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TitleResonant Nonlinear Interactions of Light with Matter
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LanguageEnglish
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Page 1

v. S. Butylkin A. E. Kaplan
Yu. G. Khronopulo E.!. Yakubovich

Resonant Nonlinear Interactions
of Light with Matter

Page 2

v. S. Butylkin A. E. Kaplan
Yu. G. Khronopulo E. I. Yakubovich

Resonant
Nonlinear Interactions
of Light with Matter
Translated by O. A. Germogenova

With 70 Figures

Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo

Page 176

moment is the same for all sublevels. If we take (d· e2) = dI2 + K20 Eo into
account (where dI2 and K20EO are, respectively, the projections on e2 of the
intrinsic and the induced dipole moments of transition 1-2) we can obtain from
(6.1.35) an approximate expression for the maximal conversion coefficient

f3max = A2maxl Al (0)
~ (K11 d12d11/kBTO)is + (Kn K20)is E A (0)

(di2)is + (2K20d12dnlkBTO)isEg + (K~O)isEg ° 1
(6.1.36)

In this formula the symbol ( ) is means the averaging that includes the isotropic
distribution function. For small values of the angular momentum it is equivalent
to a summation over the magnetic quantum numbers of the ground and excited
states (see Sect. 4.1).

In the case of polarized molecules, the term in the nominator of (6.1.36)
caused by the orientation of particles in a constant field may exceed the contri-
bution of the induced dipole moment in the doubling coefficient. Thus, when
dn '" d12 '" 10-18 cgse, polarizability K20 '" 10-24 cm3 and TO = 500 K, the ra-
tio d12 d111 K20kBTo ~ 102. To estimate f3max in such molecules it can be as-
sumed that the main contribution to the parametric part of nonlinear sus-
ceptibility only comes from resonant levels 1 and 2, i.e., that K11 '" d121d22 -
dnl/llwI. If the contribution of polarizability K20 in (6.1.36) is neglected, the
following estimate can be obtained:

f3 ~ d11 ld22 - dnl E A (0)
max kBTollw1 ° 1 (6.1.37)

Consider a gas of nonpolar molecules. For this medium the quantity
« d • e2)K) is proportional to the induced dipole moment ofthe resonant tran-
sition, and the maximal conversion coefficient is given by

(6.1.38)

According to this formula, the maximal conversion in such a gas requires the
existence of levels of matter between which the two-photon transition is allowed
and (K~oEg)is '" (di2)is, i.e., the induced dipole moment of this transition being
comparable with its intrinsic moment. It only makes sense to meet this require-
ment when the length Zmax, over which the optimal value f3max is reached, is
not greater than the gaseous cell that produces the resonant doubling (or the
length of the focal region formed during focussing of the pumping beam). Oth-
erwise, as will be shown below (see Fig. 6.4), the induced dipole moment should
be increased until the quantity Zmax becomes equal to the cell length.

165

Page 177

Expression (6.1.36) is based on the assumption that the generation of the
second harmonic leaves the energy of the first harmonic practically unchanged.
In a real situation it is necessary to take into account the two-photon absorption
of the main frequency wave and the inverse parametric conversion of the field
of double frequency in the pumping field. As one can easily see from (6.1.34),
the latter process cannot be neglected if .1'2>0.1. In order to prove this, it is
sufficient to substitute the maximal value A2max = .1' a~max into the first of
equations (6.1.34). In this case it is no longer possible to obtain a closed-form
solution of the system (6.1.34), yet there exists a method for obtaining a simple
graphical solution. Indeed, performing the change of variables al = aln and
a2 = a2n.../2, we arrive at the conclusion that the "velocity" of motion of the
point with radius-vector {! (a In, a2n) in "time" ( is equal to - "VU, where

U = 0.25(a1- 2.1'a~a2 + a~)
= 0.25(a1n - 2h.1'a~na2n + 2a~n) (6.1.39)

Plotting the equipotential curves U = const and specifying the initial conditions
it is possible to construct the phase trajectory of motion in the coordinates
a2, a~. Whence the maximal value of the double frequency a2max and the
conversion length

(max

(6.1.40)

are determined. Curve 1 in Fig.6.3 depicts the trajectory of the motion ar-
rived at this method for .1' = 0.5. When .1'2>0.1, the integral in (6.1.40) can
be computed with a sufficient accuracy if the trajectory is interpolated by a
parabola a2 = CJ - c2a~, where the coefficients CJ and C2 can be found during
the trajectory construction.

Figure 6.4 presents the qualitative dependence of the conversion coefficient
fJ on the length of the gaseous cell.

As has been demonstrated above for .1'2 <0.1 the parametric conversion of
the field of doubled frequency in the pumping field [i.e., the second term in the
rhs of the first of equations (6.1.34)] can be neglected. A similar assumption
can also be made in the more general case of the addition of frequencies, see
(6.1.33). In this approximation the solution of (6.1.33) has the form

a2 = .1'al(0)a3(OKo exp[ - (( + (o)][Ei*(( + (0) - Ei*((o)] (6.1.41)

where

166

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Operator
- dipole moment 7
- relaxation 7
Optical bistability, cavity less 318
Optical path increment 55,56

Parametric self-induced transparency 35
Perturbation theory 65,66
Phase

breakdown of 161
confinement 206,268
locking 152,153,170,173,181,188,193,
196,206,208,209,211,213,216-218,225,
229,231,232,234,236,239,240,242,243,
248-250,252-255,257,259,260,263
locking breakdown 213,249,250,
252-254,257,259
locking region 208,234,243

- plane 162,164,184
- space 174
Photodissociation 266
Plane-wave-theory 309,310,311
Polarizability 14,16,42,46,165,179,

190-192,241,263,294
of excited states 54
linear 45, 296
linear molecular 293
of matter 295
nonlinear 38,270, 271
resonance condition 17
tensor components 14
tensors 24,25,179,190,191,263

Polarization 4,12,21,22,23,24,26,37,67,70,
107,109,152,154,208,211,222,237,266,
268
amplitudes 23
average 6
eigen- 323
of incident light 304,309
intrinsic 95,108
of matter 21,208,211,222,237,266,268
nonlinear 5,36
nonresonant 25,37
resonant 37

Population
difference 67,154
inversion of 152
level 64
motion 67,70
saturation 39,152,155,159,163,251,253

Process
four-photon 206,207,240,248,258,259,
262
multiphoton 71
parametric 152,153,177,178,196,206,
207,240,248,258,259,262
parametric resonant 3,152, 153,178
parametric three-photon resonant 153
Raman 153,192,208,209,230,240
single-photon 35

- three-photon 34,152,153
- two-photon 35,50,70
Pulse, ultrashort light 268
Pumping

biharmonic 206,209,222,223,235,238
- resonant 169,172-174

Rabi
- frequency 64,65,68
- oscillations 63,65,69,108
Radiation

infrared (IR) 178,188-192,203,207,239,
268
stimulated two-photon 261,262,265,266
ultra-violet (UV) 268

Raman light scattering 13
- stimulated 12
Raman parametric conversion 209
Raman scattering 2,41,113,192

stimulated 3,42,48,152,153, 170, 178,
179,186,191,193,194,196,198,201,203,
204,205
three-photon 2

Range
infrared (IR) 153,206,209,222,259
ultraviolet (UV) 163,206,259
ultraviolet vacuum (VUV) 152,163,168,
207

Rapidly varying part 11,12
Refractive index 14,64,285,301
- nonlinearity of 153
Relaxation operator 21
Relaxation time 7,9
Resonance

first order 11,12,15
multi photon 70
second order 12,15
single-photon 2,31,40,42,46,49,53
two-photon 31,40,173
three-photon 31,40

Resonance condition 1,18,23,33,34,154,156
Resonant division of frequency 173
Resonant doubling 165
Resonant parametric interactions (RPI) 3,

34,60,268
four-photon 206,207,208,221,222,240,
254

Sagnac effect 323
Saturating

energy 57
- field 29,251
- value 293
Saturation 73,80,96,101,102,290,294,324

of absorption 64
energy 29
intensity 75
parameter 156
of populations 1,26,27,28,29,30,33,
155,206,211,223,240,251,252,293

341

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Schrodinger representation 7
Selection
- rules 114
- rules, removal of 268
Self-action 66,271,289,290,318,320
- effects 304
- of electromagnetic waves 45,270

of light 51,294,295
- at nonlinear interfaces 301
- resonanat 270
Sel~bending 284,288,290,294,295,314,325
- of beams 284,291
- length 287
- of trajectories 284
Self-defocussing 294,322,325
Self-focussing 208,210,220,270,271,273,

276,280,281,284,288,291,292,293,
294,314,318,319
beam 287
of light 322

- SRS 295,298,299,300
- strong 293
- two-dimensional 307
- weak 284,289,290
Self-induced transparency 2,164

parametric 157,177,181,184,193,194,
204
threshold for parametric 172

Self-parallax 317
Self-refraction
- of light 300
- SRS 300
Self-trapping 318-320
- cross 320
- in sodium vapor 321
Slowly varying part 15
Source, frequency-tunable 207,222,254
SRS 53,70,71,170,193,198,206-213,

215-217,219,220,222,225,232,234-238,
240,242,253-255,257-259,262,268,
295,296
anti-Stokes 58,206-212,215-220
anti-Stokes axial ASRS 210,211,
216-220
anti-Stokes component 225,242
anti-Stokes cone ASRS 210,211,216-220
electron SRS 207,209,240

- Stoke components 222,225
- threshold 193,209,222,235-238,298,299
- threshold lowering of 209,236
Stark
- effect 1,3,42,44,46,52,53,271,295
- field 31,32,35
- shifts 13,15,16,20,26,30,32,38,50,54,

58,71,91,92
- splitting 66

342

Stimulated two-photon radiation (STPR)
261,262

Substance, inverted 262
Susceptibility 36,38,41,42,45,47,48,50,

51,52,53,61,304,309
- imaginary part of 39
- linear 25,301

nonlinear 48,49,58,153,165,170,208
- nonlinear part of 45
- nonlinear resonant 45
- nonresonant 37,43

parametric 112
- proximity of 301,302
- resonant 37
Synchronism 154,157,159,160,162,163,170,

193,194,199,208,217,219,233,251,252,
263,268

Synchronization 181,188,203,204,208-211,
219,224,225,229,230,242,243,246,248,
250,252,255,257,258,263

- nonlinear 211,238

Threshold
- length 282
- power 280,283,294,296,299
Total internal reflection (TIR) 301,302,304,

306,307,308,310-314,316
- nonlinear 308
Transformation
- coefficient 235,236,245,247,252,256,

258
frequency 207,240
length 231,235,251,259
parametric 232,245,246,248
Raman 215,222,238,248,250

Transition
- probability 7,11

resonant 67,68,91,152,156,165,170,
179,201
three-photon 17
two-photon 17,165

Transparency, parametric self-induced 152,
157,162,172,177,181,184,193,194,197,
204,205,206

Tuning, frequency 209,261
Two-level model 4,10
Two-level systems 12,43,47,63,64,66,67,

70,71,271
general 1,25, 27
Two-photon absorption (TPA) 50,70,71,91,

92,232,240,241,243,245,246,248,268
- characteristic length 249
Two-photon laser (TPL) 261,262,268

Visualization 152,207,268

Wave equation 4

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