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, Hervé Leblond, Georges Boudebs

To cite this version:

Valentin Besse, Hervé Leblond, Georges Boudebs. Fifth-order nonlinear susceptibility: Effect of third-

order resonances in a classical theory. Physical Review A, American Physical Society 2015, 92 (1),

pp.013818. �10.1103/PhysRevA.92.013818�. �hal-01391971�

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PHYSICAL REVIEW A 92, 013818 (2015)

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 CS

2

.

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 ab- sorption (γ )] or nonlinear refraction [third-order (n

2

) and fifth-order (n

4

) 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 CS

2

[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 at λ = 532 nm with picosecond pulses (80 ps) following different filling factors [10]. A medium heavily investigated is CS

₂

, where in particular Ganeev et al. measured the three-photon absorption coefficient [11]. Kong et 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 CS

₂

at different wavelengths and/or in different temporal regimes provided values for n

₄

and γ , either for picosecond radiation [13] or in the femtosecond regime [4]. We observed an inversion in the sign of the higher-order nonlinear index n

₄

, depending on the wavelength used for the measurement: n

₄

is 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. V summarizes 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 density P by expanding it in a power series of the the applied optical field E [14], as

P = ε

₀

[χ

⁽¹⁾

∗ E + χ

⁽²⁾

∗ E E + χ

⁽³⁾

∗ E E E + · · · ]. (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

1050-2947/2015/92(1)/013818(10) 013818-1 ©2015 American Physical Society

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nonlinear refractive index n

₂

is related to the real part of the third-order nonlinear susceptibility Re[χ

⁽³⁾

], while the fifth-order coefficient n

₄

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 order x

⁵

. The equation satisfied by x is thus

d

²

x dt

²

+ Ŵ dx

dt + ω

²₀

x + Cx

³

+ Qx

⁵

= −e

m E

l

. (2) We assume a linear polarization, whose direction is defined by the unitary vector e

x

, and E

l

= E

l

e

x

is the electric field.

m is the mass of the electron, −e is its electric charge, and ω

₀

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.

For C = 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 = mω

²₀

x

²

2 + mC x

⁴

4 + mQ x

⁶

6 , (3)

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

We assume a monochromatic incident field E

l

= E

_l

e

^−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 E

l

. 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 amplitude E

l

, 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 term x

⁽¹⁾

satisfies the equation d

²

x

⁽¹⁾

dt

²

+ Ŵ dx

⁽¹⁾

dt + ω

²₀

x

⁽¹⁾

= −e

m [E

l

e

^−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) = ω

²₀

− ω

²

− iŴω (6) accounts for the resonance at the angular frequency ω = ω

₀

. The linear atomic dipole moment is p

_L

= −ex

⁽¹⁾

and the linear polarizability a (ω) is defined by

p

L

= a(ω)E

l

e

^−iωt

+ c.c. (7) (we use the notation a,b,c for the linear polarizability and the second- and the third-order nonlinear hyperpolarizabilities, instead of the standard notation α,β,γ , in order to avoid con- fusion with the linear and nonlinear absorption coefficients) so that

X

⁽¹⁾

(ω) = −a(ω)

e E

_l

. (8)

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

a(ω) = F

D(ω) , (9)

with

F = e

²

m . (10)

B. The cubic term

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

d

²

x

⁽³⁾

dt

²

+ Ŵ dx

⁽³⁾

dt + ω

²₀

x

⁽³⁾

= −C(x

⁽¹⁾

)

³

. (11) Expansion of the right-hand-side member shows that x

⁽³⁾

= X

⁽³⁾

(3ω)e

^−3iωt

+ X

⁽³⁾

(ω)e

^−iωt

+ c.c., and the equation satis- fied 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 yield X

⁽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 b as

p

⁽³⁾_{N L}

= b(3ω)E

_l³

e

^−3iωt

+ 3b(ω)E

_l²

E

_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 E

²_l

E

_l^∗

. (17)

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FIFTH-ORDER NONLINEAR SUSCEPTIBILITY: EFFECT . . . PHYSICAL REVIEW A 92, 013818 (2015) Using (16) and (8) in Eq. (12) yields the third-harmonic

generation term of the hyperpolarizability as

b(3ω) = R[a (ω)]

³

a(3ω), (18) in which we have set

R = −C

e

²

F = −Cm

e

⁴

. (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)]

²

X

⁽¹⁾^∗

(ω), (20) and using (17) we obtain the component of the third-order hyperpolarizability as

b(ω) = R[a(ω)]

³

a

^∗

(ω). (21) C. The fifth-order nonlinear hyperpolarizability The equation satisfied by the fifth-order correction to the electron position x

⁽⁵⁾

is

d

²

x

⁽⁵⁾

dt

²

+ Ŵ dx

⁽⁵⁾

dt + ω

²₀

x

⁽⁵⁾

= −Q(x

⁽¹⁾

)

⁵

− 3C

(x

⁽¹⁾

)

²

x

⁽³⁾

, (22) and consequently x

⁽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 amplitudes X

⁽⁵⁾

(pω) can then be written as

D(5ω)X

⁽⁵⁾

(5ω) = −Q[X

⁽¹⁾

(ω)]

⁵

− 3C[X

⁽¹⁾

(ω)]

²

X

⁽³⁾

(3ω) (24) for the fifth-harmonic generation term; we obtain

D(3ω)X

⁽⁵⁾

(3ω) = −5Q[X

⁽¹⁾

(ω)]

⁴

X

⁽¹⁾^∗

(ω)

− 3C{[X

⁽¹⁾

(ω)]

²

X

⁽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)]

²

X

⁽³⁾^∗

(ω) + 2X

⁽¹⁾

(ω)X

⁽¹⁾^∗

(ω)X

⁽³⁾

(ω)

+ [X

⁽¹⁾^∗

(ω)]

²

X

⁽³⁾

(3ω)}, (26) for the saturation of the Kerr effect and the fifth-order nonlinear absorption.

The fifth-order nonlinear dipole moment p

_{N L}⁽⁵⁾

= −ex

⁽⁵⁾

can be expanded using the fifth-order hyperpolarizability c as

p

⁽⁵⁾_{N L}

= c(5ω)E

⁵_l

e

^−5iωt

+ 5c(3ω)E

_l⁴

E

^∗_l

e

^−3iωt

+ 10c(ω)E

_l³

E

_l^∗²

e

^−iωt

+ c.c., (27)

in which the components of c are 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 E

⁵_l

, (31) X

⁽⁵⁾

(3ω) = −5c(3ω)

e E

⁴_l

E

_l^∗

, (32) X

⁽⁵⁾

(ω) = −10c(5ω)

e E

_l³

E

_l^∗²

. (33) Using the above formulas we obtain expressions for the involved fifth-order hyperpolarizability terms as

c(5ω) = T [a(ω)]

⁵

a(5ω) + 3R[a(ω)]

²

b(3ω)a(5ω), (34) c(3ω) = T [a(ω)]

⁴

a

^∗

(ω)a(3ω) +

³₅

R{3[a(ω)]

²

b(ω)a(3ω)

+ 2a(ω)a

^∗

(ω)b(3ω)a(3ω)}, (35)

c(ω) = T [a(ω)]

⁴

[a

^∗

(ω)]

²

+

₁₀³

R{3[a(ω)]

³

b

^∗

(ω)

+ 6[a(ω)]

²

a

^∗

(ω)b(ω) + a (ω)[a

^∗

(ω)]

²

b(3ω)}, (36) in which R is given by (19) and where we have set

T = −Q

e

⁴

F = −Qm

e

⁶

. (37)

In the above expressions, the resonance factors 1/D(pω) have been rewritten using the first-order hyperpolarizability a. However, the parameter C, which characterizes the strength of the third-order nonlinearity, remains in the final formula. It is indeed involved via the expression (19) of the factor R. 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)]

⁵

a(5ω) + 3 b(ω)b(3ω)a(5ω)

a(ω)a

^∗

(ω) , (38) c(3ω) = T [a(ω)]

⁴

a

^∗

(ω)a(3ω)

+ 3 5

3 b(3ω)b(ω)

a(ω) + 2 [b(3ω)]

²

a

^∗

(ω) [a(ω)]

²

, (39) c(ω) = T [a(ω)]

⁴

[a

^∗

(ω)]

²

+ 3 10

3 b(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 coefficient C either,

but use other components of the third-order hyperpolarizability

013818-3

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b and the linear polarizability a . 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 density P = P e

x

is related to the atomic dipolar moment p through P = −N ex = Np, where N is the density of atoms. Its linear part P

L

can be expressed in terms of the linear susceptibility χ

⁽¹⁾

as

P

L

= ε

₀

χ

⁽¹⁾

(ω)Ee

^−iωt

+ c.c., (41) where E is the amplitude of the macroscopic electric field.

If we identify the field E

_l

which is involved via the ex- pression of p

_L

with the macroscopic field E, we get χ

⁽¹⁾

= aN/ε

₀

. Within the same approximation, it is straightforwardly found that χ

⁽³⁾

= bN /ε

₀

and χ

⁽⁵⁾

= cN/ε

₀

. 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 field E

l

is 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 field E through

E

l

= E + P 3ε

0

, (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 dipole p using the polarizability a and hyperpolarizabilities b and c.

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

The expression for p becomes

p = aE

l

+ bE

_l³

+ cE

_l⁵

. (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 polarization P

L

= Np

_L

= N aE

l

. With respect to the macroscopic field E , the linear polarization is expressed as P

L

= ε

₀

K

⁽¹⁾

E. It yields the equation

1 − N a

3ε

₀

K

⁽¹⁾

E = N a

ε

₀

E, (44)

which is easily solved to give K

⁽¹⁾

= N a/ε

₀

1 − N a/(3ε

0

) . (45) The relation

1 + K

⁽¹⁾

3 = 1

1 − N a/3ε

0

(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 ≃ P

L

+ P

_{N L}⁽³⁾

= N

aE

l

+ bE

_l³

. (47) We expand P in the same way in a power series of the macroscopic field E as

P = ε

₀

K

⁽¹⁾

E + K

⁽³⁾

E

³

. (48) The coefficients K

⁽¹⁾

and K

⁽³⁾

are not exactly the suscepti- bilities, since the latter are defined in the frequency domain, which involves combinatorial factors when considering the harmonics of a quasimonochromatic field.

The expression (42) for E

l

is substituted into Eq. (47), as well as (48). The leading order vanishes due to (45); we keep only the terms of order E

³

and obtain the equation

1 − N a

3ε

0

K

⁽³⁾

E

³

= N b ε

₀

1 + K

⁽¹⁾

3 E

³

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

K

⁽³⁾

= N b/ε

₀

(1 − N a/3ε

₀

)

⁴

. (50) We use now the expansion

P = ε

₀

K

⁽¹⁾

E + K

⁽³⁾

E

³

+ K

⁽⁵⁾

E

⁵

+ O(E

⁷

) (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

3ε

0

K

⁽⁵⁾

E

⁵

= N b ε

₀

1 + K

⁽¹⁾

3 E

²

K

⁽³⁾

E

³

+ N c ε

₀

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/ε

₀

(1 − N a/3ε

₀

)

⁶

+ N

²

b

²

/ε

₀²

(1 − N a/3ε

₀

)

⁷

. (53)

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FIFTH-ORDER NONLINEAR SUSCEPTIBILITY: EFFECT . . . PHYSICAL REVIEW A 92, 013818 (2015) We get finally the following expression for the polarization

density:

P

ε

₀

= N a/ε

₀

1 − N a/(3ε

0

) E + N b/ε

₀

[1 − N a/(3ε

0

)]

⁴

E

³

+

N c/ε

₀

[1 − N a/(3ε

0

)]

⁶

+ N

²

b

²

/ε

₀²

[1 − N a/(3ε

0

)]

⁷

E

⁵

. (54) It is noticed that, in formula (53), the coefficient K

⁽⁵⁾

contains two terms. The first term results from the effect of the quintic hyperpolarizability c, which is natural, but the second term is the result of a cascade with twice the effect of the cubic hyperpolarizability b. 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 of χ

⁽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 quasi- monochromatic wave; hence the field is E = 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) 3ε

₀

. (56)

Since the linear susceptibility χ

⁽¹⁾

(ω) is defined by

P

L

(ω) = ε

₀

χ

⁽¹⁾

(ω)E(ω), (57) we can compute the expression for the linear susceptibility as a function of the polarizability a(ω), as

χ

⁽¹⁾

(ω) = N a(ω)

ε

₀

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

When the electric field E depends on time t, all products in the expression (43) of the dipolar moment must be replaced with convolution products. In particular, the cubic term bE

³

becomes

b ∗ EEE =

b(t

1

,t

₂

,t

₃

)E(t − t

₁

)E(t − t

₂

)E(t − t

₃

)dt

1

dt

₂

dt

₃

. (59) In the case of a field containing only discrete frequencies ω

₁

,ω

₂

, . . . ,ω

M

, 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⁾

(ω

0

) =

ω₁,ω₂,ω₃ ω₀=ω₁+ω₂+ω₃

N ε

₀

b(ω

₀

; ω

₁

,ω

₂

,ω

₃

)

3

j=0

A ω

j

3

j=1

E ω

j

.

(61) For a quasimonochromatic field (M = 2, ω

₁

= −ω

₂

= ω), we get a polarization term at the fundamental frequency ω as

P

⁽³⁾

(ω) = 3N ε

₀

b(ω)

[A(ω)]

³

A(−ω) [E(ω)]

²

E

^∗

(ω), (62) and a third-harmonic term

P

⁽³⁾

(3ω) = N ε

₀

b(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 susceptibility χ

⁽³⁾

is

P

⁽³⁾

(ω) = 3χ

⁽³⁾

(ω)[E(ω)]

²

E

^∗

(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 square b

²

of the cubic hyperpolarizability.

We denote by P

_Q⁽⁵⁾

the former and by P

_C⁽⁵⁾

the latter.

In the case of a field containing only discrete frequencies ω

j

,

the generalization of the derivation to the convolution product

can be done using Eq. (52). It gives, for the Q term, the general

013818-5

(7)

expression

P

_Q⁽⁵⁾

(ω

₀

) =

ω1,ω2,ω3,ω4,ω5

ω0=5 j=1ωj

⎡

⎣

N ε

₀

c(ω

0

; ω

₁

,ω

₂

,ω

₃

,ω

₄

,ω

₅

)

5

j=0

A(ω

j

)

5

j=1

E(ω

j

)

⎤

⎦ (70)

[P

_Q⁽⁵⁾

(ω

₀

) denotes the amplitude of P

_Q⁽⁵⁾

at the frequency ω

₀

, and so on]. The C term involving b

²

is more complicated, as

P

_C⁽⁵⁾

(ω

₀

) =

ω₁,ω₂,ω₃ ω0=3

j=1ωj

⎡

⎢

⎣

N

²

ε

₀

b(ω

0

; ω

₁

,ω

₂

,ω

₃

)

2

j=0

A(ω

j

)

2

j=1

E(ω

j

)

ω₄,ω₅,ω₆ ω3=6

j=4ωj

b(ω

3

; ω

₄

,ω

₅

,ω

₆

)

6

j=3

A(ω

j

)

6

j=4

E(ω

j

)

⎤

⎥

⎦

. (71)

If we consider the frequency-matching conditions which appear in Eq. (71), we see that ω

₃

can be substituted with ω

₄

+ ω

₅

+ ω

₆

, the matching condition becoming ω

₀

= ω

₁

+ ω

₂

+ ω

₄

+ ω

₅

+ ω

₆

. Renumbering the dummy variables, the expression (71) can be thus rewritten in the more convenient way

P

_C⁽⁵⁾

(ω

₀

) =

ω1,ω2,ω3,ω4,ω5

ω0=5 j=1ωj

⎡

⎣

N

²

ε

₀

b(ω

0

; ω

₁

,ω

₂

,ω

₃

+ ω

₄

+ ω

₅

)b(ω

3

+ ω

₄

+ ω

₅

,ω

₃

,ω

₄

,ω

₅

) A(ω

₃

+ ω

₄

+ ω

₅

)

5

j=0

A ω

j

5

j=1

E ω

j

⎤

⎦. (72)

For a quasimonochromatic field we get, for the fundamental frequency ω, i.e., the terms which express the saturation of the Kerr effect (when n

₂

and n

₄

have opposite signs),

P

⁽⁵⁾

(ω) =

10N ε

0

c(ω; ω,ω,ω, − ω, − ω)

[A(ω)]

⁴

[A(−ω)]

²

+ 3N

²

ε

₀

b(ω; ω,ω, − ω)b(−ω,ω, − ω, − ω) [A(ω)]

⁴

[A(−ω)]

³

+ 6N

²

ε

₀

b(ω; ω, − ω,ω)b(ω,ω,ω, − ω)

[A(ω)]

⁵

[A(−ω)]

²

+ N

²

ε

₀

b(ω; −ω, − ω,3ω)b(3ω,ω,ω,ω) A(3ω)[A(ω)]

⁴

[A(−ω)]

²

[E(ω)]

³

[E(−ω)]

²

, (73) where A(ω) is given by (56).

The complication comes from the intermediary frequency ω

^†

involved because of the cascaded cubic hyperpolarizabilities [ω

^†

= ω

₃

with the notations of formula (71), or ω

^†

= ω

₃

+ ω

₄

+ ω

₅

with the notations of formula (72)].

The intermediary frequency ω

^†

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 the Q term. It is worth noticing that 10% of the contribution of the cubic hyperpolarizabilities to the quintic term in the polarization at the fundamental frequency ω 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 ε

₀

c(3ω; ω,ω,ω,ω, − ω)

A(3ω)[A(ω)]

⁴

A(−ω) + 3N

²

ε

₀

b(3ω; ω,ω,ω)b(ω; ω,ω, − ω)

[A(ω)]

⁵

A(−ω)A(3ω) + 2N

²

ε

₀

b(3ω; ω, − ω,3ω)b(3ω; ω,ω,ω) [A(3ω)]

²

[A(ω)]

⁴

A(−ω)

×[E(ω)]

⁴

E(−ω). (74)

Here the intermediary frequency can be only ω or 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 ε

₀

c(5ω; ω,ω,ω,ω,ω)

A(5ω)[A(ω)]

⁵

+ N

²

ε

₀

b(5ω; ω,ω,3ω)b(3ω; ω,ω,ω) [A(ω)]

⁵

A(3ω)A(5ω)

[E(ω)]

⁵

, (75)

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

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

P

_{N L}⁽⁵⁾

= ε

₀

[χ

⁽⁵⁾

(5ω)E(ω)

⁵

e

^−5iωt

+ 5χ

⁽⁵⁾

(3ω)E(ω)

⁴

E(ω)

^∗

e

^−3iωt

+ 10χ

⁽⁵⁾

(ω)E(ω)

³

E(ω)

^∗2

e

^−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)

(8)

FIFTH-ORDER NONLINEAR SUSCEPTIBILITY: EFFECT . . . PHYSICAL REVIEW A 92, 013818 (2015) we can obtain the expressions for the fifth-order susceptibility χ

⁽⁵⁾

as a function of the cubic and quintic hyperpolarizabilities, taking into account the local field effects. For the fundamental frequency ω we get

χ

⁽⁵⁾

(ω) = N c(ω)

[A(ω)]

²

|A(ω)|

⁴

+ 3N

²

|b(ω)|

²

10A(ω)|A(ω)|

⁶

+ 3N

²

(b(ω))

²

5[A(ω)]

³

|A(ω)|

⁴

+ N

²

b(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

+ 3N

²

b(3ω)b(ω)

5[A(ω)]

⁴

|A(ω)|

²

A(3ω) + 2N

²

b(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

+ N

²

b(5ω; ω,ω,3ω)b(3ω) [A(ω)]

⁵

A(3ω)A(5ω) . (82) We used the shortcuts (14), (15) and (28)–(30) and the permutation properties of the hyperpolarizabilities b 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 hyperpolarizabities b and c.

It is seen that the local field corrections involve some nonlin- ear 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 pro- cess 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 component b(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 CS

2

at fundamental wavelengths λ = 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 frequency ω, the linear susceptibility χ

⁽¹⁾

(ω) is real. The linear polarizability a (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 susceptibil- ity 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 gen- eration 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

a

r

(3ω) − N

3ε

₀

|a(3ω)|

²

, (85) χ

_i⁽³⁾

(3ω) = N R[a(ω)]

³

[A(ω)]

³

|A(3ω)|

²

a

i

(3ω), (86) where we denote by a

r

(3ω) and a

i

(3ω) the real and imaginary parts of a (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 part χ

_i⁽³⁾

(3ω), which accounts for nonlinear energy losses in third-harmonic generation, is not zero.

We report now the fact that a(ω) and b(ω) are real in the expression (40) of c(ω), and separate its real and imaginary parts, which yields

c

r

(ω) = T [a(ω)]

⁶

+ 3 10

9 [b(ω)]

²

a(ω) + b(ω)b

r

(3ω) a(ω)

, (87) c

i

(ω) = 3

10 b(ω)b

i

(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 CS

₂

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 532/3 ≃ 177 nm, or 1064/3 ≃ 355nm, which are both close to resonance frequencies of CS

2

[21].

013818-7

(9)

B. The leading term in the fifth-order susceptibility Further, in the expression (87) of c

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 quantity A(ω) given by (56) is real under the present assumption, this yields

χ

⁽⁵⁾

(ω) = N c(ω)

[A(ω)]

⁶

+ N

²

b(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 ratio ρ of the second term (which involves b twice) divided by the first one (which involves c), 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 E

l

= E

l

(ω)e

^−iωt

+ E

_l

(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 correction x

⁽³⁾

to it obeys Eq. (11), in which the right-hand-side member

−C(x

⁽¹⁾

)

³

contains the term −3C[X

⁽¹⁾^∗

(ω)]

²

X

⁽¹⁾

(3ω)e

^−iωt

. The corresponding term in x

⁽³⁾

has the amplitude

X

⁽³⁾_R

= −3C[X

⁽¹⁾^∗

(ω)]

²

X

⁽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)]

²

E

l

(3ω), and consequently

b(ω; −ω, − ω,3ω) = Ra(ω)[a

^∗

(ω)]

²

a(3ω). (93) Using the expressions (89), (83), and (93) for the cubic and quintic hyperpolarizabilities established above, and substitut- ing for A(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 at ω/ω

0

= 1, to the right of the plot.

media and within the present approximation, χ

⁽⁵⁾

(ω) reduces to

χ

⁽⁵⁾

(ω) =

(1 − X

²

− igX)

⁶

(1 − 9X

²

− 3igX) , (96) where X = ω/ω

₀

depends only on the scale factor = 3R

²

F

²

/(10ω

¹⁴₀

) and on the relative width of the resonance g = Ŵ/ω

₀

. Assuming g = 4 × 10

⁻⁴

, we plot χ

⁽⁵⁾

(ω)/ vs X in Fig. 1.

Under the same assumptions, the nonlinear indices n

₂

and n

₄

and the fifth-order nonlinear absorption coefficient γ are related to the susceptibilities according to [4]

n

₂

= 3 4

χ

⁽³⁾

(ω)

n

²₀

ε

₀

c , (97) γ = 5ωχ

_i⁽⁵⁾

(ω)

2n

³₀

ε

²₀

c

³

, (98) and

n

₄

= 1 (2n

0

ε

₀

c)

²

5χ

_r⁽⁵⁾

(ω)

n

₀

− 9[χ

⁽³⁾

(ω)]

²

8n

³₀

, (99) where n

₀

= n

₀

(ω) is the linear index, i.e., n

²₀

(ω) = 1 + χ

⁽¹⁾

(ω). In Eq. (99) again, a natural quintic term is compared to a cascaded cubic term. In CS

2

at the wavelength λ = 532 nm, values of the nonlinear indices measured using the D4σ - Z-scan method [22] are n

₂

= (1.5 ± 0.3) × 10

⁻¹⁸

m

²

/W;

n

₄

= (1.2 ± 0.3) × 10

⁻³²

m

⁴

/W

²

[13], while the linear index is n

₀

= 1.64. Hence 9/(8n

³₀

)(χ

⁽³⁾

)

²

= 5.2 × 10

⁻⁴¹

m

⁴

/V

⁴

, while (2n

₀

ε

₀

c)

²

n

₄

= 9.1 × 10

⁻³⁷

m

⁴

/V

⁴

: the χ

⁽⁵⁾

term greatly dominates the (χ

⁽³⁾

)

²

term. This is due to the resonance of the third harmonic.

At λ = 1064, measured values are n

₀

= 1.60, n

₄

=

(2.2±0.4)×10

⁻³³

m

⁴

/W

²

, n

₂

=(4.5±1.3)×10

⁻¹⁹

m

²

/W [13],

and we find 9/(8n

³₀

)(χ

⁽³⁾

)

²

= 4.6×10

⁻⁴²

m

⁴

/V

⁴

, while

(2n

0

ε

₀

c)

²

n

₄

=1.6×10

⁻³⁷

m

⁴

/V

⁴

: the third-harmonic

resonance also occurs. We will thus neglect the (χ

⁽³⁾

)

²

term in Eq. (99).

(10)

FIFTH-ORDER NONLINEAR SUSCEPTIBILITY: EFFECT . . . PHYSICAL REVIEW A 92, 013818 (2015) TABLE I. Theoretical evaluations of the nonlinear index n

4

and three-photon absorption coefficient γ compared to values measured in CS

2

.

Theory, Theory,

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

532 nm n

4

(m

⁴

/W

²

) (1.2 ± 0.3) × 10

⁻³²

2.0 × 10

⁻³³

3.3 × 10

⁻³³

γ (m

³

/W

²

) (9.3 ± 1.9) × 10

⁻²⁶

– 7.9 × 10

⁻²⁶

1064 nm n

4

(m

⁴

/W

²

) (2.2 ± 0.4) × 10

⁻³³

1.9 × 10

⁻³⁴

3.8 × 10

⁻³⁴

γ (m

³

/W

²

) (4.6 ± 0.9) × 10

⁻²⁷

– 4.5 × 10

⁻²⁷

We can compute the ratio of the so-called three-photon absorption coefficient γ divided by the fifth-order nonlinear index n

₄

. 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

/(3ε

0

)

a

_r

(3ω) . (100) a(ω) is the Lorentzian (9), and formally the ratio χ

r

(3ω)/χ

i

(3ω) can take any value. Explicit computation shows that the term −N |a(3ω)|

²

/(3ε

₀

), 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 ω

₀

. 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 χ

r

(3ω)/χ

i

(3ω) ≃ a

r

(3ω)/a

i

(3ω). On the other hand, the maximal values of both a

i

(3ω) and a

r

(3ω) are easily computed (using the approximation that 3ω is close to the resonance frequency ω

₀

) and it is found that max(a

r

)/ max(a

i

) = 1/2.

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

n

₄

= 4π λ

χ

_i⁽⁵⁾

(ω)

χ

_r⁽⁵⁾

(ω) , (101) and consequently

n

₄

= a

r

(3ω) a

i

(3ω)

λγ

4π . (102)

Using the values γ = (9.3 ± 1.9) × 10

⁻²⁶

m

³

/W

²

at 532 nm and γ = (4.6 ± 0.9) × 10

⁻²⁷

m

³

/W

²

at 1064 nm [13], and approximating a

_r

(3ω)/a

i

(3ω) with max(a

r

)/ max(a

i

) = 1/2, we obtain n

₄

= 2.0 × 10

⁻³³

m

⁴

/W

²

and n

₄

= 1.9 × 10

⁻³⁴

m

⁴

/W

²

at 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

n

₄

= 4n

0

2 + n

²₀

9 n

²₀

− 1

²

n

²₂

N a

r

(3ω) (103) and

γ = 8π n

0

2 + n

²₀

9λ

n

²₀

− 1

²

n

²₂

N a

i

(3ω), (104)

λ being obviously the wavelength. The atomic density N can be easily evaluated from the molar mass 76.14 g/mol and density 1.26 g/cm

³

of CS

2

. A value of the polarizability at the third harmonic a(3ω) would in principle allow the application of these formulas.

The Lorentzian (9) gives the maximum values for a

i

and a

r

as max(a

i

) = 2 max(a

r

) = e

²

/(3ωε

₀

mŴ), and thus requires a value of the resonance width Ŵ, or more explicitly of the linewidth λ = 2π c/(ω

²₀

Ŵ), in which the resonance frequency is supposed to coincide with the third-harmonic frequency, ω

₀

= 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

m

³

/W

²

and n

₄

= 3.8 × 10

⁻³⁴

m

⁴

/W

²

. 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

⁻²⁶

m

³

/W

²

and n

₄

= 3.3 × 10

⁻³³

m

⁴

/W

²

. 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 a

r

(3ω) is the Lorentzian (9), is that it changes sign as 3ω crosses the resonance frequency ω

₀

.

It has indeed been observed that n

₄

measured in CS

2

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

n

₄

= −5.2 × 10

⁻³⁵

m

⁴

/W

²

[4], and at 800 nm: n

₄

= −2 × 10

⁻³⁵

m

⁴

/W

²

[12]. Between 532/3 ≃ 177 nm and 800/3 ≃ 267 nm, we find the transition

¹

_u⁺

←−

¹

_g⁺

at 197 nm [21];

between 920/3 ≃ 307 nm and 1064/3 ≃ 355 nm, the bands related to the transition

¹

_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 experi-

mental results for CS

2

. The most significant feature is the

identification of a third-harmonic resonance of the nonlinear

susceptibility χ

⁽⁵⁾

, and consequently of the fifth-order non-

linear index n

₄

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 of χ

⁽⁵⁾

are very large with respect to (χ

⁽³⁾

)

²

, to which it is

expected to be comparable. The second observation concerns

013818-9

(11)

the sign changes of n

₄

with the wavelength. They occur as the third harmonic crosses the resonance, as shown by the theoretical expression. The ratio between n

₄

and γ can be predicted from theory and is found to be in good agreement with experimental values.

Formulas allowing us in principle to predict the magnitudes of n

₄

and γ have also been provided; however, they are hardly applicable, due to the fact that it is difficult to determine the

strength of the third-harmonic resonance, and that the present simplified theory considers only a single transition.

The Lorentz model is very general, and describes a priori any material, provided that the phenomenon is due to bounded electrons, and the approximation can be made that a single transition is involved. A further condition is that no charge carrier is generated, as may occur at very large intensities or in photorefractive materials.

[1] G. L. Wood, W. W. Clark III, M. J. Miller, G. J. Salamo, and E. J. Sharp, Evaluation of passive optical limiters and switches, Proc. SPIE 1105, 154 (1989).

[2] H. Shim, M. Liu, C. Hwangbo, and G. I. Stegeman, Four- photon absorption in the single-crystal polymer bis(paratoluene) sulfonate, Opt. Lett. 23, 430 (1998); F. Smektala, C. Quemard, L. Leneindre, J. Lucas, A. Barth´el´emy, and C. De Angelis, Chalcogenide glasses with large non-linear refractive indices, J. Non-Cryst. Solids 239, 139 (1998); G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S. W. Cheong, J. S. Sanghera, and I. D. Aggarwal, Large Kerr effect in bulk Se-based chalcogenide glasses, Opt. Lett. 25, 254 (2000).

[3] V. Besse, H. Leblond, and G. Boudebs, Filamentation of light in carbon disulfide, Phys. Rev. A 89, 043840 (2014).

[4] E. L. Falc˜ao-Filho, C. B. de Ara´ujo, G. Boudebs, H. Leblond, and V. Skarka, Robust two-dimensional spatial solitons in liquid carbon disulfide, Phys. Rev. Lett. 110, 013901 (2013).

[5] H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, Kerr spatiotemporal self-focusing in a planar glass waveguide, Phys. Rev. Lett. 87, 043902 (2001).

[6] P. L. Kelley, Self-focusing of optical beams, Phys. Rev. Lett. 15, 1005 (1965).

[7] L. Berge, Wave collapse in physics: principles and applications to light and plasma waves, Phys. Rep. 303, 259 (1998).

[8] R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm, J. Opt.

A: Pure Appl. Opt. 6, 282 (2004).

[9] M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. van Stryland, Sensitive measurement of optical nonlinearities using a single beam, IEEE J. Quantum Electron. 26, 760 (1990).

[10] E. L. Falc˜ao-Filho, C. B. de Ara´ujo, and J. J. Rodrigues, Jr., High-order nonlinearities of aqueous colloids containing silver nanoparticles, J. Opt. Soc. Am. B 24, 2948 (2007).

[11] R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, Two- and three-photon absorption in CS

2

, Opt. Commun. 231, 431 (2004).

[12] D. G. Kong, Q. Chang, H.’A. Ye, Y. C. Gao, Y. X. Wang, X. R. Zhang, K. Yang, W. Z. Wu, and Y. L. Song, The fifth-order nonlinearity of CS

2

, J. Phys. B: At. Mol. Opt. Phys. 42, 065401 (2009).

[13] V. Besse, G. Boudebs, and H. Leblond, Determination of the third-and fifth-order optical nonlinearities: the general case, Appl. Phys. B: Lasers Opt. 116, 911 (2014).

[14] R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, New York, 2008).

[15] R. Miles and S. Harris, Optical third-harmonic generation in alkali metal vapors, IEEE J. Quantum Electron. 9, 470 (1973).

[16] N. Bloembergen, Nonlinear Optics, 4th ed. (World Scientific, Singapore, 1996).

[17] H. A. Lorentz, Uber ¨ die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der K¨orperdichte, Wiedem. Ann. 245, 641 (1880).

[18] H. A. Lorentz, The Theory of Electrons (B. G. Teubner, Leipzig and G. E. Stechert, New York, 1916).

[19] M. Born and E. Wolf, Principles of Optics (Cambridge Univer- sity Press, Cambridge, 1999).

[20] G. I. Stegeman, D. J. Hagan, and L. Torner, χ

⁽²⁾

cascading phe- nomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons, Opt. Quantum Electron. 28, 1691 (1996).