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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 6387–6399
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Nonlinear birefringence due to non-resonant, higher-order Kerr effect in isotropic media

George Stegeman, Dimitris G. Papazoglou, Robert Boyd, and Stelios Tzortzakis  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 6387-6399 (2011)
http://dx.doi.org/10.1364/OE.19.006387


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Abstract

The recent interpretation of experiments on the nonlinear non-resonant birefringence induced in a weak probe beam by a high intensity pump beam in air and its constituents has stimulated interest in the non-resonant birefringence due to higher-order Kerr nonlinearities. Here a simple formalism is invoked to determine the non-resonant birefringence for higher-order Kerr coefficients. Some general relations between nonlinear coefficients with arbitrary frequency inputs are also derived for isotropic media. It is shown that the previous linear extrapolations for higher-order birefringence (based on literature values of n2 and n4) are not strictly valid, although the errors introduced in the values of the reported higher- order Kerr coefficients are a few percent.

© 2011 OSA

1. Introduction

The fact that an intense beam induces a nonlinear birefringenceΔnbir(2)(I), (I – local intensity), in any medium is well-known since the early days of nonlinear optics [8

8. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).

]. Such a birefringence is usually formulated in terms of the third order susceptibility χ (3) for Kerr nonlinearities involving electronic states in a medium. For isotropic Kerr media, Δnbir(2)(I)n2I and the proportionality constant depends on how many unique eigenmodes are present. For example, the numerical factor is 1/3 for a single intense beam where-as it is 2/3 for a strong pump, weak probe geometry.

The general formulation of the nonlinear birefringence problem requires calculating the nonlinear index changes produced by a strong pump beam either for the pump itself, or for a second beam, usually a weak probe beam, with different frequency, propagation direction and/or polarization properties from those of the pump. In isotropic media this normally requires knowledge of the ratio of at least two nonlinear susceptibilities and their dispersion with frequency [8

8. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).

]. This can be a daunting problem since the number of different susceptibility terms increases rapidly with the order of the nonlinearity, i.e. with “m” inχ(m). The situation simplifies considerably for isotropic media in the non-resonant regime for the susceptibilities since there is only one independent nonlinear susceptibility for each value of m [3

3. J. Arabat and J. Etchepare, “Nonresonant fifth-order nonlinearities induced by ultrashort intense pulses,” J. Opt. Soc. Am. B 10(12), 2377–2382 (1993). [CrossRef]

,9

9. V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975). [CrossRef]

,10

10. S. V. Popov, Y. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Taylor and Francis, 1995)

]. The formulation of the nonlinear birefringence problem described here relies strongly on this fact and a formula is derived for arbitrary order nonlinearities.

The most recent interest in nonlinear birefringence due to higher-order Kerr coefficients was stimulated by experiments at ~800nm on filaments which form in air at high (>10 TW/cm2) laser intensities [1

1. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009). [CrossRef] [PubMed]

,11

11. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007). [CrossRef]

]. In order to explain their birefringence measurements, Loriot et al. assumed non-resonant nonlinear index coefficients up to n 10 (involving χ (11)). There is some controversy in the filamentation community concerning the interpretation of the measured birefringence but in this paper we simply focus on their analytical expression for the birefringence due to higher-order Kerr effects [12

12. M. Kolesik, D. Mirell, J.-C. Diels, and J. V. Moloney, “On the higher-order Kerr effect in femtosecond filaments,” Opt. Lett. 35(21), 3685–3687 (2010). [CrossRef] [PubMed]

]. They obtained their contributions to the birefringence from the well-known relations between the tensor coefficients for χ (3) and that obtained for χ (5) by Arabat and Etchepare based on an anharmonic oscillator model, and then linear extrapolation to higher-orders [3

3. J. Arabat and J. Etchepare, “Nonresonant fifth-order nonlinearities induced by ultrashort intense pulses,” J. Opt. Soc. Am. B 10(12), 2377–2382 (1993). [CrossRef]

]. Although this nonlinear oscillator model fails to reproduce accurately the frequency dispersion of the third (and presumably higher-order) nonlinearities obtained from quantum mechanics, it does give non-resonant (ω→0) results for χ (3), χ (5) and χ (7), albeit not in terms of physically measurable parameters [2

2. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express 18(3), 3011–3012 (2010). [CrossRef]

,9

9. V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975). [CrossRef]

,10

10. S. V. Popov, Y. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Taylor and Francis, 1995)

]. We are not aware of any extension to yet higher-order nonlinearities.

The most frequently cited nonlinear index coefficient n 2 in isotropic media is defined for a single intense beam (x-polarized, for example) as [8

8. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).

]:
n2(ω;ω)=14nx2ε0ceal{χxxxx(3)(ω;ω,ω,ω)+χxxxx(3)(ω;ω,ω,ω)                    +χxxxx(3)(ω;ω,ω,ω)}
(1)
Note that we have introduced a notation (ω;ω) for n 2 in which the beam which induces the nonlinear index change is the second ( + )ω argument and the first argument (-ω) identifies the eigenmode in which the index change occurs. If another eigenmode is present such as a weak “probe” beam of the same or different frequency ωp, also x-polarized but travelling at a small angle to the “pump” beam, the appropriate nonlinearity in this case is defined as
n2(ωp;ω)=14n02ε0ceal{χxxxx(3)(ωp;ωp,ω,ω)+χxxxx(3)(ωp;ωp,ω,ω)                   +χxxxx(3)(ω;ω,ωp,ω)+χxxxx(3)(ωp;ω,ωp,ω)                   +χxxxx(3)(ωp;ω,ω,ωp)+χxxxx(3)(ωp;ω,ω,ωp)}
(2)
In this paper we derive from first principles the nonlinear birefringence introduced by higher-order Kerr coefficients in the non-resonant limit in an isotropic medium. We start by reformulating the well-known relations for χ (3) in terms of combinatorial expressions which then provide a simple procedure for dealing with the higher-order Kerr nonlinearities. We find that the extrapolation used by Loriot et al. is not correct for the pump-probe geometry they considered [1

1. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009). [CrossRef] [PubMed]

].

The procedure followed here is a three-step process:

  • 1. The nonlinear polarizations PxpNL(ωp)and PypNL(ωp)are calculated in terms of the nonlinear susceptibilities by permuting the input eigenmodes via their frequencies for an isotropic medium.
  • 2. The relation between theχijk(3)(ωp;ω,ω,ωp)susceptibilities is found for an isotropic medium by permuting the polarizations. This utilizes the concept that the nonlinear polarization in an isotropic medium must be independent of the choice of axes.
  • 3. The square of the refractive indices, i.e. nx2 and ny2are calculated from the respective polarizations and the square root of each is taken to give the nonlinear birefringenceΔnbirNL=nxNLnyNL.

2. Pump-probe geometry

Here we consider the specific geometry of the Loriot et al. experiment shown in Fig. 1
Fig. 1 The pump-probe interaction geometry in reference 1. The angle between the beams was 4°.
[1

1. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009). [CrossRef] [PubMed]

]. An intense plane wave of the form
E(r,t)=12Ex(ω)eiωt+c.c.
(3)
is assumed to propagate along the z-axis in an isotropic material, i.e. the x-axis is chosen parallel to the polarization of the intense beam. A second probe beam (subscript “p”) of frequency ωp = ω is also present but propagating at a small angle from the z-axis in the y-z plane (making it a different eigenmode from the pump beam). Its polarization has equal x and y-components written as
Exp(r,t)=12Exp(ωp)eiωpt+c.c.;    Eyp(r,t)=12Eyp(ωp)eiωpt+c.c.
(4a)
and the nonlinear polarization induced in the probe beam is written as

PxpNL(r,t)=12PxpNL(ωp)eiωpt+c.c.;    PypNL(r,t)=12PypNL(ωp)eiωpt+c.c.
(4b)

3. Nonlinear polarizabilities

The third order nonlinear polarization induced by the pump beam in the molecules of the air, as experienced by the probe beam, is
Pxp(3)(ωp)=14ε0{χ¯xxxx(3)(ωp;ωp,ω,ω)+χ¯xxxx(3)(ωp;ω,ω,ωp)              +χ¯xxxx(3)(ωp;ω,ωp,ω)+χ¯xxxx(3)(ωp;ω,ω,ωp)              +χ¯xxxx(3)(ωp;ω,ωp,ω)+χ¯xxxx(3)(ωp;ωp,ω,ω)}Exp(ωp)Ex*(ω)Ex(ω).
(5)
The susceptibilitiesχ¯xxxx(3)(ωp)are values of the coefficient averaged over the constituent air molecules, i.e.
χ¯xxxx(3)(ωp;ωp,ω,ω)=qwqχxxxx,q(3)(ωp;ωp,ω,ω) etc .
(6)
here wq is the fraction of the number density corresponding to species q, i.e. nitrogen, oxygen etc. In the non-resonant limit (identified by the superscript ~), the imaginary part of the susceptibility is negligibly small, zero for ω = 0, and [8

8. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).

]
χ¯˜xxxx(3)(ωp;ωp,ω,ω)=χ¯˜xxxx(3)(ωp;ω,ωp,ω)=χ¯˜xxxx(3)(ωp;ωp,ω,ω)=χ¯˜xxxx(3)(ωp;ω,ω,ωp)=χ¯˜xxxx(3)(ωp;ω,ω,ωp)=χ¯˜xxxx(3)(ωp;ω,ωp,ω).
(7)
are all real. Therefore,
Pxp(3)(ωp)=64ε0χ¯˜xxxx(3)(ωp;ωp,ω,ω)Exp(ωp)Ex*(ω)Ex(ω).
(8)
A different way to arrive at this result is to note that there are three separate input positions for frequency in the expression for χ (3) giving 3!( = 3x2x1) different possibilities when they are permuted over the three input fields. (In nonlinear opticsEx*(ω) and Ex(ω) can be treated as separate eigenmodes because they have different frequencies in a mixing process, i.e. + ω and –ω.) Thus there are three separate NLO (nonlinear optics) eigenmodes, each of which appears just once, so that the total number of unique terms is given by 3!/1!1!1!, i.e.
Pxp(3)(ωp)=143!1!1!1!ε0χ¯˜xxxx(3)(ωp;ωp,ω,ω)Exp(ωp)|Ex(ω)|2.
(9)
Similarly, for the polarization nonlinearly induced along the y-axis by the strong x-polarized field,
Pyp(3)(ωp)=14ε0{χ¯yyxx(3)(ωp;ωp,ω,ω)+χ¯yyxx(3)(ωp;ωp,ω,ω)+χ¯yxyx(3)(ωp;ω,ωp,ω)+χ¯yxyx(3)(ωp;ω,ωp,ω)+χ¯yxxy(3)(ωp;ω,ω,ωp)+χ¯yxxy(3)(ωp;ω,ω,ωp)}Eyp(ω)Ex*(ω)Ex(ω).
(10)
In the non-resonant limit all sixχ¯˜(3)sare equal so that with 3 independent eigenmodes each of which appears only once,
Pyp(3)(ωp)=143!1!1!1!ε0χ¯˜yyxx(3)(ωp;ωp,ω,ω)Eyp(ωp)|Ex(ω)|2.
(11)
The nonlinear susceptibilities are now abbreviated so that χ¯˜xxxx(3)(ωp;ωp,ω,ω)and χ¯˜yyxx(3)(ωp;ωp,ω,ω)are written as χ¯˜xxxx(3)(ωp)and χ¯˜yyxx(3)(ωp)respectively. (This will also subsequently be extended to higher-order susceptibilities.) Furthermore, since the labeling of the axes in isotropic media is arbitrary, χ¯˜yyxx(3)(ωp)=χ¯˜xxyy(3)(ωp). Applying the same arguments as for the probe case but with two equal co-polarized fields at + ω for Px(3)(ω) but not forPy(3)(ω), the nonlinear polarizations experienced by the pump beam are
Px(3)(ω)=143!2!1!ε0χ˜xxxx(3)(ω)Ex(ω)|Ex(ω)|2=12Pxp(3)(ωp),
(12a)
Py(3)(ω)=143!1!1!1!ε0χ¯˜yyxx(3)(ω)Ey(ω)|Ex(ω)|2=Pyp(3)(ωp).
(12b)
Generalizing these results to the 2m + 1 case [4

4. Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006). [CrossRef]

],
Pxp(2m+1)(ωp)=[14m(2m+1)!m!m!ε0χ¯˜(2m+2)x(2m+1)(ωp)|Ex(ω)|2m]Exp(ωp)=(m+1)Px(2m+1)(ω),
(13)
Pyp(2m+1)(ωp)=[14m(2m+1)!m!m!ε0χ¯˜(2)y,(2m)x(2m+1)(ωp)|Ex(ω)|2m]Eyp(ωp)=Py(2m+1)(ω).
(14)
Here the co-ordinate subscripts (2m + 2)x mean that there is a total of 2m + 2 “x co-ordinates,” one referring to the output polarization of the probe, and one of the remaining 2m + 1 refers to the input probe polarization, interspersed amongst the 2m others associated with ± ω of the pump beam. The (2)y,(2m)x means that there are 2 “y” co-ordinates, one always being the first co-ordinate which refers to the output probe polarization, and the second to the input probe beam polarization interspersed amongst the 2m ± ω x-polarizations associated with the pump beam.

Therefore the total polarization for the probe beam is given by
Pxp(1)(ωp)+PxpNL(ωp)=ε0[nx2(ωp)1]Exp(ωp)=ε0[(n021)                                   +m=114m(2m+1)!m!m!χ¯˜(2m+2)x(2m+1)(ωp)|Ex(ω)|2m]Exp(ωp);
(15)
Pyp(1)(ωp)+PypNL(ωp)ε0[ny2(ωp)1]Eyp(ωp)=ε0[(n021)                                       +m=114m(2m+1)!m!m!χ¯˜yy,(2m)x(2m+1)(ωp)|Ex(ω)|2m]Eyp(ωp).
(16)
For the pump beam, the ωp on the input side is replaced by another ω and hence there are m + 1 + ω ’s but still m -ω’s so that
Px(1)(ω)+PxNL(ω)=ε0[nx2(ω)1]Ex(ω)=ε0[(n021)                             +m=114m(2m+1)!(m+1)!m!χ¯˜(2m+2)x(2m+1)(ω)|Ex(ω)|2m]Ex(ω).
(17)
ForPyNL(ω), the first 2 y’s belong to the probe beam and there are still m ω’s and m -ω’s, just like inPypNL(ωp) so that

Py(1)(ω)+Py(NL)(ω)=ε0[ny2(ω)1]Ey(ω)=ε0[(n021)                                   +m=114m(2m+1)!m!m!χ¯˜yy,(2m)x(2m+1)(ω)|Ex(ω)|2m]Ey(ω).
(18)

4. Total nonlinear birefringence

It is clear from Eqs. (15)(18), that in order to find the birefringence, the relationship between the nonlinear susceptibilitiesχ¯˜(2m+2)x(2m+1)(ω)andχ¯˜yy,(2m)x(2m+1)(ω)must be found. This depends on the symmetry properties of the medium. Even for isotropic media these are relatively complicated calculations and hence they are summarized in the Appendix along with some general results valid for all frequencies. Making the results specific to the non-resonant, isotropic medium case, Eq. (A17) is
χ¯˜(2m+2)x(2m+1)(ωp)=(2m+1)χ¯˜yy,(2m)x(2m+1)(ωp);  χ¯˜(2m+2)x(2m+1)(ω)=(2m+1)χ¯˜yy,(2m)x(2m+1)(ω).
(19)
For the pump-probe geometry in the non-resonant limit, Eq. (19) is inserted into Eqs. (16) and (18) to give
Pyp(1)(ωp)+PypNL(ωp) =ε0[ny2(ωp)1]Eyp(ωp)=ε0[(n021)                                      +m=114m(2m+1)!(2m+1)m!m!χ¯˜(2m+2)x(2m+1)(ωp)|Ex(ω)|2m]Eyp(ωp),
(20)
Py(1)(ω)+Py(NL)(ω)=ε0[ny2(ω)1]Ey(ω)=ε0[(n021)                                  +m=114m(2m+1)!(2m+1)m!m!χ¯˜(2m+2)x(2m+1)(ω)|Ex(ω)|2m]Ey(ω),
(21)
so that both the x- and y-components of the nonlinear polarization are given in terms of the same susceptibilities. Noting thatn2m(ωp;ω)=(m+1)n2m(ω;ω) from Eq. (13) and combining Eqs. (15), (17), (20), and (21) now leads directly to
nx2(ωp)=n02{1+[m=1A¯mIm]};   ny2(ωp)=n02{1+[m=11(2m+1)A¯mIm]},nx2(ω)=n02{1+[m=11m+1A¯mIm]};   ny2(ω)=n02{1+[m=11(2m+1)A¯mIm]},
(22)
in which the coefficientA¯m is given by
                                               A¯m=2n¯2m(ωp;ω)n0,A¯m=12n012mn0mcmε0m(2m+1)!m!m!1!χ¯˜(2m+2)x(2m+1)(ωp)=12n0(m+1)2mn0mcmε0m(2m+1)!m!m!1!χ¯˜(2m+2)x(2m+1)(ω).
(23)
This form was chosen so that for the individual nonlinearities m
Δn¯x(m)(ωp)=n¯2m(ωp;ω)Im.
(24)
In order to make contact with the experimental data in reference 1

1. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009). [CrossRef] [PubMed]

we focus on the nonlinear refractive indices for the pump-probe case so that the nonlinear birefringence is given by
n¯x(ωp)=n01+[m=1A¯mIm]; n¯y(ωp)=n01+[m=11(2m+1)A¯mIm];                                           ΔnbirNL(ωp)=nx(ωp)ny(ωp).
(25)
The expansion of1+bfor small b is well known from textbooks [13

13. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

], to be:
1+b=s=0(1)s(2s)!(12s)(s!)24sbs=1+12b18b2+116b35128b4+7256b5..
(26)
Therefore
Δn¯birNL(ωp)=n0s=0(1)s(2s)!(12s)(s!)24s([m=1A¯mIm]s[m=1A¯m2m+1Im]s).
(27)
The leading term (s = 1), expanded up to n 10 (largest term reported in reference 1

1. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009). [CrossRef] [PubMed]

), is
Δn¯birs=1(ωp)=n0{13A¯1I+25A¯2I2+49A¯3I3+817A¯4I4+1633A¯4I4+3265A¯5I5                  =23n¯2(ωp;ω)I+45n¯4(ωp;ω)I2+89n¯6(ωp;ω)I3                     +1617n¯8(ωp;ω)I4+3233n¯10(ωp;ω)I5
(28)
Terms with s≥2 contain products of the nonlinear coefficients. Including all of the terms up to I 5,
Δn¯birNL=23n¯2(ωp;ω)I+[45n¯4(ωp;ω)49n0n¯22(ωp;ω)]I2+[89n¯6(ωp;ω)1415n0n¯2(ωp;ω)n¯4(ωp;ω)+1327n02n¯23(ωp;ω)]I3+[1617n¯8(ωp;ω)2627n0n¯2(ωp;ω)n¯6(ωp;ω)1225n0n¯42(ωp;ω)+2215n02n¯22(ωp;ω)n¯4(ωp;ω)5081n03n¯24(ωp;ω)]I4+[3233n¯10(ωp;ω)5051n0n¯2(ωp;ω)n¯8(ωp;ω)4445n0n¯4(ωp;ω)n¯6(ωp;ω)+4027n02n¯22(ωp;ω)n¯6(ωp;ω)+3725n02n¯2(ωp;ω)n¯42(ωp;ω)6727n03n¯23(ωp;ω)n¯4(ωp;ω)+847972n04n¯25(ωp;ω)]I5.
(29)
Note that all the numerical pre-factors in this case are all less than 2.5. The products of different nonlinear coefficients are limited to 2 here. However, products of more than two nonlinear coefficients occur for higher-orders in intensity, the first one beingn¯2n¯4n¯6I6. From Eq. (29) it is evident that in a strict mathematical sense the nonlinear birefringence cannot be used as a means to measure the nonlinear coefficients higher thann¯2.There is no direct correlation between the coefficientn¯2mand the corresponding power of the intensity Im for m>1 due to the existence of the product terms. However, it makes sense to use the simplified notation of Eq. (29) if the relationn¯2m>>n¯2k1n¯2k2...n¯2k2n,m=k1+k2+...kn holds.

5. Comparison with experiments on air

Reference 1

1. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009). [CrossRef] [PubMed]

contains data measured in air and its constituents for n 2(-ωp;ω) → n 8(-ωp;ω) and also n 10(-ωp;ω) for argon. Based on their values, n¯2m(ωp;ω)>>n¯2qr(ωp;ω)n¯2uv(ωp;ω)with m = rq + vu and m≤5 is always satisfied in air. Assuming that the only nonlinear mechanism present is the Kerr effect, the nonlinear birefringence is given by the leading term, Eq. (28), which can be expressed as the series
Δn¯birNL(ωp)=m=12m2m+1n¯2m(ωp;ω)Im  .
(30)
This result should be compared with the expansion used by Loriot et al. [1

1. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009). [CrossRef] [PubMed]

]. Based on a linear extrapolation from the first two terms which Loriot et al. obtained from the literature [3

3. J. Arabat and J. Etchepare, “Nonresonant fifth-order nonlinearities induced by ultrashort intense pulses,” J. Opt. Soc. Am. B 10(12), 2377–2382 (1993). [CrossRef]

,8

8. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).

] they assumed the series
Δn¯birNL(ωp)=m=12m2m+1n¯2m(ωp;ω)Im
(31)
in their analysis of their data. Note that in both series the numerical pre-factors 2m/(2m + 1) and 2m/(2m + 1) respectively converge to unity for large m. A graphical comparison of the two expansions is given in Fig. 2
Fig. 2 Comparison between the expansion coefficients estimated by the two models. (a) Coefficients corresponding to χ(m) terms. (□) analytical model, (•) Loriot et al. estimation, dotted/dashed lines are a guide to the eye. (b) Relative error for the various coefficients of the χ(m) terms. (Dotted lines are guides to the eye).
. In Fig. 2(a) we compare the expansion terms as deduced from Eq. (30), 2m/(2m+1), to the ones derived by Loriot et al. 2m/(2m+1). As m is increased their difference is maximized for m = 11. The relative deviation of Loriot et al.’s expansion terms as compared to the analytically derived factors is depicted in Fig. 2(b). For m = 11 the relative error peaks at 6.25%. Furthermore, the Loriot et al. formulation systematically underestimates the expansion term coefficients and thus leads to an overestimation of the correspondingn¯2m(ωp;ω)coefficient for m>2.

6. Conclusions

Expressions for the non-resonant, nonlinear birefringence induced in a probe beam (frequencyωp) by a strong pump beam of the same frequency in an isotropic medium have been derived for nonlinear Kerr indices n 2 m(-ωp;ω) for arbitrary m. This was made possible by using combinatorial approaches and by assuming that in isotropic media there is only one unique value forχ(2m+1)(ωp)for each value of m which was verified previously in the literature for m = 1, 2. Some general relations for arbitrary frequency inputs were also derived.

Because the polarization, linear and nonlinear, induced in a material depends on the square of the refractive index, the nonlinear birefringence was found to depend not only on the intensity-dependent refractive index coefficients n 2 m(-ωp;ω) but also on the products of the various nonlinear index coefficients. Comparison with existing experiments in air and its constituents showed that the product terms were negligible in that case.

An analytical series was found to describe the nonlinear birefringence. This series was different from that assumed by Loriot et. al based on a linear extrapolation of two points. Since in both cases the individual numerical factors for n 2 m(-ωp;ω) converged to unity for increasing m, the errors introduced into the analysis of the data were relatively small.

Appendix A. Relationships between the nonlinear susceptibilities

Px(3)(ω4)=14ε0χxxxx(3)(ω4;ω3,ω2,ω1)E1E2E3.
(A1)

Now consider the axis system (x', y') rotated 45° from the original x-axis [8

8. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).

]. The three fields have the following components along the x'-axis and y'-axis

E1x'=12E1; E2x'=12E2; E3x'=12E3; E1y'=12E1 ; E2y'=12E2; E3y'=12E3.
(A2)

For isotropic media, χx'x'x'x'(3)(ω4;ω3,ω2,ω1)=χxxxx(3)(ω4;ω3,ω2,ω1), χxxyy(3)(ω4;ω3,ω2,ω1)=χx'x'y'y'(3)(ω4;ω3,ω2,ω1), =χyyxx(3)(ω4;ω3,ω2,ω1)etc., and hence the nonlinear polarization induced along the x'-axis is given by

Px'(3)(ω4)=14ε0[χxxxx(3)(ω4;ω3,ω2,ω1)E1x'E2x'E3x'+χxxyy(3)(ω4;ω3,ω2,ω1)E1y'E2y'E3x'                   +χxyyx(3)(ω4;ω3,ω2,ω1)E1x'E2y'E3y'+χxyxy(3)(ω4;ω3,ω2,ω1)E1y'E2x'E3y']Px'(3)(ω4)=14ε0122[χxxxx(3)(ω4;ω3,ω2,ω1)+χxxyy(3)(ω4;ω3,ω2,ω1)                      +χxyyx(3)(ω4;ω3,ω2,ω1)+χxyxy(3)(ω4;ω3,ω2,ω1)]E1E2E3.
(A3)

The nonlinear polarization Px'(3)(ω4)in Eq. (A3) can also be obtained by projecting the nonlinear polarization given by Eq. (A1) onto the x'-axis to give

Px'(3)(ω4)=14ε012χxxxx(3)(ω4,ω3,ω2,ω1)E1E2E3.
(A4)

Since Eqs. (A3) and (A4) must give the same result which is valid for any frequencies,

χxxxx(3)(ω4;ω3,ω2,ω1)=χxxyy(3)(ω4;ω3,ω2,ω1)+χxyyx(3)(ω4;ω3,ω2,ω1)                                +χxyxy(3)(ω4;ω3,ω2,ω1).
(A5)

χ˜xxyy(3)(ωp)=χ˜xyyx(3)(ωp)=χ˜xyxy(3)(ωp)    χ˜xxxx(3)(ωp)=3χ˜xxyy(3)(ωp).
(A6)

The same result holds for pump beam, i.e. χ˜xxxx(3)(ω)=3χ˜xxyy(3)(ω). Although this result is valid for a single medium, extension to multi-component air is trivial giving

χ¯˜xxxx(3)(ωp)=3χ¯˜xxyy(3)(ωp)       χ¯˜xxxx(3)(ω)=3χ¯˜xxyy(3)(ω).
(A7)

An alternate and more compact approach for arriving at the same result is to again resort to combinatorial mathematics. Since there are three input polarization components, two y'-polarized and one x'-polarized, which can be permuted among the three input eigenmodes (frequencies), there are 3! possibilities for permuting the corresponding polarization components inχ¯˜xxyy(3)(ω4;ω3,ω2,ω1). Because there must be two identical polarization components (y') and only one x′, there are 3!/2!1!unique possibilities and Eq. (A5) can be re-written in the non-resonant limit as

χ¯˜xxxx(3)(ωp)=3!2!1!χ¯˜xxyy(3)(ωp);    χ¯˜xxxx(3)(ω)=3!2!1!χ¯˜xxyy(3)(ω)
(A8)

The evaluation of the relation betweenχ¯˜xxxxxx(5)(ωp) and χ¯˜yyxxxx(5)(ωp) (and subsequently the yet higher-order susceptibilities) has additional aspects (relative to the χ (3) case) associated with the χ¯˜yyxxxx(5)(ωp)=χ¯˜xxyyxx(5)(ωp)=χ¯˜xxyyyy(5)(ωp)etc. terms. Again assuming the general case of five, parallel, co-polarized (along the x-axis) input fields namely E1, E2, E3, E4, and E5 with different frequencies ω 1, ω 2, ω 3, ω 4 and ω 5 producing the field ω 6 via χxxxxxx(5)(ω6;ω5,ω4,ω3,ω2,ω1). This produces the nonlinear polarization (along the x-axis)

Px(5)(ω4)=116ε0χxxxxxx(5)(ω6;ω5,ω4,ω3,ω2,ω1)E1E2E3E4E5.
(A9)

Now consider again the axis system (x', y') rotated 45° from the original x-axis. The five input x-polarized fields again have components along the x'-axis and y'-axis. Note that both mixed polarization terms likeχxxyyxx(5)(ω6;ω5,ω4,ω3,ω2,ω1)as well as χxxyyyy(5)(ω6;ω5,ω4,ω3,ω2,ω1) contribute to the nonlinear polarization induced along the x'-axis, Px'NL(ω6). For the first one, there are 5! input slots for the polarization of which 3 are identical (x') and the two others are also identical (y') and, for the second one, there are 4 (y') identical slots and only the x' is a single slot. Hence the number of unique combinations are 5!/3!2! and 5!/4!1! respectively for χ˜xxyyxx(5)(ω6;ω5,ω4,ω3,ω2,ω1) and for χ˜xxyyyy(5)(ω6;ω5,ω4,ω3,ω2,ω1). There are further simplifications in the non-resonant limit χ˜xxyyxx(5)(ω6;ω5,ω4,ω3,ω2,ω1)=χ˜xxyxyx(5)(ω6;ω5,ω4,ω3,ω2,ω1) etc. so that

Px'(5)(ω6)=116ε0142[χ¯˜xxxxxx(5)(ω6;ω5,ω4,ω3,ω2,ω1)+5!3!2!χ¯˜xxyyxx(5)(ω6;ω5,ω4,ω3,ω2,ω1)               +5!4!1!χ¯˜xxyyyy(5)(ω6;ω5,ω4,ω3,ω2,ω1)]E1E2E3E4E5.
(A10)

The nonlinear polarization Px'(5)(ω6)in Eq. A10 can also be obtained by projecting the nonlinear polarization given by Eq. (A9) onto the x'-axis to give

Px'NL(ω6)=116ε012χ˜xxxxxx(5)(ω6;ω5,ω4,ω3,ω2,ω1)E1E2E3E4E5.
(A11)

Again Eqs. (A10) and (A11) must yield identical results and noting again from references 2

2. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express 18(3), 3011–3012 (2010). [CrossRef]

and 8

8. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).

thatχ¯˜xxyyxx(5)(ω6;ω5,ω4,ω3,ω2,ω1) =χ¯˜xxyyyy(5)(ω6;ω5,ω4,ω3,ω2,ω1)etc. yields for the cases of interest here in the non-resonant limit

χ¯˜xxxxxx(5)(ωp)=5χ¯˜yyxxxx(5)(ωp);   χ¯˜xxxxxx(5)(ω)=5χ¯˜yyxxxx(5)(ω).
(A12)

Consider briefly the 7’th and 9’th order susceptibilities. The same procedures as for the 3′rd and 5′th order cases are used. In order to derive the relationship between the different χ¯˜xxxxxxxx(7)(ωp),χ¯˜yyxxxxxx(7)(ωp), etc. seven co-polarized input fields are considered, first in the x, y, z coordinate system and then in an axis system rotated 45° in the x-y plane. In this case, the mixed polarization terms

χxxyyxxxx(7)(ω8;ω7,ω6,ω5,ω4,ω3,ω2,ω1),χxxyyyyxx(7)(ω8;ω7,ω6,ω5,ω4,ω3,ω2,ω1)

χxxyyyyyy(7)(ω8;ω7,ω6,ω5,ω4,ω3,ω2,ω1)all contribute to the nonlinear polarizations induced along the x'-axis, Px'(7)(ω8). The number of unique combinations are 7!/5!2!, 7!/4!3!and 7!/6!1!respectively for the three cases. Thus again in the non-resonant limit

Px'NL(ω8)=164ε0182[χ¯˜xxxxxx(7)(ω8)+7!5!2!χ¯˜xxyyxxxx(7)(ω8)                 +7!4!3!χ¯˜xxyyyyxx(7)(ω8)+7!6!1!χ¯˜xxyyyyyy(7)(ω8)]E1E2E3E4E5E6E7               =164ε012χ¯˜xxxxxx(7)(ω8)E1E2E3E4E5E6E7
(A13)

Based on the preceding results, only one, unique, nonlinear susceptibility is expected for an isotropic material in the non-resonant limit for each order “2m+1” ofχ(2m+1). Therefore all the mixed polarization susceptibilities are equal which gives

χ¯˜xxxxxxxx(7)(ωp)=9χ¯˜yyxxxxxx(7)(ωp);   χ¯˜xxxxxxxx(7)(ω)=9χ¯˜yyxxxxxx(7)(ω).
(A14)

Again using the same approach, for the 9’th order susceptibility,

Px(9)(ω10)=1256ε01162[χ¯˜xxxxxxxx(9)(ω10)+9!7!2!χ¯˜xxyyxxxxxx(9)(ω10)+9!5!4!χ¯˜xxyyyyxxxx(9)(ω10)                  +9!3!6!χ¯˜xxyyyyyyxx(9)(ω10)+9!1!8!χ¯˜xxyyyyyyyy(9)(ω10)]E1E2E3E4E5E6E7E8E9               =1256ε012χ¯˜xxxxxxxx(9)(ω10)E1E2E3E4E5E6E7E8E9.
(A15)

In the non-resonant limit

χ¯˜xxxxxxxxxx(9)(ω10)=17χ¯˜yyxxxxxxxx(9)(ω10)
(A16)

These results suggest simple relations governing the relationship between the susceptibilities, namely

χ¯˜(2m+2)x(2m+1)(ωp)=(2m+1)χ¯˜yy,(2m)x(2m+1)(ωp);  χ¯˜(2m+2)x(2m+1)(ω)=(2m+1)χ¯˜yy,(2m)x(2m+1)(ω).
(A17)

For frequency inputs ω 1, ω 2, ω 3,.. ω 2 m +1 giving an output frequency ω 2 m +2 for isotropic media, the above formulas suggest the following general result:

χ¯˜(2m+2)x(2m+1)(ω2m+2)=12m[χ¯˜(2m+2)x(2m+1)(ω2m+2)+(2m+1)!(2m)!1!χ¯˜(2)y,(2m)x(2m+1)(ω2m+2)                  +(2m+1)!(2m2)!3!χ¯˜(4)y,(2m2)x,(2m+1)(ω2m+2) +(2m+1)!(2m4)!5!χ¯˜(6)y,(2m4)x(2m+1)(ω2m+2)...                  +(2m+1)!2!(2m1)!χ¯˜(2m1)y,(2)x(2m+1)(ω2m+2)].
(A18)

which gives

χ¯˜(2m+2)x(2m+1)(ω2m+2)=12m1[(2m+1)!(2m)!1!χ¯˜(2)y,(2m)x(2m+1)(ω2m+2)                +(2m+1)!(2m2)!3!χ¯˜(4)y,(2m2)x,(2m+1)(ω2m+2) +(2m+1)!(2m4)!5!χ¯˜(6)y,(2m4)x(2m+1)(ω2m+2)...                +(2m+1)!2!(2m1)!χ¯˜(2m1)y,(2)x(2m+1)(ω2m+2)].
(A19)

Acknowledgments

This work was partially supported by the European Union Marie Curie Excellence Grant “MULTIRAD” MEXT-CT-2006–042683.

References and links

1.

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009). [CrossRef] [PubMed]

2.

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express 18(3), 3011–3012 (2010). [CrossRef]

3.

J. Arabat and J. Etchepare, “Nonresonant fifth-order nonlinearities induced by ultrashort intense pulses,” J. Opt. Soc. Am. B 10(12), 2377–2382 (1993). [CrossRef]

4.

Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006). [CrossRef]

5.

For example,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(2), 282–287 (2004). [CrossRef]

6.

For example,S. Wu, X.-C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett. 61, 1919–1921 (1992). [CrossRef]

7.

E. J. Canto-Said, D. J. Hagan, J. Young, and E. W. Van Stryland, “Degenerate four-wave mixing measurements of high order nonlinearities in semiconductors,” IEEE J. Quantum Electron. 27(10), 2274–2280 (1991). [CrossRef]

8.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).

9.

V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975). [CrossRef]

10.

S. V. Popov, Y. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Taylor and Francis, 1995)

11.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007). [CrossRef]

12.

M. Kolesik, D. Mirell, J.-C. Diels, and J. V. Moloney, “On the higher-order Kerr effect in femtosecond filaments,” Opt. Lett. 35(21), 3685–3687 (2010). [CrossRef] [PubMed]

13.

M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

14.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express 17(5), 3190–3195 (2009). [CrossRef] [PubMed]

15.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81(3), 033817 (2010). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.3270) Nonlinear optics : Kerr effect
(190.5940) Nonlinear optics : Self-action effects

ToC Category:
Nonlinear Optics

History
Original Manuscript: January 14, 2011
Revised Manuscript: February 28, 2011
Manuscript Accepted: March 3, 2011
Published: March 21, 2011

Citation
George Stegeman, Dimitris G. Papazoglou, Robert Boyd, and Stelios Tzortzakis, "Nonlinear birefringence due to non-resonant, higher-order Kerr effect in isotropic media," Opt. Express 19, 6387-6399 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6387


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References

  1. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009). [CrossRef] [PubMed]
  2. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express 18(3), 3011–3012 (2010). [CrossRef]
  3. J. Arabat and J. Etchepare, “Nonresonant fifth-order nonlinearities induced by ultrashort intense pulses,” J. Opt. Soc. Am. B 10(12), 2377–2382 (1993). [CrossRef]
  4. Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006). [CrossRef]
  5. For example,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(2), 282–287 (2004). [CrossRef]
  6. For example,S. Wu, X.-C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett. 61, 1919–1921 (1992). [CrossRef]
  7. E. J. Canto-Said, D. J. Hagan, J. Young, and E. W. Van Stryland, “Degenerate four-wave mixing measurements of high order nonlinearities in semiconductors,” IEEE J. Quantum Electron. 27(10), 2274–2280 (1991). [CrossRef]
  8. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).
  9. V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975). [CrossRef]
  10. S. V. Popov, Y. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Taylor and Francis, 1995)
  11. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007). [CrossRef]
  12. M. Kolesik, D. Mirell, J.-C. Diels, and J. V. Moloney, “On the higher-order Kerr effect in femtosecond filaments,” Opt. Lett. 35(21), 3685–3687 (2010). [CrossRef] [PubMed]
  13. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).
  14. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express 17(5), 3190–3195 (2009). [CrossRef] [PubMed]
  15. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81(3), 033817 (2010). [CrossRef]

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