##### Document Text Contents

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

Page 351

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

Page 352

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

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

Page 351

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

Page 352

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