References

1

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1984, 2nd Edition), pp. 208-210.

2

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1989, 6th Edition), pp. 611-664.

3

N. Peyghambarian, S. W. Koch, and A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice Hall, New Jersey, 1993).

4

L. Roso-Franco, Phys. Rev. Lett. 55, 2149 (1985).

5

A. Sommerfeld, Optics (Academic Press, San Diego, 1949), pp. 72-75.

6

L. Roso-Franco and M. L. Pons, Opt. Lett. 15, 1230 (1990).

7

S. R. Hartmann and J. T. Massanah, Opt. Lett. 16, 1349 (1991).

8

E. Hudis and A. E. Kaplan, Opt. Lett. 19, 616 (1994).

9

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2316 (1995).

10

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, 1992), Chap. 4.

11

K. S. Yee, IEEE Transactions on Antennas and Propagation 14, 302 (1966).

12

L. Roso-Franco, J. Opt. Soc. Am. B 4, 1878 (1987).

13

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

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\clearpage \samepage \setbox\sizebox=\hbox{$E = 1/2(A_E {\cal E}exp\{i(kz-wt)\} + c.c.)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline370: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$P = 1/2(A_P {\cal P}exp\{i(kz-wt)\} + c.c.)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline372: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$A_E = \hbar (\gamma_1 \gamma_2)^{1/2} / p$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline374: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$A_P = -i N p (\gamma_1 / \gamma_2)^{1/2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline376: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$T_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline380: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \begin{equation}\frac{\partial I}{\partial z} + \frac{1}{c} \frac{\partial I}{\partial t} = - \frac{\alpha}{2} n I , \qquad \frac{\partial n}{\partial t} = - \gamma_1 n I , \end{equation} } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$I = |{\cal E}|^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline382: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\eta = t - z/v$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline384: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$I(-\infty) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline388: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$I(\infty) = F^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline390: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$n(-\infty) = 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline392: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$n(\infty) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline394: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \begin{equation}I(\eta) = \frac{F^2}{2} ( \tanh(\eta/ \eta_0) + 1 ) , \qquad n(\eta) = \frac{1}{2} ( 1 - \tanh(\eta / \eta_0) ) , \end{equation} } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$c\eta_0 = 2c/\gamma_1 F^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline396: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$c/v = 1 + \omega \psi /\gamma_1 F^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline398: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$t_0>T_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline410: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{${\cal A}=1.76pE_0t_0/\hbar=10.8\pi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline412: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$z_0=400$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline416: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\mu m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline418: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } \stepcounter{section} \end{document} n difference at various times corresponding to the results shown in Fig. 1. Figure 2(a) shows that the field penetrates progressively further into the interface as the absorption is saturated, as can be seen by comparing the field profiles at t=1.61 ps (solid line) and t=1.95 ps (dashed line) or t=2.28 ps (single-dash-dot line) (at t=2.62 ps the field has been mostly refelcted). At t=1.61 ps (solid line) Fig. 2(b) shows that n=0 before the interface signifying zero absorption, and n=1 beyond the interface signifying large absorption due to the two-level systems. At later times after the input pulse has penetrated into the interface, the population difference is depleted and the absorption front is seen to propagate into the nonlinear medium. The propagating absorption front maintains a sharp wavelength scale transition region so that the linear skin effect still occurs but now from a moving absorption front. Thus the self-reflected field must suffer a red-shift due to the Doppler-effect, akin to reflection from a mirror moving away from a source [5]. To validate this physical picture we have determined the absorption front velocity from the numerical simulation in Fig. 2(b), and the result is shown in Fig. 2(c). After initially accelerating the front reaches a maximum velocity of before decelerating back to zero velocity. The maximum wavelength shift of the reflected pulse due to the Doppler effect is then for [5], or nm for the free-space wavelength nm used here. Thus, based on the Doppler effect we expect a maximum local wavelength of nm, in good agreement with the numerical results in Fig. 1(b). The Doppler effect upon reflection from the moving absorption front can therefore explain the magnitude of the observed pulse wavelength chirp.

We are now in a position to further explain the physics underlying the pulse profiles and spectra in Fig. 1: As the leading edge of the input field penetrates into the medium the absorption front accelerates and the local wavelength of the reflected field increases, and on the trailing edge of the pulse the absorption front decelerates and the local wavelength decreases. This explains the initial rise and then decrease in the wavelength chirp in Fig. 1(b) (dashed line). Note that the field-profile in Fig. 1(b) (solid line) exhibits a minimum at the same point that the local wavelength peaks, and this begs a physical explanation. A graph of the cw intensity reflectivity of the linear interface [12], for the same parameter values used to generate Figs 1 and 2, shows that the reflectivity decreases, relative to the pulse center wavelength nm, for wavelengths red detuned from the resonance. This is so because even though the absorption decreases the skin depth increases, thus allowing more path length in the medium over which absorption of the field can occur (the situation is more complicated for blue-detuning but that does not concern us here). Thus the red-shifted peak portion of the reflected pulse in Fig. 1(b) experiences a lower reflectivity than the wings, giving rise to the double-peaked reflected field. More quantitatively, for the maximum red-shifted wavelength of nm the reflectivity is reduced to 2%. This can also be intuited by realizing that a red-shift of nm corresponds to 4.6 linewidths ( ), and a significant reduction in absorption and reflection is to be expected. Furthermore, this physical picture correctly indicates that for lower input fields the Doppler shift is reduced, in which case the differential reflection coefficient between the wings and center of the pulse need not be as large as in Fig. 1(b). In this case the reflected pulse can be single peaked, though still red-shifted.

An experimentally measurable signature of the double-peaked reflected field in Fig. 1(b) is the modulated spectrum in Fig. 1(c). The reflected field is composed of two peaks with a spacing m. Treating these as point sources in z, we expect a modulation in the wavelength spectrum with a period m, which agrees reasonably with the modulation period in Fig. 1(c). In the case that the reflected field is single-peaked the reflected spectrum is red-shifted but with no modulation.

While the slowly-varying envelope approximation (SVEA) [4, 12], cannot capture the physics of the evolution of the self-reflected wave, we can employ it to demonstrate front propagation in the medium away from the interface. In particular, we introduce the field and polarization envelopes via the definitions and , where and , to obtain the usual Maxwell-Bloch equations in the SVEA from Eqs. (1) and (2) [10]. For the case considered here with , we may ignore the population relaxation, and in the limit of small adiabatically eliminate the polarization. We then obtain the pair of coupled equations

where . These equations admit travelling wave solutions depending on the variable , where v is the front velocity [8]. Assuming a pulse profile with and , a constant, and a corresponding bleaching of the two-level system inversion from the ground-state to transparency, and , we find the front solution

where is the characteristic spatial width of the travelling wave front, and the front velocity is given by [8, 9]. For the parameters above, this gives v/c=0.026, which is smaller than the value reported above due to neglect of the self-reflected wave in the present calculation. Thus we have analytic confirmation that absorption fronts can propagate in the nonlinear medium, and we identify these with the fronts seen in our numerical simulations. Numerical integration of the full SVEA equations also clearly shows front propagation for these parameters, but with a damped, oscillatory front profile.

The nonlinear skin effect appears over a much broader range of parameters than employed here. As in the cw case we require the dimensionless parameters and F to be larger than unity so that there is a linear skin effect and a high enough level of nonlinear absorption saturation. In particular, we note that especially for longer pulses, F can be substantially smaller than the value of F=320 in the example above, though the front velocities and spectra are correspondingly reduced. For the pulse duration and material relaxation times, we require to obtain propagating absorption fronts, where the condition is imposed to avoid the regime of self-induced transparency (the input pulse area above is ) [13]. Given these restrictions, exitonic resonances in quantum well materials are prime candidates for the observation of this effect, where high absorptions and suitable relaxation times are available [3]. It remains to see whether the high levels of saturation can be achieved, and if under these condition, simple two-level polarisation dynamics can be realised.

In conclusion, we have introduced the dynamic nonlinear optical skin effect for reflection of pulses from a highly absorbing interface. This new basic effect for the electrodynamics of interfaces combines the concepts of self-r eflected waves [4] and front propagation, and is also a prime example of a nonlinear optical phenomenon where the SVEA fails and the full Maxwell equations must be employed. We have shown that the nonlinear optical skin effect arises from moving absorption fronts so that the red-shifting and spectral modulation of the reflected pulse are clear experimental signatures of the effect.

We thank Dr. Robert Indik for constructive discussions and suggestions. Ewan Wright would like to thank Profs. Ray Chiao, Alex Kaplan, John McCullen, Pierre Meystre, and Dr. John Garrison for their insightful remarks concerning this work. The authors also wish to thank the Arizona Center for Mathematical Sciences (ACMS) for support. ACMS is sponsored by AFOSR contracts F49620-94-1-0463 and F49620-94-1-0051. E. M. Wright is partially supported by the Joint Services Optical Program.

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