Fifth-order nonlinear susceptibility: Effect of third-order resonances in a classical theory

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Fifth-order nonlinear susceptibility: Effect of third-order resonances in a classical theory

Valentin Besse, Herv´e Leblond,^*and Georges Boudebs

LUNAM Universit´e, Universit´e d’Angers, Laboratoire de Photonique d’Angers, EA 4464, 2 Boulevard Lavoisier, 49000 Angers, France (Received 26 January 2015; published 10 July 2015)

We compute the fifth-order nonlinear susceptibility in the frame of a classical model based on an anharmonic oscillator, taking into account the local field corrections. A third-harmonic resonance is evidenced, which explains the strong enhancement of some measured values of the corresponding nonlinear index and its sign changes with the wavelength. The ratio between the fifth-order nonlinear index and the fifth-order nonlinear absorption is computed and is in good agreement with experimental data measured in carbon disulfide CS2.

DOI:10.1103/PhysRevA.92.013818 PACS number(s): 42.65.An

I. INTRODUCTION

Many nonlinear effects arise when specific media are exposed to intense laser beams, and in particular nonlinear absorption [two-photon absorption (β) and three-photon absorption (γ)] or nonlinear refraction [third-order (n2) and fifth-order (n4) refractive indices]. Although pure nonlinear absorption can be used to design optical limiting devices [1], and the nonlinear refractive index finds applications in ultrafast all-optical switching [2], a particular combination of these effects leads to the generation of filaments [3] or to the propagation of damped spatial solitons, which have been observed in carbon disulfide CS2 [4]. Note that, although soliton formation is often interpreted as a balance between diffraction and nonlinearity, the self-focusing phenomenon itself would not be possible in the absence of any diffraction effect, and the diffraction mechanism is a fortiori required for the relative stabilization of these structures. Similarly, catastrophic beam collapse at high powers appears when cubic nonlinearities only are taken into account [5–7]. In order to describe these complex behaviors, it is necessary to take into account higher-order nonlinearities. In the present paper, we restrict attention to centrosymmetric media and focus on the quintic nonlinearities.

Quintic nonlinear coefficients have been measured in a variety of media. In 2004, Ganeev et al. determined the fifth-order refractive index of a pseudoisocyanine solution [8]

with femtosecond radiation. Optical nonlinearities of silver nanoparticles were investigated using the Z-scan technique [9].

The fifth-order refractive index was determined atλ=532 nm with picosecond pulses (80 ps) following different filling factors [10]. A medium heavily investigated is CS2, where in particular Ganeevet al.measured the three-photon absorption coefficient [11]. Konget al.measured the fifth-order refractive index and the three-photon absorption at λ=800 nm and in the femtosecond regime (120 fs) [12]. Other studies on CS₂ at different wavelengths and/or in different temporal regimes provided values forn₄ andγ, either for picosecond radiation [13] or in the femtosecond regime [4]. We observed an inversion in the sign of the higher-order nonlinear indexn₄, depending on the wavelength used for the measurement:n₄is positive at 532 and 1064 nm and negative at 800 and 920 nm.

*Corresponding author: [email protected]

Theoretical computation of both linear and nonlinear optical susceptibilities can be performed using damped, driven oscillator models. The classical model is known as the Lorentz model, but quantum-mechanical models of two-level and multilevel atoms are also used; they allow expressions of the nonlinear susceptibilities to be obtained [14–16]. Thus, formulas exist for the second- and third-order susceptibilities.

Analytical formulas for these are, in most cases, complex quantities including interference between various resonant and nonresonant processes. Previously, the relation between the third-order and fifth-order nonlinear susceptibilities and the nonlinear optical indices has been clarified [4]. The authors used an expansion of the effective refractive index in a power series of the optical electric field. These formulas do not consider resonant and nonresonant processes. Here we propose to determine an analytical formula for the real and imaginary parts of χ⁽⁵⁾, taking into account the frequency resonances.

In Sec.II, we present the Lorentz model and the calculation of the first approximation of the susceptibility, which does not take into account the local field corrections. Since, apart from a multiplication by the density of atoms, this first-order approximation is nothing but the hyperpolarizabilities of the atoms within the nonlinear Lorentz model, we use the termi- nology of hyperpolarizability throughout the paper. The local field corrections are considered in Sec. III, concluding with expressions for the nonlinear susceptibilities. The theoretical results are discussed with respect to experimental observations in Sec. IV, which mainly deals with the third harmonic resonance of χ⁽⁵⁾. A brief conclusion in Sec.Vsummarizes the results.

II. THE FIFTH-ORDER HYPERPOLARIZABILITY A. The classical model and the linear approximation The standard macroscopic theory expresses the nonlinear optical characteristics of a medium by means of the nonlinear susceptibilities χ^(p). They are related to the polarization densityPby expanding it in a power series of the the applied optical fieldE[14], as

P=ε₀[χ⁽¹⁾∗ E+χ⁽²⁾∗ EE+χ⁽³⁾∗ EEE+ · · ·]. (1) It is well known that in the general case the products in Eq. (1) are convolutions and that the susceptibilities are tensors. We disregard here the tensorial aspect and solve the problem for a monochromatic incident wave. The third-order

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nonlinear refractive index n₂ is related to the real part of the third-order nonlinear susceptibility Re[χ⁽³⁾], while the fifth-order coefficientn₄ is linked to Re[χ⁽⁵⁾]. The nonlinear absorption coefficients β and γ are closely related to the imaginary parts of the nonlinear susceptibilities Im[χ⁽³⁾] and Im[χ⁽⁵⁾], respectively.

We consider the classical model of the elastically bound electron, i.e., the Lorentz model [17]. We denote by x the position of the electron along some axis parallel to the polarization direction of the optical wave, and consequently perpendicular to the propagation direction. The standard model for x is a harmonic oscillator, driven by the Coulomb force which acts on the electron. In order to compute the fifth-order susceptibility, we take into account the anharmonic terms in the restoring force up to orderx⁵. The equation satisfied byx is thus

d²x dt² +dx

dt +ω²₀x+Cx³+Qx⁵= −e

mEl. (2) We assume a linear polarization, whose direction is defined by the unitary vector ex, and El =Elex is the electric field.

mis the mass of the electron, −eis its electric charge, and ω₀ is the resonance angular frequency of the medium, which corresponds to the observed atomic spectral line. The classical model assumes a single resonance frequency and is equivalent (at least regarding the expression of the linear and third-order susceptibilities) to a two-level quantum model.

ForC =Q=0, Eq. (2) is nothing but the Lorentz atom model, which provides a good description of the linear optical properties of atomic vapors and nonmetallic solids. The expression of the atomic potential pertaining to Eq. (2) is

U=mω²₀x²

2 +mCx⁴

4 +mQx⁶

6 , (3)

which is centrosymmetric, and consequently even-order nonlinear susceptibilities will vanish.

We assume a monochromatic incident fieldEl =E_le⁻^iωt+ c.c., where t is the temporal variable and ω the angular frequency (c.c. stands for the complex conjugate). We also assume that the electronic displacement x remains small to ensure the validity of both the expansion of the potential U in a power series of x and that of the polarization density P in a power series of El. We seek for a solution for the electron position using a perturbation approach, expanding it as x=x⁽¹⁾+x⁽³⁾+x⁽⁵⁾+ · · ·, with x^(p) of order p with respect to the electric field amplitudeEl, which is supposed to be small with respect to the intra-atomic field. The expansion is substituted into Eq. (2), and the resulting set of equations is solved order by order.

The leading termx⁽¹⁾satisfies the equation d²x⁽¹⁾

dt² +dx⁽¹⁾

dt +ω²₀x⁽¹⁾= −e

m[Ele⁻^iωt+c.c.], (4) which is straightforwardly solved by setting x⁽¹⁾= X⁽¹⁾(ω)e⁻^iωt+c.c., and we get

X⁽¹⁾(ω)= −eE_l

mD(ω), (5)

where the denominator

D(ω)=ω²₀−ω²−iω (6) accounts for the resonance at the angular frequencyω=ω₀. The linear atomic dipole moment isp_L= −ex⁽¹⁾and the linear polarizabilitya(ω) is defined by

pL=a(ω)Ele⁻^iωt+c.c. (7) (we use the notationa,b,cfor the linear polarizability and the second- and the third-order nonlinear hyperpolarizabilities, instead of the standard notationα,β,γ, in order to avoid con- fusion with the linear and nonlinear absorption coefficients) so that

X⁽¹⁾(ω)= −a(ω)

e El. (8)

We deduce the well-known expression for the linear polarizability, which we write for convenience as

a(ω)= F

D(ω), (9)

with

F = e²

m. (10)

B. The cubic term

The amplitude of the third-order component is found from the equation

d²x⁽³⁾

dt² +dx⁽³⁾

dt +ω²₀x⁽³⁾= −C(x⁽¹⁾)³. (11) Expansion of the right-hand-side member shows thatx⁽³⁾= X⁽³⁾(3ω)e^−3iωt+X⁽³⁾(ω)e⁻^iωt+c.c., and the equation satisfied by the amplitude of the third harmonic can be expressed as

D(3ω)X⁽³⁾(3ω)= −C[X⁽¹⁾(ω)]³. (12) By substituting (8) into Eq. (12), it is straightforwardly solved to yieldX⁽³⁾(3ω). The third-order term of the nonlinear atomic dipole moment, related to the electron position as p⁽³⁾_{N L}=

−ex⁽³⁾, is expanded using the third-order hyperpolarizability bas

p⁽³⁾_{N L}=b(3ω)E_l³e^−3iωt+3b(ω)E_l²E_l^∗e⁻^iωt+c.c., (13) where we set for brevity

b(3ω)=b(3ω;ω,ω,ω) (14) and

b(ω)=b(ω;ω,ω,−ω), (15) with the same notations as in [14]. The third-order amplitudes of the position of the electron are thus expressed as functions of the hyperpolarizabilities as

X⁽³⁾(3ω)= −b(3ω)

e E_l³ (16) and

X⁽³⁾(ω)= −3b(ω)

e E²_lE_l^∗. (17)

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Using (16) and (8) in Eq. (12) yields the third-harmonic generation term of the hyperpolarizability as

b(3ω)=R[a(ω)]³a(3ω), (18) in which we have set

R= −C

e²F = −Cm

e⁴ . (19)

The term corresponding to the Kerr effect is computed in the same way; the equation is

D(ω)X⁽³⁾(ω)= −3C[X⁽¹⁾(ω)]²X⁽¹⁾^∗(ω), (20) and using (17) we obtain the component of the third-order hyperpolarizability as

b(ω)=R[a(ω)]³a^∗(ω). (21) C. The fifth-order nonlinear hyperpolarizability The equation satisfied by the fifth-order correction to the electron positionx⁽⁵⁾is

d²x⁽⁵⁾

dt² +dx⁽⁵⁾

dt +ω²₀x⁽⁵⁾

= −Q(x⁽¹⁾)⁵−3C

(x⁽¹⁾)²x⁽³⁾

, (22) and consequentlyx⁽⁵⁾can be expressed as

x⁽⁵⁾=X⁽⁵⁾(5ω)e^(−5iωt)+X⁽⁵⁾(3ω)e^(−3iωt)

+X⁽⁵⁾(ω)e⁽⁻^iωt)+c.c. (23) The equations for the amplitudesX⁽⁵⁾(pω) can then be written as

D(5ω)X⁽⁵⁾(5ω)= −Q[X⁽¹⁾(ω)]⁵−3C[X⁽¹⁾(ω)]²X⁽³⁾(3ω) (24) for the fifth-harmonic generation term; we obtain

D(3ω)X⁽⁵⁾(3ω)= −5Q[X⁽¹⁾(ω)]⁴X⁽¹⁾^∗(ω)

−3C{[X⁽¹⁾(ω)]²X⁽³⁾(ω)

+2X⁽¹⁾(ω)X⁽¹⁾^∗(ω)X⁽³⁾(3ω)} (25) for the fifth-order term of the third-harmonic generation, and

D(ω)X⁽⁵⁾(ω)= −10Q[X⁽¹⁾(ω)]³[X^(1)∗(ω)]²

−3C{[X⁽¹⁾(ω)]²X⁽³⁾^∗(ω) +2X⁽¹⁾(ω)X⁽¹⁾^∗(ω)X⁽³⁾(ω)

+[X^(1)∗(ω)]²X⁽³⁾(3ω)}, (26) for the saturation of the Kerr effect and the fifth-order nonlinear absorption.

The fifth-order nonlinear dipole momentp_{N L}⁽⁵⁾ = −ex⁽⁵⁾can be expanded using the fifth-order hyperpolarizabilitycas

p⁽⁵⁾_{N L}=c(5ω)E⁵_le⁻^5iωt+5c(3ω)E_l⁴E^∗_le⁻^3iωt

+10c(ω)E_l³E_l^∗²e⁻^iωt+c.c., (27)

in which the components ofcare more specifically

c(5ω)=c(5ω;ω,ω,ω,ω,ω), (28) c(3ω)=c(3ω;ω,ω,ω,ω,−ω), (29) c(ω)=c(ω;ω,ω,ω,−ω,−ω). (30) Comparison of Eqs. (27) and (23) gives expressions for the terms of the electron position as functions of the hyperpolarizabilities as

X⁽⁵⁾(5ω)= −c(5ω)

e E⁵_l, (31) X⁽⁵⁾(3ω)=−5c(3ω)

e E⁴_lE_l^∗, (32) X⁽⁵⁾(ω)=−10c(5ω)

e E_l³E_l^∗². (33) Using the above formulas we obtain expressions for the involved fifth-order hyperpolarizability terms as

c(5ω)=T[a(ω)]⁵a(5ω)+3R[a(ω)]²b(3ω)a(5ω), (34) c(3ω)=T[a(ω)]⁴a^∗(ω)a(3ω)+³₅R{3[a(ω)]²b(ω)a(3ω)

+2a(ω)a^∗(ω)b(3ω)a(3ω)}, (35) c(ω)=T[a(ω)]⁴[a^∗(ω)]²+₁₀³R{3[a(ω)]³b^∗(ω)

+6[a(ω)]²a^∗(ω)b(ω)+a(ω)[a^∗(ω)]²b(3ω)}, (36) in whichRis given by (19) and where we have set

T = −Q

e⁴F = −Qm

e⁶ . (37)

In the above expressions, the resonance factors 1/D(pω) have been rewritten using the first-order hyperpolarizability a. However, the parameterC, which characterizes the strength of the third-order nonlinearity, remains in the final formula. It is indeed involved via the expression (19) of the factorR. It is a drawback of the formulas, because C is an adjustable parameter in the present theory. It is possible to remove it using the expressions (18) and (21) of the third-order hyperpolarizability. We obtain

c(5ω)=T[a(ω)]⁵a(5ω)+3b(ω)b(3ω)a(5ω)

a(ω)a^∗(ω) , (38) c(3ω)=T[a(ω)]⁴a^∗(ω)a(3ω)

+3 5

3b(3ω)b(ω)

a(ω) +2[b(3ω)]²a^∗(ω) [a(ω)]²

, (39) c(ω)=T[a(ω)]⁴[a^∗(ω)]²

+ 3 10

3b(ω)b^∗(ω)

a^∗(ω) +6[b(ω)]²

a(ω) +b^∗(ω)b(3ω) a^∗(ω)

.

(40) It must be noticed that this expression is not unique. Other expressions for the fifth-order hyperpolarizability exist. These expressions do not involve the adjustable coefficientCeither, but use other components of the third-order hyperpolarizability

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band the linear polarizabilitya. Although all these expressions are fully equivalent within the frame of the model, care must be taken when applying them to a real material. We choose expressions as simple as possible.

III. LOCAL FIELD CORRECTIONS A. A first approach

1. The linear Lorentz theory

The polarization densityP=Pex is related to the atomic dipolar momentpthroughP = −N ex=Np, whereNis the density of atoms. Its linear partPLcan be expressed in terms of the linear susceptibilityχ⁽¹⁾as

PL =ε₀χ⁽¹⁾(ω)Ee⁻^iωt+c.c., (41) where E is the amplitude of the macroscopic electric field.

If we identify the field El which is involved via the expression of p_L with the macroscopic field E, we getχ⁽¹⁾= aN/ε₀. Within the same approximation, it is straightforwardly found thatχ⁽³⁾=bN /ε₀andχ⁽⁵⁾=cN/ε₀. Hence the above expressions for the polarizability and hyperpolarizabilities yield corresponding expressions for the linear and nonlinear suceptibilities.

However, it is known that the field experienced by the atoms is not the macroscopic field but the Lorentz local field (the original reference is [18], a clear demonstration can be found in [19], and application to nonlinear optics in [14]). The local fieldElis the electric field locally experienced by a molecule, which results from the action of any external source and all the molecules of the medium except the one considered. It is related to the macroscopic fieldEthrough

El= E+ P 3ε0

, (42)

P being the polarization density. For the sake of simplicity, we will disregard the vectorial character of the field and the tensorial character of the (hyper)polarizabilities, which are of no consequence for a linearly polarized wave in an isotropic medium. We will also disregard, but only in the first stage, the frequency dependence in the expression for the dipolep using the polarizability a and hyperpolarizabilities b andc.

The frequency dependence will be considered in Sec. III B.

The expression forpbecomes

p=aEl+bEl³+cEl⁵. (43) Both (hyper)polarizabilities and susceptibilities are defined as the coefficients in some power series expansion, and consequently they can be computed order by order. The linear and cubic terms can be found in [14]; however we feel it necessary to recall their derivation before computing the quintic term, for the sake of clarity.

We substitute now the expression (42) of the local field into the expression for the linear polarizationPL=Np_L =N aEl. With respect to the macroscopic fieldE, the linear polarization is expressed asPL=ε₀K⁽¹⁾E. It yields the equation

1−N a

3ε0

K⁽¹⁾E =N a

ε₀ E, (44)

which is easily solved to give K⁽¹⁾= N a/ε₀

1−N a/(3ε0). (45) The relation

1+K⁽¹⁾

3 = 1

1−N a/3ε₀ (46) will be useful in the subsequent computations.

2. Nonlinear local field corrections

We include the cubic term in the expression of the polarization, which becomes

P PL+PN L⁽³⁾ =N

aEl+bEl³ . (47) We expand P in the same way in a power series of the macroscopic fieldEas

P =ε₀

K⁽¹⁾E+K⁽³⁾E³ . (48) The coefficients K⁽¹⁾ andK⁽³⁾ are not exactly the susceptibilities, since the latter are defined in the frequency domain, which involves combinatorial factors when considering the harmonics of a quasimonochromatic field.

The expression (42) forEl is substituted into Eq. (47), as well as (48). The leading order vanishes due to (45); we keep only the terms of orderE³and obtain the equation

1−N a

3ε0

K⁽³⁾E³= N b ε₀

1+K⁽¹⁾ 3

E

3

. (49) Equation (49) is straightforwardly solved to yield

K⁽³⁾= N b/ε₀

(1−N a/3ε₀)⁴. (50) We use now the expansion

P=ε₀

K⁽¹⁾E+K⁽³⁾E³+K⁽⁵⁾E⁵

+O(E⁷) (51) of the polarization P, assuming a nonvanishing coefficient c in (43). We substitute, in the same way as above, the expression (42) for the local field into the expression of P drawn from (43), and then the expansion (51) into the obtained equation. The linear and cubic terms vanish using (45) and (50), terms of order higher than 5 are neglected, and we obtain the equation

1−N a

3ε0

K⁽⁵⁾E⁵ = N b ε₀

1+K⁽¹⁾ 3

E

2

K⁽³⁾E³

+N c ε₀

1+K⁽¹⁾ 3

E

5

. (52) The solution of Eq. (52) is straightforwardly obtained and is reduced using (45) and (50), to yield

K⁽⁵⁾= N c/ε₀

(1−N a/3ε₀)⁶ + N²b²/ε₀²

(1−N a/3ε₀)⁷. (53)

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We get finally the following expression for the polarization density:

P

ε₀ = N a/ε₀

1−N a/(3ε₀)E+ N b/ε₀ [1−N a/(3ε₀)]⁴E³ +

N c/ε₀

[1−N a/(3ε0)]⁶ + N²b²/ε₀² [1−N a/(3ε0)]⁷

E⁵. (54) It is noticed that, in formula (53), the coefficientK⁽⁵⁾contains two terms. The first term results from the effect of the quintic hyperpolarizabilityc, which is natural, but the second term is the result of a cascade with twice the effect of the cubic hyperpolarizabilityb. It is thus a quintic nonlinearity due to the combined effect of the cubic nonlinearity and local field corrections. The word “cascade” is introduced here by analogy with the well-known phenomenon ofχ⁽²⁾−χ⁽²⁾cascading in noncentrosymmetric nonlinear media [20].

B. Local field corrections in the frequency domain 1. Linear and cubic nonlinear susceptibilities

Let us first recall the known results concerning the linear and cubic polarization terms. We consider now a quasimonochromatic wave; hence the field isE=E(ω)e⁻^iωt+c.c.

According to the expression (54) for the polarization, we obtain the expression of the linear polarization amplitude as

P_L(ω)= N a(ω)

A(ω) E(ω), (55) where

A(ω)=

1−N a(ω) 3ε₀

. (56)

Since the linear susceptibilityχ⁽¹⁾(ω) is defined by

PL(ω)=ε₀χ⁽¹⁾(ω)E(ω), (57) we can compute the expression for the linear susceptibility as a function of the polarizabilitya(ω), as

χ⁽¹⁾(ω)= N a(ω)

ε₀A(ω), (58) whereA(ω) is given by (56). We proceed in an analogous way for the cubic component of the polarization.

When the electric fieldEdepends on timet, all products in the expression (43) of the dipolar moment must be replaced with convolution products. In particular, the cubic termbE³ becomes

b∗EEE=

b(t₁,t₂,t₃)E(t−t₁)E(t−t₂)E(t−t₃)dt₁dt₂dt₃. (59) In the case of a field containing only discrete frequencies ω₁,ω₂, . . . ,ωM, the generalization of the derivation given in the above section to the convolution product is easy since Eq. (49) is valid with convolution products. In this case the expression of the cubic term of the polarization density takes the form

P_{N L}⁽³⁾ =

ωj

P_{N L}⁽³⁾

ω_j e⁻^iω^j^t (60)

with

P⁽³⁾(ω0)=

ω₁,ω₂,ω₃ ω₀=ω₁+ω₂+ω₃

N ε₀b(ω₀;ω₁,ω₂,ω₃) ₃

j=0A ωj

3 j=1

E ωj .

(61) For a quasimonochromatic field (M=2,ω₁= −ω₂=ω), we get a polarization term at the fundamental frequencyωas

P⁽³⁾(ω)= 3N ε₀b(ω)

[A(ω)]³A(−ω)[E(ω)]²E^∗(ω), (62) and a third-harmonic term

P⁽³⁾(3ω)= N ε₀b(3ω)

A(3ω)[A(ω)]³[E(ω)]³, (63) in which we used the shortened notations defined by Eqs. (14) and (15).

Since the expression for the cubic polarization in terms of the third-order nonlinear susceptibilityχ⁽³⁾is

P⁽³⁾(ω)=3χ⁽³⁾(ω)[E(ω)]²E^∗(ω) (64) for the fundamental and

P⁽³⁾(3ω)=χ⁽³⁾(3ω)[E(ω)]³ (65) for the third harmonic, where we have set for brevity

χ⁽³⁾(ω)=χ⁽³⁾(ω;ω,ω,−ω) (66) and

χ⁽³⁾(3ω)=χ⁽³⁾(3ω;ω,ω,ω), (67) we obtain the expressions for the cubic nonlinear susceptibility as functions of the cubic hyperpolarizability as

χ⁽³⁾(ω)= N b(ω)

[A(ω)]²|A(ω)|² (68) for the fundamental and

χ⁽³⁾(3ω)= N b(3ω)

A(3ω)[A(ω)]³ (69) for the third harmonic, taking into account the local field effects. These results can be found, e.g., in [14]. We have merely adapted the presentation in such a way that it can be more easily generalized to the fifth order.

2. The fifth-order nonlinear susceptibility

It is seen from expression (54) for the polarization that the quintic term contains two components, one of which is proportional to the quintic hyperpolarizability c and the other involves the cascading of twice the cubic effect and is proportional to the squareb²of the cubic hyperpolarizability.

We denote byPQ⁽⁵⁾the former and byPC⁽⁵⁾the latter.

In the case of a field containing only discrete frequenciesωj, the generalization of the derivation to the convolution product can be done using Eq. (52). It gives, for theQterm, the general

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expression

P_Q⁽⁵⁾(ω₀)=

ω1,ω2,ω3,ω4,ω5

ω0=5 j=1ωj

⎡

⎣N ε₀c(ω0;ω₁,ω₂,ω₃,ω₄,ω₅) ₅

j=0A(ωj)

5 j=1

E(ωj)

⎤

⎦ (70)

[P_Q⁽⁵⁾(ω₀) denotes the amplitude ofPQ⁽⁵⁾at the frequencyω₀, and so on]. TheCterm involvingb²is more complicated, as

P_C⁽⁵⁾(ω0)=

ω₁,ω₂,ω₃ ω0=3

j=1ωj

⎡

⎢⎢

⎢⎣

N²ε₀b(ω₀;ω₁,ω₂,ω₃) ₂

j=0A(ωj)

2 j=1

E(ωj)

ω₄,ω₅,ω₆ ω3=6

j=4ωj

b(ω₃;ω₄,ω₅,ω₆) ₆

j=3A(ωj) 6 j=4

E(ωj)

⎤

⎥⎥

⎥⎦. (71)

If we consider the frequency-matching conditions which appear in Eq. (71), we see thatω₃can be substituted withω₄+ω₅+ ω₆, the matching condition becomingω₀=ω₁+ω₂+ω₄+ω₅+ω₆. Renumbering the dummy variables, the expression (71) can be thus rewritten in the more convenient way

P_C⁽⁵⁾(ω₀)=

ω1,ω2,ω3,ω4,ω5

ω0=5 j=1ωj

⎡

⎣N²ε₀b(ω0;ω₁,ω₂,ω₃+ω₄+ω₅)b(ω3+ω₄+ω₅,ω₃,ω₄,ω₅) A(ω3+ω₄+ω₅)₅

j=0A ωj

5 j=1

E ω_j

⎤

⎦. (72)

For a quasimonochromatic field we get, for the fundamental frequencyω, i.e., the terms which express the saturation of the Kerr effect (whenn₂andn₄have opposite signs),

P⁽⁵⁾(ω)=

10N ε0c(ω;ω,ω,ω,−ω,−ω)

[A(ω)]⁴[A(−ω)]² +3N²ε₀b(ω;ω,ω,−ω)b(−ω,ω,−ω,−ω) [A(ω)]⁴[A(−ω)]³

+6N²ε₀b(ω;ω,−ω,ω)b(ω,ω,ω,−ω)

[A(ω)]⁵[A(−ω)]² +N²ε₀b(ω;−ω,−ω,3ω)b(3ω,ω,ω,ω) A(3ω)[A(ω)]⁴[A(−ω)]²

[E(ω)]³[E(−ω)]², (73) whereA(ω) is given by (56).

The complication comes from the intermediary frequency ω^† involved because of the cascaded cubic hyperpolarizabilities [ω^†=ω₃with the notations of formula (71), orω^†=ω₃+ω₄+ω₅ with the notations of formula (72)].

The intermediary frequencyω^†can indeed be either−ω,+ω, or 3ω. The combinatorial factors are easily computed; they are respectively 3, 6, and 1, the sum being 10, which is the combinatorial factor involved in theQterm. It is worth noticing that 10% of the contribution of the cubic hyperpolarizabilities to the quintic term in the polarization at the fundamental frequencyω comes from a cascade effect involving the third-harmonic frequency 3ω.

The quintic term of the polarization at the third-harmonic frequency is P⁽⁵⁾(3ω)=

5N ε0c(3ω;ω,ω,ω,ω,−ω)

A(3ω)[A(ω)]⁴A(−ω) +3N²ε₀b(3ω;ω,ω,ω)b(ω;ω,ω,−ω)

[A(ω)]⁵A(−ω)A(3ω) +2N²ε₀b(3ω;ω,−ω,3ω)b(3ω;ω,ω,ω) [A(3ω)]²[A(ω)]⁴A(−ω)

×[E(ω)]⁴E(−ω). (74)

Here the intermediary frequency can be only ωor 3ω, with combinatorial factors 3 and 2 respectively, yielding a total of 5.

Finally, the quintic term of the polarization at the fifth-harmonic frequency is P⁽⁵⁾(5ω)=

N ε₀c(5ω;ω,ω,ω,ω,ω)

A(5ω)[A(ω)]⁵ +N²ε₀b(5ω;ω,ω,3ω)b(3ω;ω,ω,ω) [A(ω)]⁵A(3ω)A(5ω)

[E(ω)]⁵, (75)

for which the intermediary frequency involved by the cascading of the cubic hyperpolarizabilities can be 3ωonly.

Since, in the present case of an input quasimonochromatic wave, the quintic nonlinear polarization is expressed in terms of susceptibilities as

PN L⁽⁵⁾ =ε₀[χ⁽⁵⁾(5ω)E(ω)⁵e^−5iωt+5χ⁽⁵⁾(3ω)E(ω)⁴E(ω)^∗e^−3iωt+10χ⁽⁵⁾(ω)E(ω)³E(ω)^∗²e⁻^iωt+c.c.], (76) in which we have set for brevity

χ⁽⁵⁾(5ω)=χ⁽⁵(5ω;ω,ω,ω,ω,ω), (77) χ⁽⁵⁾(3ω)=χ⁽⁵(3ω;ω,ω,ω,ω,−ω), (78) χ⁽⁵⁾(ω)=χ⁽⁵(ω;ω,ω,ω,−ω,−ω), (79)

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we can obtain the expressions for the fifth-order susceptibilityχ⁽⁵⁾as a function of the cubic and quintic hyperpolarizabilities, taking into account the local field effects. For the fundamental frequencyωwe get

χ⁽⁵⁾(ω)= N c(ω)

[A(ω)]²|A(ω)|⁴ + 3N²|b(ω)|²

10A(ω)|A(ω)|⁶ + 3N²(b(ω))²

5[A(ω)]³|A(ω)|⁴ +N²b(ω;−ω,−ω,3ω)b(3ω)

10A(3ω)[A(ω)]²|A(ω)|⁴ , (80) for the third-harmonic frequency 3ω,

χ⁽⁵⁾(3ω)= N c(3ω)

A(3ω)[A(ω)]²|A(ω)|² + 3N²b(3ω)b(ω)

5[A(ω)]⁴|A(ω)|²A(3ω) +2N²b(3ω;ω,−ω,3ω)b(3ω)

5[A(3ω)]²[A(ω)]³|A(ω)|² , (81)

and finally for the fifth-harmonic frequency 5ω, χ⁽⁵⁾(5ω)= N c(5ω)

A(5ω)[A(ω)]⁵ +N²b(5ω;ω,ω,3ω)b(3ω) [A(ω)]⁵A(3ω)A(5ω) .(82) We used the shortcuts (14), (15) and (28)–(30) and the permutation properties of the hyperpolarizabilitiesb and c.

Notice that relation (82) is valid beyond the approximation of the Lorentz model which yields expressions (18), (21), and (38)–(40) of the hyperpolarizabitiesbandc.

It is seen that the local field corrections involve some nonlinear interactions which do not intervene in the expressions of the corresponding hyperpolarizabilities (or of the susceptibilities if local field corrections are not taken into account). The interaction of the third harmonic of the polarization with the fundamental wave through the local field yields the following:

(a) a fifth-harmonic component through the process 3ω+ω+ω−→5ω, corresponding to the component b(5ω;ω,ω,3ω) in Eq. (82),

(b) a third-harmonic component through the process 3ω+ ω−ω−→3ω, corresponding to the componentb(3ω;ω,− ω,3ω) in Eq. (81),

(c) and a fundamental harmonic component through the process 3ω−ω−ω−→ω, corresponding to the component b(ω;−ω,−ω,3ω) in Eq. (80).

IV. RESONANCE OF THE THIRD HARMONIC A special situation of practical interest is that when the fundamental frequency is far from the resonance, but the third harmonic is close to it. Indeed, many experiments are performed in the transparency domain, i.e., in a situation where the fundamental frequency of the wave is far from any resonance. However, in many materials, the transparency range is not so large that the third harmonic will still belong to it. In this case, the third harmonic may be expected to be resonant.

This happens, e.g., in CS2 at fundamental wavelengthsλ= 532 nm or 1064 nm. The formulas derived above have several consequences, which directly relate to observations, in such a situation.

A. The order of the dominant nonlinear absorption The above assumptions imply that the linear absorption of the fundamental is negligible, which is expressed mathemati- cally by the fact that, at frequencyω, the linear susceptibility χ⁽¹⁾(ω) is real. The linear polarizabilitya(ω) is also real, since they are related through relation (58).

From Eq. (21) it is seen that

b(ω)=R[a(ω)]⁴ (83) is real too. Consequently the third-order nonlinear susceptibility at the fundamental frequency becomes, according to (68),

χ⁽³⁾(ω)= N R[a(ω)]⁴

[A(ω)]⁴ . (84)

It is real too, and hence the third-order nonlinear absorption vanishes, as does the linear absorption.

The real and imaginary parts of the third-harmonic generation component of the nonlinear susceptibility are thus, according to (18) and (69),

χ_r⁽³⁾(3ω)= N R[a(ω)]³ [A(ω)]³|A(3ω)|²

ar(3ω)− N

3ε₀|a(3ω)|²

, (85) χ_i⁽³⁾(3ω)= N R[a(ω)]³

[A(ω)]³|A(3ω)|²a_i(3ω), (86) where we denote byar(3ω) andai(3ω) the real and imaginary parts ofa(3ω). Analogous notation will be used throughout the paper. We obtain thus simplified expressions for these components of the third-order susceptibility. It is seen that the imaginary partχ_i⁽³⁾(3ω), which accounts for nonlinear energy losses in third-harmonic generation, is not zero.

We report now the fact thata(ω) andb(ω) are real in the expression (40) ofc(ω), and separate its real and imaginary parts, which yields

cr(ω)=T[a(ω)]⁶+ 3 10

9[b(ω)]²

a(ω) +b(ω)br(3ω) a(ω)

, (87) ci(ω)= 3

10

b(ω)bi(3ω)

a(ω) . (88)

It is seen that c_i(ω) is not zero, i.e., fifth-order nonlinear absorption will occur.

Hence, if the fundamental is off resonant and the third harmonic is resonant, the third-order nonlinear absorption is zero but the fifth-order one is not. It has been seen indeed in [13], that the third-order nonlinear absorption of CS₂ is very low, and that the medium exhibits only fifth-order nonlinear absorption, at both wavelengths 532 and 1064 nm.

The wavelength of the third harmonic is 532/3177 nm, or 1064/3 355nm, which are both close to resonance frequencies of CS₂[21].

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B. The leading term in the fifth-order susceptibility Further, in the expression (87) ofc_r(ω), we can neglect the off-resonant terms, and consequently

c(ω) 3 10

b(ω)b(3ω)

a(ω) . (89)

In the same way we neglect all off-resonant terms in the expression (80) of the fifth-order susceptibility at fundamental frequency, and retain only resonant terms. Taking into account the fact that the quantityA(ω) given by (56) is real under the present assumption, this yields

χ⁽⁵⁾(ω)= N c(ω)

[A(ω)]⁶ +N²b(ω;−ω,−ω,3ω)b(3ω) 10A(3ω)[A(ω)]⁶ .(90) The first term contains a third-harmonic resonance, incorpo- rated in the expression (89) of the quintic hyperpolarizability c(ω). The second term also contains such a resonance, due to the cascaded cubic hyperpolarizabilities involving the third harmonic. The question naturally arises as to which resonance is the strongest one. Therefore we compute the ratioρof the second term (which involvesbtwice) divided by the first one (which involvesc), as

ρ =N b(ω;−ω,−ω,3ω)b(3ω)

10A(3ω)c(ω) . (91)

The expression of the cubic hyperpolarizability term b(ω;−ω,−ω,3ω) is missing. It can be derived using the same procedure as in Sec. II. The input field must involve a term oscillating at frequency 3ω, as El =El(ω)e⁻^iωt+ El(3ω)e^−3iωt+c.c. The linear term x⁽¹⁾ of the electron position is expressed as in Sec. II, except that it involves components at frequencies±3ω, and the cubic correctionx⁽³⁾ to it obeys Eq. (11), in which the right-hand-side member

−C(x⁽¹⁾)³ contains the term −3C[X⁽¹⁾^∗(ω)]²X⁽¹⁾(3ω)e⁻^iωt. The corresponding term inx⁽³⁾has the amplitude

X⁽³⁾_R = −3C[X⁽¹⁾^∗(ω)]²X⁽¹⁾(3ω)

D(ω) . (92)

On the other hand, it is clear that X⁽³⁾_R =(−3/e)b(ω;−ω,

−ω,3ω)[E_l^∗(ω)]²E_l(3ω), and consequently

b(ω;−ω,−ω,3ω)=Ra(ω)[a^∗(ω)]²a(3ω). (93) Using the expressions (89), (83), and (93) for the cubic and quintic hyperpolarizabilities established above, and substituting forA(3ω) its expression (56), we get

ρ = N a(3ω)

3−N a(3ω), (94) and consequently 1+ρ=1/A(3ω). Making use of Eq. (89) again, and of the expressions (68) and (69) which relate the susceptibilities to the (hyper)polarizabilities, we obtain

χ⁽⁵⁾(ω)= 3 10

χ⁽³⁾(ω)χ⁽³⁾(3ω)

χ⁽¹⁾(ω) , (95) i.e., the same relation holds as we would have drawn from Eq. (89) without taking local field corrections into account.

If local field corrections are neglected, which occurs in dilute

FIG. 1. (Color online) Normalized fifth-order susceptibility vs normalized angular frequency, in the approximation of the third- harmonic resonance. Blue solid line, real part; red dotted line, imaginary part. The fundamental resonance occurs atω/ω0=1, to the right of the plot.

media and within the present approximation,χ⁽⁵⁾(ω) reduces to

χ⁽⁵⁾(ω)=

(1−X²−igX)⁶(1−9X²−3igX), (96) where X=ω/ω₀ depends only on the scale factor = 3R²F²/(10ω¹⁴₀ ) and on the relative width of the resonance g=/ω₀. Assumingg=4×10⁻⁴, we plotχ⁽⁵⁾(ω)/vsX in Fig.1.

Under the same assumptions, the nonlinear indicesn₂and n₄ and the fifth-order nonlinear absorption coefficientγ are related to the susceptibilities according to [4]

n₂ =3 4

χ⁽³⁾(ω)

n²₀ε₀c , (97) γ = 5ωχ_i⁽⁵⁾(ω)

2n³₀ε²₀c³ , (98) and

n₄ = 1 (2n0ε₀c)²

5χ_r⁽⁵⁾(ω)

n₀ −9[χ⁽³⁾(ω)]² 8n³₀

, (99) where n₀=n₀(ω) is the linear index, i.e., n²₀(ω)=1+ χ⁽¹⁾(ω). In Eq. (99) again, a natural quintic term is compared to a cascaded cubic term. In CS2at the wavelengthλ=532 nm, values of the nonlinear indices measured using the D4σ- Z-scan method [22] are n₂=(1.5±0.3)×10⁻¹⁸ m²/W;

n₄=(1.2±0.3)×10⁻³²m⁴/W²[13], while the linear index is n₀=1.64. Hence 9/(8n³₀)(χ⁽³⁾)²=5.2×10⁻⁴¹m⁴/V⁴, while (2n₀ε₀c)²n₄=9.1×10⁻³⁷ m⁴/V⁴: theχ⁽⁵⁾term greatly dominates the (χ⁽³⁾)²term. This is due to the resonance of the third harmonic.

At λ=1064, measured values are n₀=1.60, n₄= (2.2±0.4)×10⁻³³ m⁴/W²,n₂=(4.5±1.3)×10⁻¹⁹m²/W [13], and we find 9/(8n³₀)(χ⁽³⁾)²=4.6×10⁻⁴²m⁴/V⁴, while (2n0ε₀c)²n₄=1.6×10⁻³⁷m⁴/V⁴: the third-harmonic resonance also occurs. We will thus neglect the (χ⁽³⁾)² term in Eq. (99).

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TABLE I. Theoretical evaluations of the nonlinear indexn4and three-photon absorption coefficientγcompared to values measured in CS2.

Theory, Theory,

Experiment [13] from Eq. (102) from Eqs. (103),(104)

532 nm n4 (m⁴/W²) (1.2±0.3)×10⁻³² 2.0×10⁻³³ 3.3×10⁻³³

γ (m³/W²) (9.3±1.9)×10⁻²⁶ – 7.9×10⁻²⁶

1064 nm n4 (m⁴/W²) (2.2±0.4)×10⁻³³ 1.9×10⁻³⁴ 3.8×10⁻³⁴

γ (m³/W²) (4.6±0.9)×10⁻²⁷ – 4.5×10⁻²⁷

We can compute the ratio of the so-called three-photon absorption coefficientγ divided by the fifth-order nonlinear index n₄. According to Eq. (95), and then to Eqs. (85) and (86), since there is neither linear nor cubic nonlinear absorption,

χ_r⁽⁵⁾(ω)

χ_i⁽⁵⁾(ω) =χ_r⁽³⁾(3ω)

χ_i⁽³⁾(3ω) =a_r(3ω)−N|a(3ω)|²/(3ε₀)

a_r(3ω) . (100) a(ω) is the Lorentzian (9), and formally the ratio χ_r(3ω)/χi(3ω) can take any value. Explicit computation shows that the term −N|a(3ω)|²/(3ε₀), although non-negligible, affects the value of the ratio χ_r⁽³⁾(3ω)/χ_i⁽³⁾(3ω) merely as would do a shift in the resonance frequency ω₀. We are not interested here in the value of the resonance frequency, but intend to estimate the intensity of the resonance peak.

Therefore, this term does not change the value of the ratio, and we will disregard it for the sake of simplicity, i.e., we use the approximation χr(3ω)/χi(3ω)ar(3ω)/ai(3ω). On the other hand, the maximal values of both ai(3ω) and ar(3ω) are easily computed (using the approximation that 3ω is close to the resonance frequency ω₀) and it is found that max(ar)/max(ai)=1/2.

From Eqs. (98) and (99), we find that γ

n₄ = 4π λ

χ_i⁽⁵⁾(ω)

χr⁽⁵⁾(ω), (101) and consequently

n₄=ar(3ω) ai(3ω)

λγ

4π. (102)

Using the valuesγ =(9.3±1.9)×10⁻²⁶ m³/W² at 532 nm and γ =(4.6±0.9)×10⁻²⁷m³/W² at 1064 nm [13], and approximatinga_r(3ω)/ai(3ω) with max(ar)/max(ai)= 1/2, we obtain n₄=2.0×10⁻³³m⁴/W² and n₄ =1.9× 10⁻³⁴m⁴/W²at 532 and 1064 nm, respectively. These values are lower than the measured values by less than one order of magnitude, which is in quite good agreement within the accuracy of both measurements and theory.

Using (97)–(99), we can express relation (95) in terms of indices, which in turn yields

n₄ =4n0

2+n²₀ 9

n²₀−1 ²

n²₂N a_r(3ω) (103) and

γ = 8π n₀ 2+n²₀ 9λ

n²₀−1 ²

n²₂N ai(3ω), (104)

λbeing obviously the wavelength. The atomic densityN can be easily evaluated from the molar mass 76.14 g/mol and density 1.26 g/cm³of CS2. A value of the polarizability at the third harmonica(3ω) would in principle allow the application of these formulas.

The Lorentzian (9) gives the maximum values fora_i and aras max(ai)=2 max(ar)=e²/(3ωε₀m), and thus requires a value of the resonance width, or more explicitly of the linewidthλ=2π c/(ω²₀), in which the resonance frequency is supposed to coincide with the third-harmonic frequency, ω₀=3ω. For λ=1064 nm, assuming λ=1.4 ˚A allows us to retrieve values close to the experimental measures, as γ =4.5×10⁻²⁷ m³/W² and n₄=3.8×10⁻³⁴ m⁴/W². Retrieving values comparable to the experimental measures at λ=532 nm requires a narrower resonance; e.g., with λ=0.2 ˚A, we obtain γ =7.9×10⁻²⁶m³/W² and n₄= 3.3×10⁻³³m⁴/W². The measured values and theoretical evaluations are summarized in Table I. In the real medium, obviously, there is not a single sharp resonance, but a set of transition lines. Possibly many of them contribute, each with a weaker weight than in this theoretical estimation.

The most important consequence of Eq. (103), where ar(3ω) is the Lorentzian (9), is that it changes sign as 3ω crosses the resonance frequencyω₀.

It has indeed been observed that n₄ measured in CS2

takes positive values at wavelengths λ=532 nm, and at λ=1064, as mentioned above, but negative values at 920 nm:

n₄ = −5.2×10⁻³⁵m⁴/W² [4], and at 800 nm: n₄= −2× 10⁻³⁵ m⁴/W² [12]. Between 532/3177 nm and 800/3 267 nm, we find the transition¹_u⁺←−¹_g⁺at 197 nm [21];

between 920/3307 nm and 1064/3355 nm, the bands related to the transition¹_u⁻←−¹⁺_g at 355.2 nm seem to be involved, although this transition is forbidden.

V. CONCLUSIONS

In summary, we have derived analytical expressions for the fifth-order optical susceptibility based on anharmonic corrections to the Lorentz model. The local field corrections were taken into account; however, they do not play an essential role.

The theoretical predictions were compared with experimental results for CS2. The most significant feature is the identification of a third-harmonic resonance of the nonlinear susceptibility χ⁽⁵⁾, and consequently of the fifth-order nonlinear index n₄ and the three-photon absorption coefficient γ. This resonance gives an explanation for two important observations: The first one is the fact that some measured values ofχ⁽⁵⁾are very large with respect to (χ⁽³⁾)², to which it is expected to be comparable. The second observation concerns