Nonlinear optics: brief background
Transcript of Nonlinear optics: brief background
IPT 544000-Selected Topics in Ultrafast Optics
Chen-Bin Huang 1/22
Second-Order Ultrafast Nonlinear Optics1
Foreword:
Nonlinear optical effects play a very important role in ultrafast optics. We will first
introduce the formalism of nonlinear optics and then focus on second order nonlinear
processes, including second harmonic generation (SHG), sum frequency generation
(SFG), difference frequency generation (DFG), and optical parametric amplification
(OPA). These processes are important for conversion of short pulses from
mode-locked lasers to new frequency ranges and, particularly for the case of second
harmonic generation, for pulse measurements.
Nonlinear optics: brief background
Maxwell's equations in vacuum are linear. Nonlinear effects arise in materials, since
the material response may be nonlinear in the applied field. The nonlinear material
response can couple back to the optical field, giving rise to nonlinear optics. For a
nonlinear medium, the electric flux density (sometimes referred as the electric
displacement field) is expressed as
PED 0 , (1)
where the polarization density is expressed in the time-domain (in scalar form) as
...)( 3)3(2)2()1(0 EEEP , (2)
where (k) denotes the k-th order susceptibility.
In general, we can have two input electric fields at different frequencies
.].)(~
)(~
[2
121
21 cceEeEE tjtj . (3)
1 Special acknowledgement to Prof. S.-D. Yang for his notes covering a majority of this topic.
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Insert this back to Eq. (2) and only look at the term quadratic with fields,
.}.
*)(~
)(~
),:0(*)(~
)(~
),:0(
*)(~
)(~
),:(2
)(~
)(~
),:(2
)(~
)(~
),:2(
)(~
)(~
),:2({4
2222)2(
1111)2(
)(212121
)2(
)(212121
)2(
222222
)2(
211111
)2(0)2(
21
21
2
1
cc
EEEE
eEE
eEE
eEE
eEEP
tj
tj
tj
tjNL
(4)
we can see that more frequency components are generated through the field
interaction with the nonlinear polarization.
Now in vectorial form, we may write the optical field and polarization as
.].)(~
)(~
[ˆ2
1 3
121
21
i
tji
tjii cceEeEu E (5a)
.].)(~
[ˆ2
1 3
1
)(21
)2( 21
i
tjiiNL ccePu P , (5b)
where
)(~
)(~
),:(2
)( 2
3
1,12121
)2(021
)2(
kkj
jijki EED
P
. (6)
and D is the degeneracy factor (D=2 for sum-frequency generation, while D=1 for
second harmonic generation).
Comments:
1. We have assumed that the process is off-resonance, so that the nonlinear
susceptibility can be taken as real.
2. We have assumed the medium is dispersionless, so the nonlinear
susceptibility is frequency independent.
3. Nonlinear polarization is having a fast response, and the nonlinear processes
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are treated as instantaneous.
4. When crystal symmetries are considered, many elements within the dielectric
tensor reduce to zero. For crystal showing inversion symmetry, second order
susceptibility is essentially zero.
5. It is common to use the engineering notation: )(2
2
1ijkijkd . For common
materials, dijk ranges from {10-13 to 10-11} m/V.
Forced wave equation
We now express the electric flux density of a nonlinear medium as
NLPED )(1 , (7)
where the EPEE 02
)1(0)1( n and NLP represent the linear and nonlinear
polarization densities, respectively.
Assumptions made for our medium:
1. Isotropic in its linear properties, so )1( is a scalar.
2. Homogeneous, so that the linear and nonlinear susceptibilities are not space
dependent.
3. Source free and nonmagnetic.
From our assumptions above, the wave propagation equation reduces to
2
2
02
2
)1(02
ttNL
PE
E . (8)
Now for a linearly polarized plane wave and the excited nonlinear polarization
propagating in the z-direction ( ),(ˆ tzeeE , ),(ˆ tzpp NLNL P with their own unit
vectors), Eq. (8) is simplified to:
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)ˆˆ(),(),(),(2
2
02
2
)1(02
2
petzpt
tzet
tzez NL
. (9)
The Fourier transform of Eq. (9) with respect to time is then (note here the
Fourier-transform relation of temporal derivatives)
)ˆˆ(),(),(),( 02
)1(02
2
2
pezPzEzEz NL , (10)
where )},({),( tzeFzE and )},({),( tzpFzP NLNL .
Comments:
1. The above derivation assumes the nonlinear polarization is weak, so we are
using a perturbation approach.
2. )1( is a constant in Eq. (9), but can be frequency-dependent in Eq. (10). We
see that the frequency domain formalism is good when dispersion is
considered.
3. The nonlinear polarization take a generalized form, we have not yet made
any assumption of its dependence on the optical field.
4. In Eq. (10), we can see that the nonlinear polarization can only drive the field
of the same frequency.
Frequency-domain formulation
We express the optical field and the nonlinear polarization as:
0
])([ ..),(2
ˆ2
1),( ccez
detze zktj
(11a)
0..),(
2ˆ
2
1),( ccez
dptzp tj
NLNL
, (11b)
where cnk )()()( )1(0 is the dispersive wave number.
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Inserting Eq. (11) and the dispersive wave number into Eq. (10), assuming pe ˆˆ ,
zjkNL ezz
zjkz
z)(
02
2
2
),(),()(2),(
. (12)
If the change in the electric field envelope due to the effect of nonlinear propagation is
small on the order of a few optical wavelengths, then we may invoke the slowly
varying envelope approximation (SVEA):
22
2
2 kz
kz
, (13)
which leads us to the result of the forced wave equation in the frequency domain
zjkNL ez
n
cjz
z)(0 ),(
)(2),(
. (14)
The nonlinear polarization shows up as a source term which modifies or excites the
complex spectral amplitude of the electric field.
(What is the implication of SVEA?)
Comments:
1. The dispersion of electric field is fully taken into account by the term
zjke )( ; while the dispersion for the nonlinear polarization is embedded
within ),( zNL .
2. Explicit separation of the zjke )( term helps to eliminate the )1(02
term in Eq. (10).
3. We have not yet made any assumption on the pulse duration, so Eq. (14) is
valid even for very short pulses (very broad spectrum).
4. ),( z and ),( zNL are spectral envelopes centered at 0 .
Time-domain formulation for the FWE
In the time domain formulation, we express the field and the nonlinear polarization in
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terms of slowly varying envelope functions multiplied by a carrier as:
eetzatze zktj ˆ}),(Re{),( )( 00 , (15a)
petzptzP tjNLNL ˆ}),(~Re{),( 0
. (15b)
The E-field base-band spectral envelop is related to the time-domain envelope
through
dtetzatzaFzA tj ~),()},({)~,( , (16)
where 0~ . Substituting Eq. (16) into Eq. (9) and assuming pe ˆˆ , we get
zjkNLNL
NL
tj
et
p
t
pjp
ez
A
z
AjkAkk
d
0
2
2
0200
2
2
020
2
2
22
~~
~
))((~ ~
. (17)
To simplify Eq. (17), we introduce the following assumptions:
1. We invoke SVEA in both space and time:
Akz
Ak
z
A 2002
2
2
and at
a
t
a 2002
2
2
(18)
2. We assume no backward traveling waves, so that 00 2)( kkk .
3. No group velocity dispersion: so that gv
kk
kk
~~)( 00
.
After the inverse Fourier transform to the left-hand side of Eq. (17) and all the above
assumptions, we have the resulting forced wave equation in the time domain:
zjkNL
g
etzpn
cj
t
tza
vz
tza0),(~
2
),(1),(
0
00
. (19)
Comments:
1. Without nonlinear polarization, the pulse propagates with a group velocity
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without distortion.
2. In the derivation of Eq. (19), higher order dispersions are neglected, therefore
this equation might not be valid for very short pulses.
Continuous-wave second harmonic generation
Basic formulations
Let e(z,t) consist of two CW waves at frequencies of 0 and 20:
..)()(2
1),( )2(
2)( 200 ccezEezEtze zktjzktj
, (20)
where E(z)and E(z)are complex amplitudes, and k= k(0), k=k(20) are the
propagation constants. The 2nd-order nonlinear polarization pNL(z,t) induced by e(z,t)
and crystal nonlinearity is:
),(2),( 20 tzedtzp effNL , (21)
where the effective nonlinear coefficient deff is determined by the field polarization
and material properties. Substituting Eq. (20) into Eq. (21), we derive two nonlinear
polarization temporal envelopes driving CW waves at 0 and 20, respectively:
zkkj
effNL ezEzEdtzp 2)()(2),(~
20, , (22a)
zkj
effNL ezEdtzp 22
02, )(),(~ . (22b)
Substituting these two equations into Eq. (19) gives rise to a system of coupled
equations:
zkjeff eEE
cn
dj
z
E )(2
0
(23a)
zkjeff eE
cn
dj
z
E )(2
2
02
, (23b)
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where
kkk 22 , (23c)
is the wave vector mismatch between the fundamental-harmonic (FH) and
second-harmonic (SH) waves, and n=n(0), n=n(20).
When the FH pump is non-depleted (i.e. E(z)E(0)=E), Eqs. (23a-b) are
decoupled:
zkjeEjzE
z)(2
2 )(
, and cn
deff
2
0 , (24)
direct integration of Eq. (24) from z=0 to z=L results in:
222 2
sinc)(kL
j
eLk
LEjLE . (25)
(1) When k=0:
Perfect phase matching, we have LEjLE 22 )( , SH intensity I2(L)=
2220 )(
2
1LEcn = 222
IL , and the CW SHG efficiency is formulated as:
ILI
ISHG
222 , 22
30
2202
2
nnc
deff . (26)
Since Eq. (26) is a monotonically increasing function of L, energy is always
converted from FH wave into SH wave. We also note that under perfect phase
match condition, nn 2 .
(2) k0:
Phase mismatch exists, Eq. (26) will be degraded by a sinc2 factor: SHG
2
sinc22 LkL
2sin 2 Lk
. SHG efficiency becomes a periodic function
of L, the direction of energy conversion will be reversed for every coherence
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length lc /k.
Phase matching using birefringence
Here we consider phase matching using optical birefringent crystals. Due to different
crystal lattices, some crystals have indices of refraction that depend on the e-field
polarization. For a uniaxial crystal, there exists one direction (called the optical axis),
where a plane wave only sees the same index of refraction no matter how it is
polarized. Light propagating other than on the optical axis partially sees a constant
index (ordinary ray, O-ray) and partially see an angle-dependent index (extraordinary
ray, or E-ray).
Mathematically, birefringence can be incorporated into the linear polarization as:
2
2
2
0
00
00
00
e
o
o
ij
n
n
n
, (27)
where on is the index seen by the ordinary ray, while en is the index seem by the
extraordinary ray.
For the O-ray, the propagation constant is c
nk o .
For the E-ray, the propagation constant is c
nk e )( . The angular dependent
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indices generate an index ellipsoid expressed as:
2
2
2
2
2
sincos
)(
1
eoe nnn
. (28)
Now we focus on phase matching in the case where oe nn (positive uniaxial
crystals), using the following figure:
Type I matching ( oee ):
For perfect phase matching, we require that ),()2( 00 eo nn . This requires
phase matching by angle tuning of the crystal, from Eq. (28), we have:
]
)(
1
)(
1[
])(
1
)2(
1[
sin
02
02
02
02
2
oe
oop
nn
nn
. (29)
Type II matching ( oeo ):
The phase matching condition in this case is simply:
),()()2(2 000 eoo nnn . (30)
Due to birefringence, the Poynting vector of e-ray is typically non-parallel to the
wave-vector. We will discuss briefly in class over different focusing and walk-off
effects in the SHG power.
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Ultrashort pulse second harmonic generation
Basic formulation
We denote the input field and the second harmonic fields as:
..),(2
1
2
1),(
0
])([ ccdezEtze zktj
, (31a)
..),(2
1
2
1),(
0
])([22
2 ccdezEtze zktj
. (31b)
The Fourier transform of Eq. (21) is:
),(),(2),( 0 zEzEdzP effNL . (32)
If the FH pump is non-depleted, )(),0(),( EEzE , we only need to
analyze the evolution of SH signal driven by nonlinear polarization centered at 20.
We can evaluate the positive frequency components first by inserting Eq. (31a) into
Eq. (32):
0
)()(02, )()(
2),( zkkj
effNL eEEd
dzP
. (33)
Substituting it into Eq. (14), we get the forced wave equation for SH signal:
0
),(2 )()(
2),( zkjeEE
djzE
z
, (34)
where is defined in Eq. (24). The wave vector mismatch is defined as:
)()()(),( 2 kkkk , (35)
which is a function of both (SH band) and ’(FH band) in general.
Comment:
1. Eqs. (34-35) mean that every SH frequency component is driven by
infinitely many combinations of FH field components at frequencies ' and
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', i.e. viewed as a result of the sum-frequency generations by the entire
FH spectrum instead of by a single frequency component at /2.
2. The nonlinear polarization driving the SH fields takes the form of the
auto-convolution of the FH fields. Therefore, if for transform-limited FH
pulses, we can expect the SH spectral envelope to be slightly broader than
that of the FH field.
General solution of ultrafast SHG
By integrating Eq. (34) (assuming no SH at the input) from – 2L to 2
L , we obtain
the general output SH spectrum:
0 2 2
)',(sinc)()(
2),(
LkLEE
djLE
(36)
If the GVD effect is negligible, k(,’) can be reduced to a function of only:
)(2
2)(2)2(),(
0
02
2002
00
kk
kkkkk
, (37)
where
)(
1
)2(
1
00
gg vv. (38)
Phase mismatch depends on both phase velocity mismatch and the inverse group
velocity mismatch (GVM) between FH and SH pulses. This will allow the terms
within the bracket of Eq. (36) being pulled out from the integration, leading to a
transfer function relation:
),(),( 22 LHPjLE NL , (39)
where
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)()( EEPNL (40)
and
2]2[sinc),( 02 sTLLH , (41)
represent the nonlinear polarization and PM spectra for SH signal, respectively. The
GVM walk-off Ts within the nonlinear crystal of length L, and the frequency detuning
of H2(L,) due to carrier wave vector mismatch (k 0) are defined as:
LTs , k . (42)
Comments:
1. Not all of the SH frequencies can be driven by the nonlinear polarization
even under phase-matching condition. This is true since phase matching does
not guarantee in GVM. Thus in the real world, the SH spectrum can by
substantially narrower than the FH spectrum.
2. We have neglected higher-order dispersion terms, which mean the FH pulse
broadening during propagation within the crystal is neglected.
Ultrafast SHG for thin and thick crystals
Since only spectral envelopes are important for short pulses, we focus on the
baseband representations of Eqs. (39-41):
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),(~
)(~
),(~
2 LHPjLA NL , (43)
where )2,(),(~
022 LELA , )()(~
0 EA , ))(~~
()(~ AAPNL ,
]2/)(sinc[),(~ sTLLH .
We will only examine the waveform and efficiency of pulsed SHG by two phase
matched extreme cases:
(1) Thin crystal (long pulses):
With a broad PM bandwidth: LLHLH )0,(~
),(~ , Eq. (43) is approximated by
LPjLA NL )(~
),(~
2 , whose temporal representation is:
)()},(~
{),( 22
12 tLajLAtLa F , (44a)
)()( 2222 tILtI . (44b)
The output SH pulse energy U2= dttLIAeff ),(2 =effeff tA
UL
222
, where Aeff is the
effective beam cross-sectional area, teff dttIdttI )()( 22
is the effective
FH pulse duration. For FH pulses with average power P , repetition trep, the
pulsed SHG efficiency is formulated as:
,22
222
, peakeff
rep
effthinSHG ILP
t
t
A
L
U
U
. (45)
(2) Thick crystal (short pulses):
For narrow PM bandwidth: Eq. (43) is approximated by: ),(~
2 LA
),(~
)0(~ LHPj NL , whose temporal representation is:
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s
NL
T
tPjtLa
)0(
~),(2 (46a)
s
NL
T
tPstLI
2
2
2
)0(~
),( , (46b)
where 222
00 2cnds eff , and the rectangular function in Eq. (46b) is defined as:
(x)=
otherwise ,0 ;
2
1 for ,1 x . The output SH pulse energy
U2= dttLIAeff ),(2 = sNL
eff TP
sA2
2)0(
=
22 )( dtta
sLAeff
. If the FH pulse has
no nonlinear temporal phase, I(t)= 2/)(20 tacn , U2= 2
2
U
A
L
eff
. The
pulsed SHG efficiency becomes:
s
effpeak
effthickSHG T
tIL
A
UL
U
U
,
222
2, . (47)
Comments:
1. By Eq. (44), the SH field is simply the square of its FH input, as long as the
PM bandwidth is much broader than the FH spectrum.
2. Comparing Eq. (45) with Eq. (26), enhancement in ultrashort pulse SHG is
much more efficient than CW source by the ratio of repetition time to the
pulse duration.
3. By Eq. (46), GVM walkoff can seriously stretch the SH field to a
rectangular function of width Ts (regardless of the shape of FH pulse), if
the PM bandwidth is too narrow. In this case, FH pulse only determines the
SH pulse energy.
4. Comparing Eq. (47) with Eq. (45), narrowband SHG efficiency scales
linearly with crystal length L (instead of L2), and is reduced by a factor of
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teff /Ts.
5. When SHG pulse duration is not important, high SHG efficiency can be
achieved using the thick crystal approach.
We will also discuss briefly quasi-phase matching for both CW and pulsed fields
in class.
Quasi Phase Matching (QPM)
We have derived under the assumption of perfectly phase matching, the second
harmonic intensity grows as a quadratic relation to the crystal length. On the other
hand, for non-perfectly phase matched case, the second harmonic intensity oscillates.
Perfect phase matching is difficult to achieve in practice due to fabrication tolerances,
in the class, we will briefly introduce the technique of quasi-phase matching for SHG
yield enhancement. The basic idea is to implement periodic spatial modulation to the
nonlinear coefficients, so that in the sense of turning )(zdd effeff , phase mismatch
may be canceled via the spatial modulation periodicity.
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Three-wave interactions
Basic formulations
Thus far we restricted our discussions to SHG, in which the frequencies of the
interacting waves satisfy 111 2 . The generalized frequency conversion
process in second order nonlinearity involves three interacting three waves satisfying
the relation 321 . In our notation, the highest frequency term is denoted by 3.
To obtain higher conversion efficiency, phase matching or momentum conservation
also needs to be satisfied: 321 kkk .
We now express the total optical field as three individual frequency components
3
1
)( ..),(2
1),(
i
zktji ccetzatze ii . (48)
Using the formula for nonlinear polarization as expressed in Eq. (21), we show the
terms driving at frequencies of 1, 2, and 3 as
.}.
{),(])([*
13])([*
23
])([210
132231
213
cceaaeaa
eaadtzpzkktjzkktj
zkktjeffNL
. (49)
Insert this expression into the time-domain nonlinear wave equation (Eq. (19)), we
obtain basic equations for three-wave parametric interactions:
kzj
g
kzj
g
kzj
g
eaajt
a
vz
a
eaajt
a
vz
a
eaajt
a
vz
a
2133
3,
3
*132
2
2,
2
*231
1
1,
1
1
1
1
, (50a)
with
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cn
d
i
effii 2
and )( 213 kkkk . (50b)
For CW fields, the time derivatives can be omitted, and this leads to the
Manley-Rowe relation, a manifestation of conservation of energy:
z
I
z
I
z
I
3
3
2
2
1
1
111
. (51)
Coming back to pulsed fields, we explicitly show the phase mismatch in the
frequency domain by rewriting iii . By assuming the crystal initially
satisfies 213 we obtain the relation
)(),,( 22
21
1
13
3
3321
kkk
k . (52)
Now with energy conservation and igi
i
vk
,
1
in mind, we may further express the
phase mismatch as
12,1,
32,3,
)11
()11
( gggg vvvv
k . (53)
Comments:
1. We can see that phase mismatch depends on the frequency variation of both
inputs.
2. For perfect phase matching, the lower frequency variation is tracking the
variation of the higher frequency field.
Sum frequency generation (SFG)
For sum-frequency generations, the two lower frequency inputs fields are used to
generate an up-converted signal field ( 321 ). Refer to Eq. (50), here we
assume the inputs are non-depleted, and phase-matched.
First we focus on the case where group-velocity walk-off=0, and the
up-converted signal can be expressed as
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)()(1
2,2
1,13
3
3,
3
ggg v
zta
v
ztaj
t
a
vz
a
. (54)
And the intensity (in the retarded time-frame moving along with group velocity
3,'3 gvztt ) at the output end is
2'32
'31
3213
0
223'
33 )()(2
),( LtItInnnc
dtLI eff
. (55)
This result is very similar to the quasi-CW limit as we discussed for SHG.
Now we take group velocity walk-off 0 , and the signal field is expressed as
)()(),(
2321313
'33 ztaztaj
z
tza
(56a)
)11
(,, jgig
ij vv . (56b)
The general solution takes the form of
)()(),( 23'320 13
'313
'33 ztaztdzajtLa
L . (57)
In order to obtain better insight, let’s assume one of the inputs is a very short
pulse (a flat-top delta function), )()(1 tta , and the solution for the signal is
13
'312
213
'3
13
3'33
)(
2
1sq),(
ta
L
tjtLa . (58)
Comments:
1. The up-converted signal is a windowed, temporally scaled version of )(2 ta .
2. This scheme is useful for pulse measurement, where the cross-correlation of
the two lower frequencies yields up-converted signal that tracks the
group-velocity difference of the two inputs.
Now we discuss the case where group-velocity walk-off between the two inputs
is large ( 212 L , where 2 is the pulse duration of )(2 ta ). In this limit, it is
interesting to find that the output signal takes the form of
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13
'312
213
3'33
)(),(
ta
jtLa . (59)
Comments:
1. The up-converted signal is independent of short input )(1 ta !
2. The up-converted signal is a true scaled version of )(2 ta in the time-domain,
and be stretched, compressed, and even time-reversed.
Cross-correlation measurements of intensity profiles of (a) input signal field and (b,c)
output fields in sum frequency generation experiment using 10-mm LBO crystal. The
duration of the reference pulse is 130 fs (not shown). Inverse group velocities are 5289
fs/mm for the input o-wave, 5435 fs/mm for the input e-wave, and 5472 fs/mm for the
sum frequency wave. (b) a1 is an o-wave; a2 is an e-wave. (c) a1 is an e-wave; a2 is an
o-wave.
Difference frequency generation (DFG)
Similar to how we treated for SFG, but now the signal is changed to a2. We quickly
obtain the expression for our signal intensity when inputs are non-depleted and
phase-matched:
2'23
'21
3213
0
222'
22 )()(2
),( LtItInnnc
dtLI eff
. (60)
We now turn to the case when group velocity walk-off 0 , we have
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)()(*),( 23'230 12
'212
'22 ztaztdzajtLa
L , (61)
where 2,'2 gvztt . Again we assume input a1 is very short and can be
approximated as a delta function:
12
'213
312
'2
12
2'22
)(
2
1sq),(
ta
L
tjtLa . (62)
Again we see the signal is a scaled version of the longer input field.
Electric field profiles of 13.7 μm pulses generated by short pulse difference frequency
generation in GaSe, using ∼15 fs input pulses at 780 nm. (a) Input: a single pulse. (b)
Input: a pulse pair separated by 60 fs. The output field profiles are measured by
electro-optic sampling (see chapter 10). The group velocity mismatches are estimated to
be |η13| ≈ 160 fs/mm; |η12| ≈ 850 fs/mm.
Optical parametric amplification (OPA)
So far we have assumed the input field amplitudes are fixed during the nonlinear
interactions. In this part, we consider the case where the amplitudes of the two lower
frequency fields a1, a2 can vary, while a3 is strong and remains non-depleted. This
leads to the phenomenon called optical parametric amplification.
Since a3 can be viewed as constant, we are left with two coupled equations
between a1 and a2. It is easier to see the effect of OPA using CW fields:
*0*
0
* 2
1
32
31
2
1
a
a
aj
aj
a
a
z
, (63)
where we have assume that 0k and 2 is real. The solutions to Eq. (63) take the
IPT 544000-Selected Topics in Ultrafast Optics
Chen-Bin Huang 22/22
form of ]exp[~11 zaa and ]exp[~
22 zaa . So the linear differential equation set is
reduced to
0** 2
1
32
31
a
a
aj
aj
. (64)
To obtain non-trivial solution, we require that 0det , and this gives
321 a . (65)
We can see that indeed is real, so amplification with exponential growth is
permitted.
We now focus on the case when initially there is only one lower frequency field
at the input ( 0)0(2 a ). With this boundary condition, the solutions yield
)sinh()0(*)(
)cosh()0()(
13
3
1
22
11
zaa
ajza
zaza
, (66)
where we note interestingly the indicated the phase sensitivity requirement between
the pump (a3) and the signal field (a1). We can now express the field intensities as
)(sinh)0()(
)(cosh)0()(
21
1
22
211
zIzI
zIzI
, (67)
and we see that the difference in photon numbers at and remains fixed throughout the
process, consistent with the Manley-Rowe relation.