Nonlinearly Induced Relaxation to the Ground State in a Two-Level System

We report the experimental demonstration of a nonlinear process in a two-level system, in which the amplitude of the excited state decays, transferring irreversibly a large fraction of its energy to the ground state, while shedding a part of it into radiation states. The experiments where preformed in a nonlinear optical waveguide, supporting two or three modes. The process is general, and is expected to occur in other nonlinear few level systems such as nonlinear quantum wells and Bose-Einstein condensates.

Figure 1

Analogy between the quantum well system (a) and the optical waveguide system (b). $V(x)$ is the well’s potential, $x$ is a spatial coordinate, and $E_{g,e}$ are the energy eigenvalues of the ground and excited states. $n(x)$ is the refractive-index profile and ${\beta }_{g,e}$, are the propagation constants eigenvalues of the ground and excited states. (c) Intensity modal shapes of the ground (top) and excited (bottom) states of the waveguide.

Figure 2

Experimental setup. (a) The ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ slab-waveguide system used in our experiments. The light is vertically confined inside a high index layer (the slab core), sandwiched between two low-index clad layers. An effective 1D waveguide is defined by etching into the top clad layer of the slab waveguide. Top-cladding, core, and bottom-cladding widths of the slab waveguide are $1.5\mu \mathrm{m}$, $1.5\mu \mathrm{m}$, and $4\mu \mathrm{m}$, respectively, producing a vertical index step of 0.03. (b) Experimental system. Using the cylindrical telescope, the horizontal width of the beam is tuned so that it best matches that of the excited waveguide mode.

Figure 3

Experimentally obtained modes of the double-moded waveguide used in our experiments. (a) Intensity distribution of mode 1 (solid line) and mode 2 (dashed line). (b) Corresponding modal-field distributions. Waveguide width is $16\mu \mathrm{m}$, with an index-step of 0.0007 obtained by etching $0.83\mu \mathrm{m}$ deep into the top cladding of the slab-waveguide sample.

Figure 4

Experimental results for the double-moded waveguide of Fig. 3. Two sets of measurements are shown, of the same waveguide, under slightly different launching conditions ($\pi$-step position), expressed by a different modal content of the excitation at the waveguide input. (a), (b), (c): first set, (d), (e), (f): second set. (a), (d): the modal content at the waveguide output, expressed in terms of the intensity fractions of mode 1 (squares) and mode 2 (circles), as a function of the output (average) power. The intensity fraction of mode 1 in the linear case is about 0.1 and 0.2, for set 1 and set 2, respectively, growing to about 0.45 and 0.75 at maximal power, in the process of ground-state selection. (b), (e): cross sections of the intensity profile at the waveguide output, for low power (i.e., linear, in solid line) and maximal power (dashed line). (c), (f): photographs of the intensity measured at the waveguide output, for low power (upper panel) and maximal power (lower panel).

Figure 5

Experimental results for a three-moded waveguide. Two sets of measurements are shown, of the same waveguide, under different launching conditions, expressed in a different modal content of the excitation. (a), (b): first set, (c), (d): second set. (a), (c): Cross sections of the intensity at the waveguide output, for low power (i.e., linear, solid line) and high power (dashed line). At low power, the intensity profile exhibits three lobes, indicating substantial mode 3 content. At high power the three lobes disappear, indicating reduced mode 3 content, after being transformed into lower waveguide modes. (b), (d): Photographs of the intensity measured at the waveguide output, for low power (upper panel) and high power (lower panel). Sample parameters are the same as that of the double-moded waveguide, except of the waveguide width being $30\mu \mathrm{m}$.

Figure 6

Simulation results of a tapered waveguide exhibiting a cascaded GSS process (illustrated in inset). (a) Low-power (upper panel) and high-power (lower panel) propagation in the waveguide. (b) Waveguide power, normalized to the input power, as a function of propagation distance, for low power (solid line) and high power (dashed line). Initial waveguide width is $25\mu \mathrm{m}$ (with three guided modes), gradually decreasing to $3\mu \mathrm{m}$ (where the waveguide is single moded). Modal intensity fraction at the input is 0.05, 0.475, and 0.475 for modes 1, 2, and 3, respectively. At low power, 95% of the power is radiated out of the waveguide in two steps, corresponding to the widths in which the tapered-waveguide loses each guided mode. At 1500 Watt, 42% of the input power reaches the waveguide end, being occupied completely in the ground state. Index step is 0.001 and overall taper length is 10 cm.