Thick-medium model of transverse pattern formation in optically excited cold two-level atoms with a feedback mirror

We study a pattern-forming instability in a laser-driven optically thick cloud of cold two-level atoms with a planar feedback mirror. We develop a theoretical model, enabling a full analysis of transverse patterns in a medium with saturable nonlinearity, taking into account diffraction within the medium, and both the transmission and reflection gratings. The focus of the analysis is on the combined treatment of nonlinear propagation in a diffractively and optically thick medium and the boundary condition given by feedback. We demonstrate explicitly how diffraction within the medium breaks the degeneracy of Talbot modes inherent in thin-slice models. We predict the existence of envelope curves bounding all possible pattern-formation thresholds and illustrate their interaction with threshold curves by experimental observation of a sudden transition between length scales as mirror displacement is varied.

Figure 1

Schematic of the SFM configuration. A linearly polarized beam is sent into an atomic cloud modeled as a thick slab of length $L$ (blue) with a nonlinear susceptibility ${\chi }_{NL}$. The transmitted beam is retroreflected by a mirror (M) with an adjustable displacement $DL$ beyond the end of the medium. The forward ($F$) and backward ($B$) propagating beams interfere inside the cloud. The experimental parameters are a cloud of ${}^{87}\mathrm{Rb}$ atoms with $T=200\mu \mathrm{K}$ driven at a detuning of $\delta >0$ from the $F=2\to F^{\prime }=3$ transition of the $D_2$ line, an optical density (base $e$) in line center equal to 210, and an effective sample size (FWHM of cloud) $L=8.5\mathrm{mm}$ [9].

Figure 2

Dependence of zero-order intensities $p,q$, scaled to saturation intensity $I_{s\delta }$, on the longitudinal coordinate $z$, scaled to medium length $L$, in a two-level medium with on-resonance optical density equal to 210 (see Fig. 1): forward $p\left(z\right)$ (dashed lines) and backward $q\left(z\right)$ (solid lines) for several cases. The four lower curves are for output $p\left(1\right)=0.3$, $\delta /\mathrm{\Gamma }=5$, and an $R=1$ mirror so that $q\left(1\right)=0.3$: Top (green dashed) and lowest (red solid) curves are for $h=0$, i.e., no reflection grating. The inner curves are for $h=1$: upper (blue dashed) and lower (light orange solid). The two uppermost (black dashed and orange solid) curves are for output $p\left(1\right)=1.2$, larger detuning $\delta /\mathrm{\Gamma }=10$, and $h=1$, to illustrate a case where absorption effects might be considered negligible, leading to a quasi-Kerr approximation to the two-level response.

Figure 3

Two-level instability domain ($\delta >0$) reported in Ref. [9]. Diffracted power $P_d$ is measured as a function of $\delta >0$ (note that $\mathrm{\Delta }=2\delta /\mathrm{\Gamma }$) and input intensity $I$. Note the logarithmic horizontal scale. The dotted loops indicate maximal instability domains calculated in the thin-medium approximation as described in Ref. [9]: closed circles, domain calculated from (14), i.e., with all reflection gratings included ($h=1$); open circles, domain calculated from (11), i.e., with no reflection gratings ($h=0$). Both dotted traces are rescaled to absolute values of intensity and detuning.

Figure 4

Illustration of transverse gain circles (see the text) calculated from (15) for different input intensities. The parameters here are OD equal to 210, $\delta /\mathrm{\Gamma }=2$, and $\theta =2$. The unit circle centered on the origin is the locus of the feedback phase as mirror displacement $D$ is varied. Lying on it, the red dot is the feedback phase for the particular case $D=-1.3$. The displaced circles are the loci of transverse gain for three cases: (a) The smallest gain circle (red solid) lies wholly inside the unit circle and so the system is always below threshold for this case (scaled input intensity $p_0=7.90564$), (b) the middle gain circle (green dashed) touches the unit circle and so the system reaches threshold for one value of $D$ ($p_0=8.1266$), and (c) the largest gain circle (blue dotted) intersects the unit circle at two well-spaced points and so the system is above threshold for a wide range of $D$ values, including $D=-1.3$ ($p_0=8.29754$). Points on the arc of the touching circle corresponding to $g_r$ (blue) and $g_i$ (brown) are also shown. Its center is also marked with a green dot.

Figure 5

Threshold and envelope curves (scaled to $I_{s\delta }$) calculated from (15) for a two-level system with all gratings included ($h=1$) with an $R=1$ feedback mirror. The scaled input intensity $p_0$ is plotted against diffraction parameter $\theta =Q^{2}L/2k$. The outer (blue) curve is the envelope curve, the limiting threshold for any mirror displacement; the inner (orange) curve is the threshold curve for mirror displacement $D=0$, which, close to its maximum, touches the envelope curve. It also touches the envelope at low values of $p_0$, in fact almost coinciding with the envelope curve over a wide range of $\theta$. The envelope curve has finite intercepts with the $\theta =0$ axis (see the text for discussion). The other parameters are an OD of 210 and $\delta /\mathrm{\Gamma }=1.5$.

Figure 6

Threshold and envelope curves (scaled to $I_{s\delta }$) calculated from (15) for the same conditions as in Fig. 5, except that the feedback mirror displacement is $D=-1.3$, which corresponds to the experimental results of Fig. 3. The scaled input intensity $p_0$ is plotted (here on a logarithmic scale, for clarity) against the diffraction parameter $\theta =Q^{2}L/2k$, which is continued to negative $\theta$ (see the text) so as to present results for red, as well as blue, atomic tuning. The envelope curve, the continuation to negative $\theta$ of that in Fig. 5, shows a large red-blue tuning asymmetry. Inside the envelope is a set of discrete closed threshold loops for $D=-1.3$, each of which touches the envelope above and below.

Figure 7

Threshold and envelope curves (scaled to $I_{s\delta }$) calculated from (15) for the same conditions as in Fig. 6, except $\delta /\mathrm{\Gamma }=5$. The scaled input intensity $p_0$ is plotted (again on a logarithmic scale, for clarity) against the diffraction parameter $\theta =Q^{2}L/2k$, which is continued to negative $\theta$ (see the text) so as to present results for red, as well as blue, atomic tuning. Upper and lower portions of both envelope and threshold curves are well separated for large $\theta$, asymptotically corresponding to phase-conjugate oscillation thresholds.

Figure 8

Threshold and envelope curves (scaled to $I_{s\delta }$) calculated from (15) for the same conditions as in Fig. 6, except $\delta /\mathrm{\Gamma }=13.1$. Note that $p_0$ is here plotted on a linear scale.

Figure 9

Two-level instability domain, range of threshold input intensity $p_0$ in terms of $\delta /\mathrm{\Gamma }$, with a logarithmic horizontal scale, as in Fig. 3. The larger loop (black circles) is as presented in Ref. [9], calculated in the thin-medium approximation for $R=1$, but identifiable as the $\theta =0$ envelope (see the text); the smaller loop (red solid) is calculated from (15), i.e., with all reflection gratings included ($h=1$) and for $R=0.95$ and $D=-1.3$. The OD is equal to 210. The contour plot loops show experimental data of Fig. 3, for comparison.

Figure 10

Threshold and envelope curves (scaled to $I_{s\delta }$): blue dashed curves, envelope curves calculated from (23) for a two-level medium described by ${\widehat{A}}_{\mathrm{all}}$, with $h=1$, a quasi-Kerr coefficient $|{\alpha }_{l}L\mathrm{\Delta }|=8$, and detuning $\delta /\mathrm{\Gamma }=13.1$; orange solid curves, threshold curves with a feedback mirror at negative effective distance ($D=-1.3$) from the end of the medium, which touches the envelope curves.

Figure 11

Two-level instability domain, range of threshold input intensity in terms of $\delta /\mathrm{\Gamma }$, with a logarithmic horizontal scale. The red solid closed loop is that calculated from (15) with absorption and all reflection gratings included ($h=1$). Here $R=0.95$, $D=-1.3$, and the OD is equal to 210, as in Fig. 9. The black dotted open curve is calculated for the same parameters from (22), as derived from the quasi-Kerr model equations Eq. (18). The latter curve is calculated only for $\delta /\mathrm{\Gamma }>7.5$, because the neglect of absorption in Eq. (18) is untenable at small detunings.

Figure 12

Threshold intensity (in units of ${\alpha }_{l}L\mathrm{\Delta }p/2$) vs diffraction parameter $\theta =Q^{2}L/2k$. Blue dashed curves are envelope curves calculated from (23) for a Kerr medium with $h=0$. Positive and negative intensity values correspond to self-focusing and self-defocusing Kerr media, respectively. Also shown are threshold curves with a feedback mirror at positive effective distance from the end of the medium. The black solid curves with fewer wiggles show $D=2.5$ and the orange solid curves with more wiggles show $D=10.0$. In both cases the threshold curves touch the envelope curves and are confined by them.

Figure 13

Pattern period (arbitrary units) vs mirror displacement $D$ at fixed intensity $s=0.085$. Threshold curves were calculated from (22) for a two-level medium described by ${\widehat{A}}_{\mathrm{all}}$, with $h=1$. The quasi-Kerr coefficient ${\alpha }_{l}L\mathrm{\Delta }=13.94$, corresponding to blue detuning. For optical density 210 [9], this corresponds to detuning $\mathrm{\Delta }=2\delta /\mathrm{\Gamma }=15$.

Figure 14

(a) Diffracted power (experiment, left axis) and predicted threshold intensity ($p_0$, scaled to $I_{s\delta }$) vs scaled mirror displacement $D$. The cloud thickness is $L=9\mathrm{mm}$. (b) Pattern period $\mathrm{\Lambda }$ vs mirror displacement. In physical units, the $x$ axis corresponds to $-60$ mm to +40 mm measured from the center of the cloud. The parameters are blue detuning and $\mathrm{\Delta }=15$ (see [9]). The diffracted power is normalized to its maximal value. The red closed dots show the experimental data for the first Talbot balloon (lowest wave number) and gray circles the experimental data for the second Talbot balloon [next highest wave number excited in (a) enhanced by factor of 5]. The red and gray curves are the corresponding theoretical predictions and are calculated from (22) using the all-grating two-level model. The inset shows the measured $D$ period as a function of the pattern size (stars), together with the Talbot effect prediction (line).

Figure 15

(a) Predicted threshold intensity ($p_0$, scaled to $I_{s\delta }$), (b) experimentally observed diffracted power (normalized to its maximal value), and (c) pattern period vs mirror displacement $D$. The lower curve in (a) is for the first (lowest wave number) and the upper curve is for the second (next highest wave number excited) Talbot balloon, with both curves calculated for a self-focusing Kerr medium with $h=1$ described by ${\widehat{A}}_{\mathrm{Kerr}}$. The experimental parameters are an effective-medium length $L=3.2\mathrm{mm}$, beam intensity $I=18\mathrm{mW}/{\mathrm{cm}}^2$, and detuning $\mathrm{\Delta }=-14$. Red closed circles show the experimental data for the first Talbot balloon and blue open circles the experimental data for the second Talbot balloon. The two curves in (c) are the corresponding theoretical predictions for the first (upper red) and second (lower blue) Talbot balloons, calculated as in (a). The insets show far-field patterns obtained at the mirror positions indicated illustrating the length scale competition. In unscaled parameters, the $x$ axis corresponds to $-12.8$ mm to +10.2 mm measured from cloud center.

Figure 16

(a) Pattern length scale (characterized by diffraction parameter $\theta$) and (b) threshold intensity ($p_0$, scaled to $I_{s\delta }$) vs mirror displacement $D$ for a self-focusing Kerr medium with $h=1$ described by ${\widehat{A}}_{\mathrm{Kerr}}$. The red solid curve shows the minimum threshold, the blue dashed curve the lowest-wave-number (first Talbot), and the blue dotted curve the second-lowest-wave-number (second Talbot) balloon.