) cos 2 (0) (2 )
2 2P t E E t P P = + = +
[GaAs. CdTe, InAs, KDP, ADP, LiNbO3, LiTaO3, ]
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SHG does not occur in isotropic, centrosymmetry crystals
22 0 2P E
2
2
,-
.
is orIf isotropic centrosymmetricboth E and E give the same P polarization
that means the molecules are not polarized by the sencond effect
+
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Second harmonic generationSecond harmonic generation
From Fundamentals of Photonics (Bahaa E. A. Saleh)
P2
E
P2(t)
E(t)
2 22 0 2 0 0 2 0 2 2
1 1( ) cos 2 (0) (2 )
2 2P t E E t P P = + = +
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Second harmonic generationSecond harmonic generation
cosoE E t
( )22 0 22
0 2
1 cos 22
1
2
o
o
P E t
E
=+
1 2P P P+
1 0 1 cosoP E t
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Second harmonic generationSecond harmonic generation
From Fundamentals of Photonics (Bahaa E. A. Saleh)
2 / 2)(
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Second harmonic generationSecond harmonic generation
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Phase matching (index matching) in SHGPhase matching (index matching) in SHG
( )
2
2
2
2
22
2 0
k k k
n nc c
n nc
= =
= =
Output intensity after second harmonic generation
2 2sin , 22
L kI c k k k
=
Phase matching : k=0
O-ray surface
(n)
E-ray surface
(n2)
Optic axis
Direction of
Matching
(k=0)
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Frequency mixing by three-wave mixingFrequency mixing by three-wave mixing
frequency up-converter
parametric amplifier
parametric oscil lator
( )1 2 3 +
( )3 1 2 3 2 1
( )3 1 2 3 2 1 + ( )2 idler, or parameter,
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Parametric interactionParametric interaction
{ } { }
( ) ( ) ( ) ( )
1 2
1 1 2 2
1 1 1 2 2 2
22 0 2
2 2
( ) ( )
cos cos
1 1
exp( ) exp( ) exp( ) exp( )2 2
2 21 1 1 2 2 2 1 3 1 3+ , + , ,
o o
o o
E E E
E t E t
E i t i t E i t i t
P E
= += += + + +
= += ==
=
Frequency conservation Momentum (phase) matching
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Third-order nonlinear effectThird-order nonlinear effect
In media possessing centrosymmetry, the second-order nonlinear
term is absent since the polarization must reverse exactly when the
electric field is reversed.
The dominant nonlinearity is then of third order,
33 0 3P E
The third-order nonlinear material is called a Kerr medium.
P3
E
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Optical Kerr effect
2( )n I n n I+
Self-phase modulation)()()()()()(
2
3 InEIEEP
Index modulation by optical Intensity
)()( 000 nLkInnn =+=+=
{ } { } 00 )()( nxInxInnn >+= Self-focusing, Self-guiding (Spatial solitons)
{ } { } 00 )()( nxInxInnn
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2( )n I n n I+Self-phase modulation
The phase shift incurred by an optical beam of power P and
cross-sectional area A, traveling a distance L in the medium,
Self-focusing (Optical Kerr lens)
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In a linear medium,
wave is spreading.In an optical Kerr medium,
wave can be self-guided.= Self-guided beam
Spatial Solitons
: Nonlinear Schrodinger equation
(Also, see 19.8)
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Raman Gain
: Raman Gain
Coefficient
Coupling of light to the high-frequency vibrational modes of the medium,
which act as an energy source providing the gain.
For low-loss media, the Raman gain may exceed the loss at reasonable levels
of power, so that the medium can act as an opt ical amplifier.
Fiber Raman lasers
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Four-wave mixing (third-order nonlinearity)Four-wave mixing (third-order nonlinearity)
Superposition of three waves of angular frequencies 1, 2, and 3
33 0 3P E (as sum of 63 = 216 terms)
{ }{ }
4 1 2 3
3 4 1 2
3 4 1 2
G G G G
If
k k k k
= + + = +
+ = +
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Optical phase conjugationOptical phase conjugation
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Phase conjugate mirror (PCM)
Conventional mirror
PCM
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Optical phase conjugationOptical phase conjugation
When all four waves are of the same frequency degenerated four-wave mixing (DFWM)
Assuming further that two waves (3,4) are uniform plane waves traveling in opposite
directions,
Nonlinear
Medium
(DFWM)
A1Probe beam
A3Pump beam
A4Pump beam
A2PC beam
*2 3 4 1A A A A : conjugated version of wave 1
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Note: Phase Conjugation and Time Reversal
( ) ( ) ( )kztietE = rr Re,1
( ) ( ) ( )kztietE += rr Re,2
( ) ( ) ( )2 , Rei t kz
E t e
= r r
Phase conjugation
Time reversal
Incident
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Degenerate Four-Wave Mixing as a Form of Real-Time Holography
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Image restoration by phase conjugationImage restoration by phase conjugation
Optical reciprocity.
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19.4. Coupled-wave theory of three-wave mixing19.4. Coupled-wave theory of three-wave mixing
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by equating terms on both sides at each of the frequencies w1, w2, and w3, separately,
Coupled-wave Equationsin three-wave mixing
Homework :
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Mixing of Three Collinear Uniform Plane Waves
Homework :
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Second-harmonic generation (SHG)
Z = 0
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Second-harmonic generation (SHG)
photons of wave 1 are converted to half as many photons of wave 3. photon numbers are conserved.
Photon flux densities
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Second-harmonic generation (SHG)
Efficiency of second-harmonic generation
To maximize the efficiency, we must confine the wave to the smallest possible area A
and the largest possible interaction length L.
This is best accomplished with waveguides (planar or channel waveguides or fibers).
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Second-harmonic generation (SHG)
Effect of Phase Mismatch
For weak-coupl ing case
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19.6. Anisotropic nonlinear media19.6. Anisotropic nonlinear media
Three-Wave Mixing in Anisotropic Second-Order Nonlinear Media
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Phase Matching in Three-Wave Mixing
Thus the fundamental wave is an extraordinary waveand the second-harmonic wave is an ordinary wave.
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