Abstract

We report a theoretical and experimental study on the real and imaginary part of the third-order nonlinear optical susceptibility at 532 nm and 7 ns pulse for high-purity silica samples containing Au nanoparticles prepared by ion implantation. We present a method for measuring the magnitude and sign of refractive and absorptive nonlinearities based on four-wave mixing (FWM). This method is derived from a comparison of the light intensities of incident and self-diffracted polarized waves. In the nanosecond regime the samples exhibit saturable absorption and it seems that a thermal effect is the mechanism responsible of nonlinearity of index.

© 2007 Optical Society of America

1. Introduction

The nonlinear optical properties of different metallic nanoparticles have been investigated extensively [1-4]. These materials are being improved continuously to exhibit a stronger and faster optical response, controlled shape and size, easy fabrication, durability and thermo-mechanical stability [5]. Both the optical properties and the spatial distribution of metallic nanoparticles prepared by ion implantation in silica exhibit a very good stability [6]. We present results for silica samples containing Au nanoparticles and showing a considerable nonlinear optical response. This property makes them attractive for potential applications such as optical limitation, optical coupling, optical modulation and waveguiding.

For most of the optical materials the refractive index can be expressed as n=n0+n2I, where n0 represents the linear refractive index in a weak field, I is the light intensity in the media and n2 is the nonlinear refractive index. In a similar way, the intensity dependent absorption coefficient is defined as: α(I)=α0+βI, where α0 and β represent the linear and nonlinear absorption coefficients, respectively [7].

The magnitude of the third-order nonlinear optical susceptibility χ(3), can be expressed as,

χ(3)=(Reχ(3))2+(Imχ(3))2.

A relationship of n2 and β with χ(3) (esu) has been established:

χ(3)=n0c7.91×102n2+in02cλπ2β,

where c is the light velocity in the vacuum, λ is the wavelength of the light, and n2 and β are expressed in m2/W and m/W, respectively.

The parameters n2 and β depend on the different components of the tensor of third-order nonlinear optical susceptibility χ(3). The search of n2 and β has generated an important number of works. Different techniques have been developed to measure them, such as Z-scan [8], four-wave mixing [9], third harmonic generation [10], nonlinear interferometry [11], ellipse rotation [12], nonlinear ellipsometry [13], Raman spectroscopy [14], holography [15], measure of beam distortion [16], photoacoustic experiments [17] or spectrally resolve two-beam coupling [18]. In this work we present a method based on a FWM technique to measure the sign and magnitude of n2 and β in a nonlinear optical material and we identified the physical mechanisms of optical nonlinearity exhibited by our samples.

2. Theory of coupling and self-diffraction of waves

We consider a thin isotropic nonlinear media in interaction with two optical waves associated with the amplitudes E1 and E2, with the same optical frequencies, linear polarization, and mutually coherent, but with arbitrary intensity. The vectors of the incident and self-diffracted waves are represented in Fig. 1, where z is the propagation direction, ±θ is the geometric incident angle for the propagation vectors k1 and k2 ; k3 and k4 correspond to the selfdiffracted waves; Δk is the difference in propagation constants between the zero and the first order of diffraction.

 

Fig. 1. Diagram of the propagation vectors of the incident and self-diffracted beams.

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Consider the two circular components of polarization of the electric field E⃗+ and E⃗- :

E=E++E,

for an optically isotropic sample the index of refraction for each component of polarization is:

n±n0+2πn0(AE±2+(A+B)E2),

where A=χ(3)1122 and B=χ(3)1212, are components of the tensor χ(3) [7].

We write the equation describing the amplitude of the electric field E along the nonlinear absorptive media with length D as:

E(x,D)=T̂(x)E(x,0),

with:

T̂(x)=[exp(iΨ++α(I)z2)00exp(iΨ+α(I)z2)],

where,

α(I)=α0+2βn(ε0μ0)12(E2),
Ψ±(x)=Ψ±(0)+Ψ±(1)cos2πxΛ,
Ψ±(0)=4π2Dn0λ[A(E1±2+E2±2)+(A+B)(E12+E22)],
Ψ±(1)=4π2Dn0λ[AE1±E2±*+(A+B)E1E2*].

The amplitude of transmitted and self-diffracted waves is calculated considering E(x,D) as a Fourier series, using the expression exp(iΨcosKx)=∑imJm(Ψ)exp(iKx), where K=2π/Λ is the wave number of the induced grating, Jm(Ψ) is a Bessel function of order m, and a property of the Bessel functions J-m=(-1)m Jm. We take in account that the intensity of the self-diffracted waves is much lower than the intensity of the incident waves. We obtained:

E1±(z)=[J0(Ψ±(1))E1±+iJ1(Ψ±(1))E2±]exp(iΨ±(0)+α0z2+βn(ε0μ0)12E2z)
E2±(z)=[J0(Ψ±(1))E2±iJ1(Ψ±(1))E1±]exp(iΨ±(0)+α0z2+βn(ε0μ0)12E2z)
E3±(z)=[iJ1(Ψ±(1))E1±J2(Ψ±(1))E2±]exp(iΨ±(0)+α0z2+βn(ε0μ0)12E2z)
E4±(z)=[iJ1(Ψ±(1))E2±J2(Ψ±(1))E1±]exp(iΨ±(0)+α0z2+βn(ε0μ0)12E2z)

where E and E are the amplitudes of the circular components of the incident beams; E and E are the amplitudes of the self-diffracted waves (see Fig. 1).

3. Experiment

For the preparation of the nanoparticles, a high-purity silica glass plate was implanted at room temperature with 2 MeV Au2+ ions at a fluence of around 2.8×1016 ions/cm2. After implantation the sample was thermally annealed in an oxidizing atmosphere (air) at a temperature of 1100°C for 1 hr [19]. The concentration depth distribution of Au and the ion fluences were determined by Rutherford Backscattering Spectrometry (RBS) measurements using a 2.54 MeV 4He+ beam. Ion implantation and RBS analysis were performed at the IFUNAM ’s 3MV Tandem accelerator (NEC 9SDH-2 Pelletron). The maximum of the Au depth profile distribution is located approximately 0.5 µm beneath the surface sample.

For the FWM experiment we use a frequency doubled Q-switched laser, with pulses of 7 ns FWHM and linear polarization. The maximum pulse energy in our experiment was 4 mJ with a wavelength λ=532 nm. The intensity rate I1:I2 was 1.5:1. The radius of the beam waist at the focus in the sample was measured to be 0.15 mm. Figure 2 shows the experimental set up, where L represents the focusing system of lens, BM is a beam splitter, P is a half-wave plate, M1 and M2 are mirrors, MNL is the sample, and FD1-4 are photodetectors.

 

Fig. 2. Experimental set up.

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4. Results and discussion

We observe that our samples behave like waveguides in the near infrared spectrum. However, the linear absorption is high at the UV region and at the surface plasmon resonance band associated with the presence of Au nanoparticles in the silica. Figure 3 shows the linear optical absorption spectrum for our sample and one can clearly observe the surface plasmon resonance band (520 nm) associated with the Au nanoparticles in the silica sample.

 

Fig. 3. Linear optical absorption of a silica sample containing Au nanoparticles as a function of the wavelength.

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To increase the accuracy of the measurements of nonlinear parameters of the silica with Au nanoparticles, we first calibrated our measurement system using CS2 with D=1 mm, as nonlinear medium with well known third-order nonlinear susceptibility |χ(3)|=1.9×10-12 esu [7]. The calibration allows us to reduce the measurement uncertainty associated mainly with the determination of the beam diameter. The intensities of the diffracted beams behind the silica sample containing Au nanoparticles with D=1 µm were measured when the incident waves E1 and E2 had parallel linear polarizations. The intensities measured by the four photodetectors behind the sample were I1=300 MW/cm2, I2=200 MW/cm2, I3=90 W/cm2, and I4=60 W/cm2.

Also we obtained the intensity I3 of the self-diffracted wave E3 in the FWM experiment, but in this case the polarization of the wave E1 was fixed, and the polarization of the wave E2 was rotated by the half-wave plate P. Each point in Fig. 4 corresponds to an average of 20 shots in the same experimental conditions, when the polarization of the incident wave was controlled by P (see Fig. 2).

 

Fig. 4. Intensity of I3 as a function of the angle ϕ between planes of polarization of the incident waves.

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When the nature of the sample presents an isotropic mechanism of nonlinearity of index, then B=0 in Eq. (4). In this case, according to Eq. (4), we can notice that in a FWM experiment with incident waves with orthogonal linear polarizations, there is no induced circular birefringence, neither intensity fringes, so nonlinear absorption can be considered isotropic [20]. Under these circumstances, as we can observe in Fig. 4, there is no intensity of self-diffraction when the angle of planes of polarization of the incident waves is 90°.

With Eqs. (11-14) and our experimental data we get for our sample: |χ(3)|=2.2×10-9 esu, n2=-2×10-15 m2/W, β=-5×10-8 m/W. Using these results and Eq. (2) we obtain: Reχ(3)=-1.5×10-9 esu and Imχ(3)=-1.6×10-9 esu. We noticed that the error bars for the magnitude of the nonlinear parameters obtained decrease from ±20% to ±2.5% in our results when we obtained higher self-diffracted intensities in the experiment, but in the best case it is necessary to be careful with the occurrence of optical damage in the sample.

From our calculations using Eqs. (11-14) we obtained a negative sign for n2 in our samples. Thus, we can say that a thermal effect is responsible for their nonlinearity of index, because that means that the change of index is produced by a positive isotropic change of density. The sign of β leads us to identify the absorption mechanism in the samples, being in this case saturable absorption. We expected this result taking in account the magnitude of the intensities of transmitted and self-diffracted waves in our experiments and our observations of laser ablation in similar samples. To have a comparative result we measured the transmittance of the sample that can be expressed as:

I0Ii=exp[(α0+βIi)z].

The data obtained are shown in Fig. 5. The adjusted curve was calculated from Eq. (15) using the parameters obtained from calculations by using Eq. (2). As we can see there is good agreement between the experimental and theoretical results.

 

Fig. 5. Transmitted intensity as a function of the incident intensity in the sample.

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5. Conclusions

Using a four-wave mixing experiment we obtained the sign and a significant magnitude of n2 and β for silica samples containing Au nanoparticles. To discriminate the physical mechanism of n2 from a straightforward measure of |χ(3)|, we consider the two cases of induced birefringence generated: when the polarizations of the incident waves are parallel and when they are mutually orthogonal. We indicated the presence of a saturable absorption and a thermal effect responsible for the mechanism of nonlinearity in our samples. The method presented here can be useful for optical measurements of several materials, and with some modifications it can also detect the time of response of the third order nonlinearity.

Acknowledgments

The authors would like to thank K. López and F.J. Jaimes for the accelerator operation and J.G. Morales for his assistance in the annealing of the samples. This work was partially supported by DGAPA-UNAM projects IN-108407 and IN-119706, and the CONACyT grants No. 50504 and No. 42626-F.

References and links

1. S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D.K. Hensley, “Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3,” Nucl. Instrum. Methods Phys. Res. B , 166167, 750–757 (2000). [CrossRef]  

2. F. Hache, D. Ricard, C. Flytzanis, and U. Kreibig, “The optical Kerr effect in small metal particles and metal colloids: The case of gold,” Appl. Phys. A 47, 347–357 (1988). [CrossRef]  

3. I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, “Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method,” J. Appl. Phys. 79, 1244–1249 (1996). [CrossRef]  

4. H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, “Origin of third-order optical nonlinearity in Au:SiO2 composite films on femtosecond and picosecond time scales,” Opt. Lett. 23, 388–390 (1998). [CrossRef]  

5. F. Gonella and Mazzoldi, Handbook of nanostructured materials and nanotechnology (Academic Press, San Diego), Vol. 4(Optical properties), Chap. 2 (Metal nanoclusters composite glasses), (2000).

6. J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, “Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions” Nucl. Instrum. Methods Phys. Res. B 161163, 1058–1063 (2000). [CrossRef]  

7. R. W. Boyd, Nonlinear Optics, Academic Press, San Diego (1992).

8. M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990). [CrossRef]  

9. S. R. Friberg and P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. 23, 2089–2094 (1987). [CrossRef]  

10. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in electric field strength,” Phys. Rev. 137, A801–A818 (1965). [CrossRef]  

11. M. J. Moran, C. Y. Shen, and R, L. Carman, “Interferometric measurement of nonlinear refractive-index coefficient relative to CS2 in laser-system related materials,” IEEE J. Quantum Electron. 11, 259 (1975). [CrossRef]  

12. A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. 9, 1064–1071 (1973). [CrossRef]  

13. R. Volle, V. Boucher, K. Dorkenoo, R. Chevalier, and X. Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443–451 (2000). [CrossRef]  

14. K. Akihama, T. Asai, and S. Yamasaki, “Measurement method of nonresonant third-order susceptibility by using off-resonant coherent anti-Stokes Raman spectroscopy,” Appl. Opt. 32, 7434–7441 (1993). [CrossRef]   [PubMed]  

15. L. Rodríguez, C. Simos, M. Sylla, O. Marcano, and X. Phu, “New holographic technique for third-order optical properties measurement,” Opt. Commun. 247, 453–460 (2005). [CrossRef]  

16. W. E. Williams, M. J. Soileau, and E. W. V Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984). [CrossRef]  

17. Y. Bae, J. J. Song, and Y.B. Kim, “Photoacustic study of two-photon absorption in hexagonal ZnS,” J. Appl. Phys. 53, 615–619 (1982). [CrossRef]  

18. I. Kang, T. Krauss, and F. Wise, “Sensitive measurement of nonlinear refraction and two-photon absorption by spectrally-resolved two-beam coupling,” Opt. Lett. 22, 1077–1079 (1997). [CrossRef]   [PubMed]  

19. A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, and E. Muñoz, “Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres,” Nucl. Instrum. Methods Phys. Res. B 191, 333–336 (2002). [CrossRef]  

20. C. Torres-Torres and A. V. Khomenko, “Vectorial self-diffraction of two degenerated waves in media with optical Kerr effect” (in Spanish), Revista Mexicana de Física , 51, 162–167 (2005).

References

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  1. S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
    [CrossRef]
  2. F. Hache, D. Ricard, C. Flytzanis, and U. Kreibig, "The optical Kerr effect in small metal particles and metal colloids: The case of gold," Appl. Phys. A 47, 347-357 (1988).
    [CrossRef]
  3. I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, "Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method," J. Appl. Phys. 79, 1244-1249 (1996).
    [CrossRef]
  4. H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, "Origin of third-order optical nonlinearity in Au:SiO2 composite films on femtosecond and picosecond time scales," Opt. Lett. 23, 388-390 (1998).
    [CrossRef]
  5. F. Gonella and Mazzoldi, Handbook of Nanostructured Materials and Nanotechnology (Academic Press, San Diego), Vol. 4 (Optical properties), Chap. 2 (Metal nanoclusters composite glasses), (2000).
  6. J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, "Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions" Nucl. Instrum. Methods Phys. Res. B 161-163, 1058-1063 (2000).
    [CrossRef]
  7. R. W. Boyd, Nonlinear Optics, (Academic Press, San Diego, 1992).
  8. M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, E. W. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  9. S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. 23, 2089-2094 (1987).
    [CrossRef]
  10. P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in electric field strength," Phys. Rev. 137, A801-A818 (1965).
    [CrossRef]
  11. M. J. Moran, C. Y. Shen, and R. L. Carman, "Interferometric measurement of nonlinear refractive-index coefficient relative to CS2 in laser-system related materials," IEEE J. Quantum Electron. 11, 259 (1975).
    [CrossRef]
  12. A. Owyoung, "Ellipse rotation studies in laser host materials," IEEE J. Quantum Electron. 9, 1064-1071 (1973).
    [CrossRef]
  13. R. Volle, V. Boucher, K. Dorkenoo, R. Chevalier, and X. Phu, "Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method," Opt. Commun. 182, 443-451 (2000).
    [CrossRef]
  14. K. Akihama, T. Asai, and S. Yamasaki, "Measurement method of nonresonant third-order susceptibility by using off-resonant coherent anti-Stokes Raman spectroscopy," Appl. Opt. 32, 7434-7441 (1993).
    [CrossRef] [PubMed]
  15. L. Rodríguez, C. Simos, M. Sylla, O. Marcano, and X. Phu, "New holographic technique for third-order optical properties measurement," Opt. Commun. 247, 453-460 (2005).
    [CrossRef]
  16. W. E. Williams, M. J. Soileau, and E. W. V Stryland, "Optical switching and n2 measurements in CS2," Opt. Commun. 50, 256-260 (1984).
    [CrossRef]
  17. Y. Bae, J. J. Song, and Y.B. Kim, "Photoacustic study of two-photon absorption in hexagonal ZnS," J. Appl. Phys. 53, 615-619 (1982).
    [CrossRef]
  18. I. Kang, T. Krauss, and F. Wise, "Sensitive measurement of nonlinear refraction and two-photon absorption by spectrally-resolved two-beam coupling," Opt. Lett. 22, 1077-1079 (1997).
    [CrossRef] [PubMed]
  19. A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
    [CrossRef]
  20. C. Torres-Torres and A. V. Khomenko, "Vectorial self-diffraction of two degenerated waves in media with optical Kerr effect" (in Spanish), Revista Mex. de Física,  51, 162-167 (2005).

2005 (2)

L. Rodríguez, C. Simos, M. Sylla, O. Marcano, and X. Phu, "New holographic technique for third-order optical properties measurement," Opt. Commun. 247, 453-460 (2005).
[CrossRef]

C. Torres-Torres and A. V. Khomenko, "Vectorial self-diffraction of two degenerated waves in media with optical Kerr effect" (in Spanish), Revista Mex. de Física,  51, 162-167 (2005).

2002 (1)

A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
[CrossRef]

2000 (3)

J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, "Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions" Nucl. Instrum. Methods Phys. Res. B 161-163, 1058-1063 (2000).
[CrossRef]

R. Volle, V. Boucher, K. Dorkenoo, R. Chevalier, and X. Phu, "Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method," Opt. Commun. 182, 443-451 (2000).
[CrossRef]

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

1998 (1)

1997 (1)

1996 (1)

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, "Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method," J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

1993 (1)

1990 (1)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, E. W. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

1988 (1)

F. Hache, D. Ricard, C. Flytzanis, and U. Kreibig, "The optical Kerr effect in small metal particles and metal colloids: The case of gold," Appl. Phys. A 47, 347-357 (1988).
[CrossRef]

1987 (1)

S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. 23, 2089-2094 (1987).
[CrossRef]

1984 (1)

W. E. Williams, M. J. Soileau, and E. W. V Stryland, "Optical switching and n2 measurements in CS2," Opt. Commun. 50, 256-260 (1984).
[CrossRef]

1982 (1)

Y. Bae, J. J. Song, and Y.B. Kim, "Photoacustic study of two-photon absorption in hexagonal ZnS," J. Appl. Phys. 53, 615-619 (1982).
[CrossRef]

1975 (1)

M. J. Moran, C. Y. Shen, and R. L. Carman, "Interferometric measurement of nonlinear refractive-index coefficient relative to CS2 in laser-system related materials," IEEE J. Quantum Electron. 11, 259 (1975).
[CrossRef]

1973 (1)

A. Owyoung, "Ellipse rotation studies in laser host materials," IEEE J. Quantum Electron. 9, 1064-1071 (1973).
[CrossRef]

1965 (1)

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

Akihama, K.

Asai, T.

Bae, Y.

Y. Bae, J. J. Song, and Y.B. Kim, "Photoacustic study of two-photon absorption in hexagonal ZnS," J. Appl. Phys. 53, 615-619 (1982).
[CrossRef]

Banks, C.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Boucher, V.

R. Volle, V. Boucher, K. Dorkenoo, R. Chevalier, and X. Phu, "Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method," Opt. Commun. 182, 443-451 (2000).
[CrossRef]

Carman, R. L.

M. J. Moran, C. Y. Shen, and R. L. Carman, "Interferometric measurement of nonlinear refractive-index coefficient relative to CS2 in laser-system related materials," IEEE J. Quantum Electron. 11, 259 (1975).
[CrossRef]

Cheang-Wong, J. C.

A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
[CrossRef]

J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, "Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions" Nucl. Instrum. Methods Phys. Res. B 161-163, 1058-1063 (2000).
[CrossRef]

Chevalier, R.

R. Volle, V. Boucher, K. Dorkenoo, R. Chevalier, and X. Phu, "Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method," Opt. Commun. 182, 443-451 (2000).
[CrossRef]

Crespo-Sosa, A.

A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
[CrossRef]

J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, "Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions" Nucl. Instrum. Methods Phys. Res. B 161-163, 1058-1063 (2000).
[CrossRef]

Curley, M. J.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Dorkenoo, K.

R. Volle, V. Boucher, K. Dorkenoo, R. Chevalier, and X. Phu, "Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method," Opt. Commun. 182, 443-451 (2000).
[CrossRef]

Espejel-Morales, R.

J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, "Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions" Nucl. Instrum. Methods Phys. Res. B 161-163, 1058-1063 (2000).
[CrossRef]

Flytzanis, C.

F. Hache, D. Ricard, C. Flytzanis, and U. Kreibig, "The optical Kerr effect in small metal particles and metal colloids: The case of gold," Appl. Phys. A 47, 347-357 (1988).
[CrossRef]

Friberg, S. R.

S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. 23, 2089-2094 (1987).
[CrossRef]

Fu, J. S.

Hache, F.

F. Hache, D. Ricard, C. Flytzanis, and U. Kreibig, "The optical Kerr effect in small metal particles and metal colloids: The case of gold," Appl. Phys. A 47, 347-357 (1988).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, E. W. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Hensley, D. K.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Hernández, J. M.

A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
[CrossRef]

J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, "Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions" Nucl. Instrum. Methods Phys. Res. B 161-163, 1058-1063 (2000).
[CrossRef]

Ila, D.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Kang, I.

Khomenko, A. V.

C. Torres-Torres and A. V. Khomenko, "Vectorial self-diffraction of two degenerated waves in media with optical Kerr effect" (in Spanish), Revista Mex. de Física,  51, 162-167 (2005).

Kim, Y.B.

Y. Bae, J. J. Song, and Y.B. Kim, "Photoacustic study of two-photon absorption in hexagonal ZnS," J. Appl. Phys. 53, 615-619 (1982).
[CrossRef]

Krauss, T.

Kreibig, U.

F. Hache, D. Ricard, C. Flytzanis, and U. Kreibig, "The optical Kerr effect in small metal particles and metal colloids: The case of gold," Appl. Phys. A 47, 347-357 (1988).
[CrossRef]

Liao, H. B.

Maker, P. D.

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

Manabe, Y.

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, "Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method," J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

Marcano, O.

L. Rodríguez, C. Simos, M. Sylla, O. Marcano, and X. Phu, "New holographic technique for third-order optical properties measurement," Opt. Commun. 247, 453-460 (2005).
[CrossRef]

Moran, M. J.

M. J. Moran, C. Y. Shen, and R. L. Carman, "Interferometric measurement of nonlinear refractive-index coefficient relative to CS2 in laser-system related materials," IEEE J. Quantum Electron. 11, 259 (1975).
[CrossRef]

Muñoz, E.

A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
[CrossRef]

J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, "Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions" Nucl. Instrum. Methods Phys. Res. B 161-163, 1058-1063 (2000).
[CrossRef]

Nakamura, A.

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, "Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method," J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

Oliver, A.

A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
[CrossRef]

J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, "Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions" Nucl. Instrum. Methods Phys. Res. B 161-163, 1058-1063 (2000).
[CrossRef]

Owyoung, A.

A. Owyoung, "Ellipse rotation studies in laser host materials," IEEE J. Quantum Electron. 9, 1064-1071 (1973).
[CrossRef]

Phu, X.

L. Rodríguez, C. Simos, M. Sylla, O. Marcano, and X. Phu, "New holographic technique for third-order optical properties measurement," Opt. Commun. 247, 453-460 (2005).
[CrossRef]

R. Volle, V. Boucher, K. Dorkenoo, R. Chevalier, and X. Phu, "Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method," Opt. Commun. 182, 443-451 (2000).
[CrossRef]

Poker, D. B.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Ricard, D.

F. Hache, D. Ricard, C. Flytzanis, and U. Kreibig, "The optical Kerr effect in small metal particles and metal colloids: The case of gold," Appl. Phys. A 47, 347-357 (1988).
[CrossRef]

Rodríguez, L.

L. Rodríguez, C. Simos, M. Sylla, O. Marcano, and X. Phu, "New holographic technique for third-order optical properties measurement," Opt. Commun. 247, 453-460 (2005).
[CrossRef]

Rodríguez-Fernández, L.

A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
[CrossRef]

Roiz, J.

A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, E. W. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Sarkisov, S. S.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Sasaki, S.

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, "Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method," J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, E. W. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Shen, C. Y.

M. J. Moran, C. Y. Shen, and R. L. Carman, "Interferometric measurement of nonlinear refractive-index coefficient relative to CS2 in laser-system related materials," IEEE J. Quantum Electron. 11, 259 (1975).
[CrossRef]

Simos, C.

L. Rodríguez, C. Simos, M. Sylla, O. Marcano, and X. Phu, "New holographic technique for third-order optical properties measurement," Opt. Commun. 247, 453-460 (2005).
[CrossRef]

Smith, P. W.

S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. 23, 2089-2094 (1987).
[CrossRef]

Soileau, M. J.

W. E. Williams, M. J. Soileau, and E. W. V Stryland, "Optical switching and n2 measurements in CS2," Opt. Commun. 50, 256-260 (1984).
[CrossRef]

Song, J. J.

Y. Bae, J. J. Song, and Y.B. Kim, "Photoacustic study of two-photon absorption in hexagonal ZnS," J. Appl. Phys. 53, 615-619 (1982).
[CrossRef]

Stryland, E. W. V

W. E. Williams, M. J. Soileau, and E. W. V Stryland, "Optical switching and n2 measurements in CS2," Opt. Commun. 50, 256-260 (1984).
[CrossRef]

Stryland, E. W. V.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, E. W. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Svetchnikov, V. L.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Sylla, M.

L. Rodríguez, C. Simos, M. Sylla, O. Marcano, and X. Phu, "New holographic technique for third-order optical properties measurement," Opt. Commun. 247, 453-460 (2005).
[CrossRef]

Tanahashi, I.

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, "Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method," J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

Terhune, R. W.

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

Tohda, T.

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, "Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method," J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

Torres-Torres, C.

C. Torres-Torres and A. V. Khomenko, "Vectorial self-diffraction of two degenerated waves in media with optical Kerr effect" (in Spanish), Revista Mex. de Física,  51, 162-167 (2005).

Volle, R.

R. Volle, V. Boucher, K. Dorkenoo, R. Chevalier, and X. Phu, "Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method," Opt. Commun. 182, 443-451 (2000).
[CrossRef]

Wang, H.

Wang, J. -C.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Wei, T.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, E. W. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Williams, E. K.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Williams, W. E.

W. E. Williams, M. J. Soileau, and E. W. V Stryland, "Optical switching and n2 measurements in CS2," Opt. Commun. 50, 256-260 (1984).
[CrossRef]

Wise, F.

Wong, G. K. L.

Wong, K. S.

Xiao, R. F.

Yamasaki, S.

Zandbergen, H. W.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Zykov, G. A.

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. A (1)

F. Hache, D. Ricard, C. Flytzanis, and U. Kreibig, "The optical Kerr effect in small metal particles and metal colloids: The case of gold," Appl. Phys. A 47, 347-357 (1988).
[CrossRef]

IEEE J. Quantum Electron. (4)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, E. W. V. Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. 23, 2089-2094 (1987).
[CrossRef]

M. J. Moran, C. Y. Shen, and R. L. Carman, "Interferometric measurement of nonlinear refractive-index coefficient relative to CS2 in laser-system related materials," IEEE J. Quantum Electron. 11, 259 (1975).
[CrossRef]

A. Owyoung, "Ellipse rotation studies in laser host materials," IEEE J. Quantum Electron. 9, 1064-1071 (1973).
[CrossRef]

J. Appl. Phys. (2)

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, "Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method," J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

Y. Bae, J. J. Song, and Y.B. Kim, "Photoacustic study of two-photon absorption in hexagonal ZnS," J. Appl. Phys. 53, 615-619 (1982).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. B (3)

A. Oliver, J. C. Cheang-Wong, J. Roiz, L. Rodríguez-Fernández, J. M. Hernández, A. Crespo-Sosa, E. Muñoz, "Metallic nanoparticle formation in ion-implanted silica after annealing in reducing or oxidizing atmospheres," Nucl. Instrum. Methods Phys. Res. B 191, 333-336 (2002).
[CrossRef]

S. S. Sarkisov, M. J. Curley, E. K. Williams, D. Ila, V. L. Svetchnikov, H. W. Zandbergen, G. A. Zykov, C. Banks, J. -C. Wang, D. B. Poker, and D. K. Hensley, "Nonlinear optical waveguides produced by MeV ion implantation in LiNbO3," Nucl. Instrum. Methods Phys. Res. B,  166-167, 750-757 (2000).
[CrossRef]

J. C. Cheang-Wong, A. Oliver, A. Crespo-Sosa, J. M. Hernández, E. Muñoz, and R. Espejel-Morales, "Dependence of the optical properties on the ion implanted depth profiles in fused quartz after a sequential implantation with Si and Au ions" Nucl. Instrum. Methods Phys. Res. B 161-163, 1058-1063 (2000).
[CrossRef]

Opt. Commun. (3)

R. Volle, V. Boucher, K. Dorkenoo, R. Chevalier, and X. Phu, "Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method," Opt. Commun. 182, 443-451 (2000).
[CrossRef]

L. Rodríguez, C. Simos, M. Sylla, O. Marcano, and X. Phu, "New holographic technique for third-order optical properties measurement," Opt. Commun. 247, 453-460 (2005).
[CrossRef]

W. E. Williams, M. J. Soileau, and E. W. V Stryland, "Optical switching and n2 measurements in CS2," Opt. Commun. 50, 256-260 (1984).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

Revista Mex. de Física (1)

C. Torres-Torres and A. V. Khomenko, "Vectorial self-diffraction of two degenerated waves in media with optical Kerr effect" (in Spanish), Revista Mex. de Física,  51, 162-167 (2005).

Other (2)

R. W. Boyd, Nonlinear Optics, (Academic Press, San Diego, 1992).

F. Gonella and Mazzoldi, Handbook of Nanostructured Materials and Nanotechnology (Academic Press, San Diego), Vol. 4 (Optical properties), Chap. 2 (Metal nanoclusters composite glasses), (2000).

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Figures (5)

Fig. 1.
Fig. 1.

Diagram of the propagation vectors of the incident and self-diffracted beams.

Fig. 2.
Fig. 2.

Experimental set up.

Fig. 3.
Fig. 3.

Linear optical absorption of a silica sample containing Au nanoparticles as a function of the wavelength.

Fig. 4.
Fig. 4.

Intensity of I3 as a function of the angle ϕ between planes of polarization of the incident waves.

Fig. 5.
Fig. 5.

Transmitted intensity as a function of the incident intensity in the sample.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

χ ( 3 ) = ( Re χ ( 3 ) ) 2 + ( Im χ ( 3 ) ) 2 .
χ ( 3 ) = n 0 c 7.91 × 10 2 n 2 + i n 0 2 c λ π 2 β ,
E = E + + E ,
n ± n 0 + 2 π n 0 ( A E ± 2 + ( A + B ) E 2 ) ,
E ( x , D ) = T ̂ ( x ) E ( x , 0 ) ,
T ̂ ( x ) = [ exp ( i Ψ + + α ( I ) z 2 ) 0 0 exp ( i Ψ + α ( I ) z 2 ) ] ,
α ( I ) = α 0 + 2 β n ( ε 0 μ 0 ) 1 2 ( E 2 ) ,
Ψ ± ( x ) = Ψ ± ( 0 ) + Ψ ± ( 1 ) cos 2 π x Λ ,
Ψ ± ( 0 ) = 4 π 2 D n 0 λ [ A ( E 1 ± 2 + E 2 ± 2 ) + ( A + B ) ( E 1 2 + E 2 2 ) ] ,
Ψ ± ( 1 ) = 4 π 2 D n 0 λ [ A E 1 ± E 2 ± * + ( A + B ) E 1 E 2 * ] .
E 1 ± ( z ) = [ J 0 ( Ψ ± ( 1 ) ) E 1 ± + i J 1 ( Ψ ± ( 1 ) ) E 2 ± ] exp ( i Ψ ± ( 0 ) + α 0 z 2 + β n ( ε 0 μ 0 ) 1 2 E 2 z )
E 2 ± ( z ) = [ J 0 ( Ψ ± ( 1 ) ) E 2 ± i J 1 ( Ψ ± ( 1 ) ) E 1 ± ] exp ( i Ψ ± ( 0 ) + α 0 z 2 + β n ( ε 0 μ 0 ) 1 2 E 2 z )
E 3 ± ( z ) = [ iJ 1 ( Ψ ± ( 1 ) ) E 1 ± J 2 ( Ψ ± ( 1 ) ) E 2 ± ] exp ( i Ψ ± ( 0 ) + α 0 z 2 + β n ( ε 0 μ 0 ) 1 2 E 2 z )
E 4 ± ( z ) = [ i J 1 ( Ψ ± ( 1 ) ) E 2 ± J 2 ( Ψ ± ( 1 ) ) E 1 ± ] exp ( i Ψ ± ( 0 ) + α 0 z 2 + β n ( ε 0 μ 0 ) 1 2 E 2 z )
I 0 I i = exp [ ( α 0 + β I i ) z ] .

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