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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 24 — Nov. 28, 2005
  • pp: 9788–9795
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Energy transfer between incident laser and elastically backscattered waves in nonlinear absorption media

Richard L. Sutherland  »View Author Affiliations


Optics Express, Vol. 13, Issue 24, pp. 9788-9795 (2005)
http://dx.doi.org/10.1364/OPEX.13.009788


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Abstract

A model of energy transfer by nearly degenerate two-beam coupling in media exhibiting two-photon and excited state absorption is presented and discussed. The two beams include an incident laser, which has a nonlinear time dependent phase, and an elastically backscattered wave derived from the incident laser. Superposition of these two waves creates an index modulation, arising from a spatially modulated, two-photon-created excited-state population, which couples the forward and backward propagating waves. Reflectance of the stimulated wave and transmittance of the incident laser are computed, including the effects of two-photon absorption and excited state absorption as well as energy transfer between the two beams, and the relevance of the results to experimental measurements is discussed. The backscattered wave has a frequency that is unshifted with respect to that of the incident laser, and the small signal gain is proportional to the square of the incident laser intensity. The different effects due to multimode and chirped waves are also discussed.

© 2005 Optical Society of America

1. Introduction

Two-beam coupling (TBC) has a rich history in nonlinear optics. Early studies established two conditions for one-way energy transfer between two electromagnetic waves in a Kerr medium: 1) the frequencies must be nondegenerate, and 2) the nonlinearity of the medium must have a finite response time [1

1 . Y. Silberberg and I. Bar Joseph , “ Optical instabilities in a nonlinear Kerr medium ,” J. Opt. Soc. Am. B 1 , 662 – 670 ( 1984 ). [CrossRef]

]. The nonlinear refractive index is usually assumed to obey a simple Debye relaxation model. Energy transfer under these conditions is always from the high frequency beam to the low frequency beam for a positive Kerr nonlinearity. Silberberg and Joseph showed that counter-propagating beams in such a Kerr medium can exhibit optical instabilities and self-oscillation [1

1 . Y. Silberberg and I. Bar Joseph , “ Optical instabilities in a nonlinear Kerr medium ,” J. Opt. Soc. Am. B 1 , 662 – 670 ( 1984 ). [CrossRef]

,2

2 . Y. Silberberg and I. Bar Joseph , “ Instabilities, self-oscillation, and chaos in simple nonlinear optical interaction ,” Phys. Rev. Lett. 48 , 1541 – 1543 ( 1982 ). [CrossRef]

], while Yeh gave an exact solution for energy transfer between two co-propagating beams including the effect of linear absorption [3

3 . P. Yeh , “ Exact solution of a nonlinear model of two-wave mixing in Kerr media ,” J. Opt. Soc. Am. B 3 , 747 – 750 ( 1986 ). [CrossRef]

]. In each of these studies, both beams had non-zero input values.

TBC also describes several stimulated scattering phenomena, such as stimulated Raman (SRS), Brillouin (SBS), and Rayleigh-wing scattering (SRWS). The Stokes (down-shifted frequency) wave in these phenomena arises internally from scattered light and experiences gain at the expense of the incident laser beam. The Stokes frequency shift is a normal mode of the medium (e.g., vibrational) for SRS, an acoustic frequency for SBS, and the inverse reorientation time of an anisotropic molecule for SRWS. SRS, SBS, and SRWS are resonant phenomena, indicating that the nonlinear medium has a finite response time. In each of these stimulated scattering effects the small-signal gain of the Stokes wave is proportional to the laser intensity [4

4 . R. Boyd , Nonlinear Optics ( Academic Press, New York , 1992 ).

].

Recently, He et al. reported the observation of stimulated backscattering in a two-photon absorption (TPA) medium, where the frequency of the stimulated wave was identical to the incident laser frequency (within the resolution of their interferometer), and the small-signal gain was quadratic in the incident laser intensity [5

5 . G. S. He , T. -C. Lin , and P. N. Prasad , “ Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation ,” Opt. Express 12 , 5952 – 5961 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-5952 . [CrossRef] [PubMed]

,6

6 . G. S. He , C. Lu , Q. Zheng , P. N. Prasad , P. Zerom , R. W. Boyd , and M. Samoc , “ Stimulated Rayleigh-Bragg scattering in two-photon absorbing media ,” Phys. Rev. A 71 , 063810 ( 2005 ). [CrossRef]

], in contrast to the stimulated light scattering phenomena described above. They considered a stimulated thermal Rayleigh scattering model based on TPA-enhanced temperature and density fluctuations, but ruled this out due to (a) the broad linewidth of their pump laser, which would severely reduce the gain of the stimulated wave, and (b) the fact that the peak gain in this theory is for an anti-Stokes-shifted wave, contrary to their experimental results. They concluded that the stimulated wave was due to a Bragg grating formed by the superposition of the incident laser beam and an elastically (Rayleigh) backscattered wave, creating an index grating via a TPA-resonance enhanced nonlinear index coefficient n 2, i.e., a Kerr effect. The stimulated backscattered wave is then due to reflection of the laser from this grating.

Here I present a simple model that is in agreement with the main results of the observed stimulated backscattering in TPA media: the frequency of the stimulated wave is unshifted with respect to that of the incident laser, and the small-signal gain is quadratic in the laser intensity. Two assumptions are made: 1) the incident laser has a nonlinear time dependent phase, consistent with a relatively broad linewidth, and 2) TPA produces a non-negligible population of an excited state of the medium [7

7 . R. L. Sutherland , M. C. Brant , J. Heinrichs , J. E. Rogers , J. E. Slagle , D. G. McLean , and P. A. Fleitz “ Excited state characterization and effective three-photon absorption model of two-photon-induced excited state absorption in organic push-pull charge-transfer chromophores ,” J. Opt. Soc. Am. B 22 , 1939 – 1948 ( 2005 ). [CrossRef]

,8

8 . J. E. Rogers , J. E. Slagle , D. G. McLean , R. L. Sutherland , B. Sankaran , R. Kannan , L. -S. Tan , and P. A. Fleitz , “ Understanding the one-photon photophysical properties of a two-photon absorbing chromophore ,” J. Phys. Chem. A 108 , 5514 – 5520 ( 2004 ). [CrossRef]

].

2. Model

The interaction of the two chirped waves is illustrated schematically in Fig. 1. Let the total field in a medium of length d be described by

Ezt=AL(z,t+τ)exp{i[kzωtb(t+τ)2]}
+AS(z,tτ)exp{i[kzωtb(tτ)2]}+c.c.
(1)

where k = /c, n is the linear refractive index, and 2τ= 2n(d - z)/c is the relative time delay at position z and time t between the forward propagating laser wave (L) and the backward propagating scattered wave (S). I assume that the scattered wave originates at z = d by some elastic scattering process, so AS (d,t) = √AL (d,t) where η is a constant << 1. Consequently, the scattered wave has the same spectral composition as the incident laser wave. However, for zd a lower frequency part of the scattered wave is always interacting with a higher frequency part of the incident wave. The total polarization of the medium is given by

P=ε0(χg(1)+(NeN)Δχ(1)+3χ(3)E2)E
(2)

where χ (n) is the n-th order susceptibility, and the angular brackets indicate an average over a time longer than an optical period but shorter than (2)-1. I have assumed that TPA produces a single excited state of number density Ne << N, the total number density of the nonlinear chromophore. I have also assumed an isotropic medium with linearly polarized light for simplicity, and both Δχ(1) =χe(1) - χg(1) and χ (3) are complex quantities (g signifies the ground state of the medium). The excited state decays back to the ground state with a time constant Te , so Ne obeys the following kinetic equation:

Net=σ2NI22ħωNeTe
(3)

where σ 2 = β/N is the TPA cross section (β is the TPA coefficient), and I(z,t) = 2ε 0 ncE 2(z,t〉 is the total intensity. Let the amplitudes be slowly varying in time compared to Te . Equation (3) can then be integrated to yield

Fig. 1. Schematic diagram of the chirped pulse interaction in a nonlinear medium. The backscattered pulse is assumed to be derived from the incident laser pulse at the back of the medium where it has the same frequency as the incident pulse. The color schematically indicates the frequency chirp, with blue representing higher frequencies and red representing lower frequencies.
NeN=C0+{C1exp[i(2kz4bτt)]+C2exp[i(4kz8bτt)]+c.c.},
(4a)
C0=σ2Te2ħω(IL2+IS2+4ILIS),
(4b)
C1=σ2Te2ħω(IL+IS)(4ε0ncALAS*1i4Te),
(4c)
C2=σ2Te2ħω(2ε0ncALAS*)21i8Te,
(4d)

where ε 0 is the free-space permittivity. I will make the assumption that (4bτTe )2 << 1.

Following a procedure directly analogous to that of Yeh [3

3 . P. Yeh , “ Exact solution of a nonlinear model of two-wave mixing in Kerr media ,” J. Opt. Soc. Am. B 3 , 747 – 750 ( 1986 ). [CrossRef]

] and Boyd [4

4 . R. Boyd , Nonlinear Optics ( Academic Press, New York , 1992 ).

] for non-degenerate TBC, Eqs. (4a)–(4d) are substituted into Eq. (2), and then both Eqs. (1) and (2) are substituted into Maxwell’s wave equation in the slowly varying amplitude approximation. Matching up synchronous terms, the following coupled-wave equations for the laser and backscattered intensities are derived:

dILdz=g(1zd)(IL+IS)ILISγeff(IL2+3IS2+6ILIS)ILβ(IL+2IS)IL
(5a)
dISdz=g(1zd)(IL+IS)ILIS+γeff(3IL2+IS2+6ILIS)IS+β(2IL+IS)IS
(5b)

where g and γeff are the backward wave gain and effective three-photon absorption (3PA) coefficients, respectively, with

g=8bTeωΔχR(1)dc2Isat2,
(6)
γeff=NΔσIsat2,
(7)

where ΔχR(1) = Re(Δχ (1)), Δσ = σe -σg is the difference between the linear absorption cross sections of the excited and ground states, and Isat = (2ħω/σ 2 Te )1/2 is the two-photon saturation intensity. Note that g = 0 when b = 0 (no chirp). Hence, without chirp there can be no growth of the backward scattered wave. The gain g also carries the sign of b (positive or negative chirp). For a negatively chirped pulse, the scattered wave will be attenuated. For the rest of the paper, I will assume that b > 0.

3. Results and discussion

Let us examine Eqs. (5a) and (5b) in the case when IS << IL ≈ constant. It can be seen that the backward stimulated wave will experience exponential growth when IL>2β(12g3γeff). This defines the threshold condition for stimulated backscattering when there is no linear absorption. From Eq. (5b), the small signal gain G, given in terms of ΔIS = IS (0)-IS (d), is G = ΔIS /IS (d) ∝ IL2 for input intensities where the gain dominates TPA. Also, the reflectance of the scattered wave is defined by R = IS (0)/IL (0), and the change ΔR=R-η = ΔIS /ILIS (d)ILGIS (d)/IL . These results are in agreement with the data presented by He et al. for the measured small-signal gain and reflectance in a 0.01-M solution of PRL 802 in THF [6

6 . G. S. He , C. Lu , Q. Zheng , P. N. Prasad , P. Zerom , R. W. Boyd , and M. Samoc , “ Stimulated Rayleigh-Bragg scattering in two-photon absorbing media ,” Phys. Rev. A 71 , 063810 ( 2005 ). [CrossRef]

].

Under conditions where the nonlinear absorption terms (β and γeff ) can be ignored, Eqs. (5a) and (5b) can be solved analytically. The result can be expressed as

η0=R[(1R)(1R+η0)+2η02]12(1+R)2exp(Γ)(1R+2η01R+η0)32,
(8)

where η 0 = IS (d)/IL (0), and

Γ=12(1R)2gIL2(0)d.
(9)

Note that in general η 0η, although when the gain is sufficiently small η 0η. Also, when the gain is not too large so that (1 - R) >> η 0, Eq. (8) can be simplified to the following approximation:

η0R(1R)(1+R)2exp(Γ).
(10)

It is interesting to compare this result to the case of SBS, for which the denominator of Eq. (10) becomes exp(Γ)-,R with Γ→(1-R)gBIL (0)d , where gB is the Brillouin gain coefficient [4

4 . R. Boyd , Nonlinear Optics ( Academic Press, New York , 1992 ).

]. Thus, in the case of negligible loss by absorption, the backscattered reflectance as a function of incident laser intensity will look similar to the Brillouin reflectance. A plot of the reflectance given by Eq. (10) for a constant input IS (d)/IL (0), over a range of Γ where the approximation is valid, is shown in Fig. 2.

Numerical solutions of Eqs. (5a) and (5b) are also given in Fig. 2. These illustrate the agreement of the approximation given by Eq. (10) with the exact result, and the effect of including the nonlinear absorption terms. Figure 3 gives IS (0) as a function of IL (0). Parameters have been adjusted to yield results close to the experimental data of He et al [5

5 . G. S. He , T. -C. Lin , and P. N. Prasad , “ Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation ,” Opt. Express 12 , 5952 – 5961 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-5952 . [CrossRef] [PubMed]

,6

6 . G. S. He , C. Lu , Q. Zheng , P. N. Prasad , P. Zerom , R. W. Boyd , and M. Samoc , “ Stimulated Rayleigh-Bragg scattering in two-photon absorbing media ,” Phys. Rev. A 71 , 063810 ( 2005 ). [CrossRef]

]. The coefficients used for the solid curve in Fig. 3 are β = 9.46 cm/GW (the value quoted in Ref 6

6 . G. S. He , C. Lu , Q. Zheng , P. N. Prasad , P. Zerom , R. W. Boyd , and M. Samoc , “ Stimulated Rayleigh-Bragg scattering in two-photon absorbing media ,” Phys. Rev. A 71 , 063810 ( 2005 ). [CrossRef]

), η = 0.04, g = 1800 cm3/GW2, and γeff = 120 cm3/GW2, with d = 1 cm. In Figure 4 the corresponding plot for the laser transmittance is given. Here the transmittance is the ratio of output power to input power. To compute this, I assumed a Gaussian dependence for the input intensity and then integrated the output intensity over the area of the cylindrically symmetric beam. The result is compared with what would be expected for pure TPA (no TBC or ESA). The departure from pure TPA in this case is due primarily to energy transfer from the laser to the backscattered beam. Figure 3 also shows a somewhat different result (dashed curve) yielding comparable backscattered intensity. Often the effective TPA cross section measured in the nanosecond regime is a factor ~ 100 larger than the intrinsic cross section (usually measured in the femtosecond regime) [7

7 . R. L. Sutherland , M. C. Brant , J. Heinrichs , J. E. Rogers , J. E. Slagle , D. G. McLean , and P. A. Fleitz “ Excited state characterization and effective three-photon absorption model of two-photon-induced excited state absorption in organic push-pull charge-transfer chromophores ,” J. Opt. Soc. Am. B 22 , 1939 – 1948 ( 2005 ). [CrossRef]

]. Thus, for these calculations I chose β = 0.09 cm/GW. The other parameters are η = 0.02, g = 2860 cm3/GW2, and γeff = 300 cm3/GW2. TPA is low in this case, but the loss due to ESA significantly slows down the growth of the backscattered wave at higher intensities.

To get an order of magnitude of the numbers involved, consider the experiment of He et al. [5

5 . G. S. He , T. -C. Lin , and P. N. Prasad , “ Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation ,” Opt. Express 12 , 5952 – 5961 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-5952 . [CrossRef] [PubMed]

,6

6 . G. S. He , C. Lu , Q. Zheng , P. N. Prasad , P. Zerom , R. W. Boyd , and M. Samoc , “ Stimulated Rayleigh-Bragg scattering in two-photon absorbing media ,” Phys. Rev. A 71 , 063810 ( 2005 ). [CrossRef]

] and the results shown in Figs. 3 and 4 for β= 9.46 cm/GW. For a laser wavelength of 532 nm and Te ~ 1 ns [5

5 . G. S. He , T. -C. Lin , and P. N. Prasad , “ Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation ,” Opt. Express 12 , 5952 – 5961 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-5952 . [CrossRef] [PubMed]

,8

8 . J. E. Rogers , J. E. Slagle , D. G. McLean , R. L. Sutherland , B. Sankaran , R. Kannan , L. -S. Tan , and P. A. Fleitz , “ Understanding the one-photon photophysical properties of a two-photon absorbing chromophore ,” J. Phys. Chem. A 108 , 5514 – 5520 ( 2004 ). [CrossRef]

], Isat ~ 700 MW/cm2. For a severely chirped pulse the laser linewidth is Δω L ~ 2L where τL is the laser pulse width. For Δω L/2π~ 24 GHz (0.8 cm-1), τL ~ 10 ns, and N = 6 × 1018 cm-3 (0.01-M concentration), Eqs. (6) and (7) yield ΔχR(1) ~ 4 × 10-3 and Δσ ~ 1 × 10-17 cm2. I note that this numerical example contravenes the approximation made earlier for which (4bτTe )2 is small compared to 1. To account for deviations due to this, the gain and ESA terms in Eqs. (5a) and (5b) would need to be modified by the inclusion of the factor [1 + (4b τTe )2]-1. For example, in the negligible nonlinear absorption case, Eq. (9) would be multiplied by a factor ln(1 + α)/α, where α= (4bTend/c)2, which is ~ 1 for α<< 1. In the present numerical example, this factor is ~ 0.5. Consequently, the estimates for ΔχR(1) and Δσ should be increased by a factor ~ 2.

Fig. 2. Backscattered reflectance as a function of exponential gain factor for various values of nonlinear absorption coefficients. IS (d)/IL (0) = 10-2 in all cases. Circles give values for R calculated by the approximation in Eq. (10).
Fig. 3. Backscattered intensity as a function of incident laser intensity.
Fig. 4. Nonlinear transmittance of the incident laser as a function of incident intensity.

Notice that Re(χ (3)), or n 2, does not appear in the gain coefficient of Eq. (6). The reason for this is that χ (3) was assumed to have an instantaneous response [Im(χ (3)) ∝ β]. There is no two-beam coupling in a Kerr medium with an instantaneous response [3

3 . P. Yeh , “ Exact solution of a nonlinear model of two-wave mixing in Kerr media ,” J. Opt. Soc. Am. B 3 , 747 – 750 ( 1986 ). [CrossRef]

,4

4 . R. Boyd , Nonlinear Optics ( Academic Press, New York , 1992 ).

]. It is possible however, under TPA resonant conditions, that n 2 could have a finite response time (i.e., χ (3) is complex with a finite damping coefficient [4

4 . R. Boyd , Nonlinear Optics ( Academic Press, New York , 1992 ).

]). However, if this is included in the development of the coupled-wave intensity equations, it would lead to a term in the gain G that is proportional to IL , not IL2, which would be inconsistent with the experimental results of He et al. [6

6 . G. S. He , C. Lu , Q. Zheng , P. N. Prasad , P. Zerom , R. W. Boyd , and M. Samoc , “ Stimulated Rayleigh-Bragg scattering in two-photon absorbing media ,” Phys. Rev. A 71 , 063810 ( 2005 ). [CrossRef]

]. It is also quite likely that the Kerr refractive term would be much smaller than the term proportional to ΔχR(1).

It should be noted at this stage that linear chirp is not a unique requirement for the energy transfer between incident laser and elastically backscattered waves. Higher order chirp terms in the nonlinear time dependent phase have been ignored for simplicity but will make additional contributions to the beam coupling. Another potential mechanism involves multimode beams. In this case, though, the spectral nature of the backscattered beam will not match that of the incident beam. Take the simple case where the incident laser wave consists of two modes: ω 0 and ω 1 = ω 0 + δω > ω 0 (δω << ω 0). Employing the mechanism involving a TPA-populated excited state described above, there will be energy transfer from the ω 1 mode of the incident laser beam to the ω 0 mode of the backscattered beam [4

4 . R. Boyd , Nonlinear Optics ( Academic Press, New York , 1992 ).

]. Likewise, the ω 1 mode of the backscattered wave will yield its energy to the ω 0 mode of the laser wave. The backscattered wave will thus be single-mode and its spectrum conspicuously different from the incident laser. However, if the modes in the incident laser beam are equally chirped, both modes of the backscattered wave will be amplified, but not equally. When the laser linewidth is determined primarily by the linear chirp (2L >> δω), the spectrum of the backscattered wave will superficially resemble that of the incident laser, but the energy distribution amongst the modes will differ. This will generally be the case also when the number of modes is greater than two.

4. Conclusions

In summary, I have presented a model of two-beam coupling in a nonlinear absorption medium whereby energy is transferred from an incident laser beam to an elastically backscattered beam. In the mechanism proposed, the incident laser has a nonlinear time dependent phase and populates an excited state of the medium by two-photon absorption. The incident and backscattered waves superpose to form an interference pattern, leading to the formation of a Bragg grating. The Bragg grating consists of an index modulation resulting from a modulation of the excited state population. This grating is out of phase with the interference pattern which forms it due to the finite lifetime of the two-photon-populated excited state and the frequency chirp of the two waves. Energy flows one-way from the higher frequency part to the lower frequency part of the coupled waves. Complete energy conversion to the backscattered wave is prohibited, however, in part because of loss due to two-photon and excited state absorption. Nevertheless, the power spectrum of the backscattered wave can be nearly identical to that of the incident wave.

Acknowledgments

I gratefully acknowledge the Air Force Office of Scientific Research (AFOSR/NL) for their support of this work as well as support from the Air Force Research Laboratory, Materials and Manufacturing Directorate (AFRL/ML).

References and links

1 .

Y. Silberberg and I. Bar Joseph , “ Optical instabilities in a nonlinear Kerr medium ,” J. Opt. Soc. Am. B 1 , 662 – 670 ( 1984 ). [CrossRef]

2 .

Y. Silberberg and I. Bar Joseph , “ Instabilities, self-oscillation, and chaos in simple nonlinear optical interaction ,” Phys. Rev. Lett. 48 , 1541 – 1543 ( 1982 ). [CrossRef]

3 .

P. Yeh , “ Exact solution of a nonlinear model of two-wave mixing in Kerr media ,” J. Opt. Soc. Am. B 3 , 747 – 750 ( 1986 ). [CrossRef]

4 .

R. Boyd , Nonlinear Optics ( Academic Press, New York , 1992 ).

5 .

G. S. He , T. -C. Lin , and P. N. Prasad , “ Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation ,” Opt. Express 12 , 5952 – 5961 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-5952 . [CrossRef] [PubMed]

6 .

G. S. He , C. Lu , Q. Zheng , P. N. Prasad , P. Zerom , R. W. Boyd , and M. Samoc , “ Stimulated Rayleigh-Bragg scattering in two-photon absorbing media ,” Phys. Rev. A 71 , 063810 ( 2005 ). [CrossRef]

7 .

R. L. Sutherland , M. C. Brant , J. Heinrichs , J. E. Rogers , J. E. Slagle , D. G. McLean , and P. A. Fleitz “ Excited state characterization and effective three-photon absorption model of two-photon-induced excited state absorption in organic push-pull charge-transfer chromophores ,” J. Opt. Soc. Am. B 22 , 1939 – 1948 ( 2005 ). [CrossRef]

8 .

J. E. Rogers , J. E. Slagle , D. G. McLean , R. L. Sutherland , B. Sankaran , R. Kannan , L. -S. Tan , and P. A. Fleitz , “ Understanding the one-photon photophysical properties of a two-photon absorbing chromophore ,” J. Phys. Chem. A 108 , 5514 – 5520 ( 2004 ). [CrossRef]

9 .

N. Tang and R. L. Sutherland , “ Time-domain theory for pump-probe experiments with chirped pulses ,” J. Opt. Soc. Am. B 14 , 3412 – 3423 ( 1997 ). [CrossRef]

10 .

A. Dogariu , T. Xia , D. J. Hagan , A. A. Said , E. W. Van Stryland , and N. Bloembergen , “ Purely refractive transient energy transfer by stimulated Rayleigh-wing scattering , J. Opt. Soc. Am. B 14 , 796 – 803 ( 1997 ). [CrossRef]

11 .

A. Dogariu and D. J. Hagan , “ Low frequency Raman gain measurements using chirped pulses , Opt. Express 1 , 73 – 76 ( 1997 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-1-3-73 . [CrossRef] [PubMed]

12 .

A. Yariv , Quantum Electronics, Third Ed. ( John Wiley & Sons, New York , 1989 ).

OCIS Codes
(190.4180) Nonlinear optics : Multiphoton processes
(190.5890) Nonlinear optics : Scattering, stimulated
(190.7070) Nonlinear optics : Two-wave mixing

ToC Category:
Research Papers

History
Original Manuscript: September 8, 2005
Revised Manuscript: September 7, 2005
Published: November 28, 2005

Citation
Richard Sutherland, "Energy transfer between incident laser and elastically backscattered waves in nonlinear absorption media," Opt. Express 13, 9788-9795 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-24-9788


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References

  1. Y. Silberberg and I. Bar Joseph, “Optical instabilities in a nonlinear Kerr medium,” J. Opt. Soc. Am. B 1, 662-670 (1984). [CrossRef]
  2. Y. Silberberg and I. Bar Joseph, “Instabilities, self-oscillation, and chaos in simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541-1543 (1982). [CrossRef]
  3. P. Yeh, “Exact solution of a nonlinear model of two-wave mixing in Kerr media,” J. Opt. Soc. Am. B 3, 747-750 (1986). [CrossRef]
  4. R. Boyd, Nonlinear Optics (Academic Press, New York, 1992).
  5. G. S. He, T. –C. Lin, and P. N. Prasad, “Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation,” Opt. Express 12, 5952-5961 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-5952">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-5952</a>. [CrossRef] [PubMed]
  6. G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, “Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,” Phys. Rev. A 71, 063810 (2005). [CrossRef]
  7. R. L. Sutherland, M. C. Brant, J. Heinrichs, J. E. Rogers, J. E. Slagle, D. G. McLean, and P. A. Fleitz, “Excited state characterization and effective three-photon absorption model of two-photon-induced excited state absorption in organic push-pull charge-transfer chromophores,” J. Opt. Soc. Am. B 22, 1939-1948 (2005). [CrossRef]
  8. J. E. Rogers, J. E. Slagle, D. G. McLean, R. L. Sutherland, B. Sankaran, R. Kannan, L. –S. Tan, and P. A. Fleitz, “Understanding the one-photon photophysical properties of a two-photon absorbing chromophore,” J. Phys. Chem. A 108, 5514-5520 (2004). [CrossRef]
  9. N. Tang and R. L. Sutherland, “Time-domain theory for pump-probe experiments with chirped pulses,” J. Opt. Soc. Am. B 14, 3412-3423 (1997). [CrossRef]
  10. A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer by stimulated Rayleigh-wing scattering, J. Opt. Soc. Am. B 14, 796-803 (1997). [CrossRef]
  11. A. Dogariu and D. J. Hagan, “Low frequency Raman gain measurements using chirped pulses, Opt. Express 1, 73-76 (1997), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-1-3-73">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-1-3-73</a>. [CrossRef] [PubMed]
  12. A. Yariv, Quantum Electronics, Third Ed. (John Wiley & Sons, New York, 1989).

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