Waves in intensity coherence of evolving intense twin beams

Strong correlations between the signal and idler beams imprinted during their generation dominantly determine the properties of twin beams. They are also responsible for the waves in intensity coherence observed in the wave-vector space of a twin beam propagating in a nonlinear crystal in the regime with pump depletion. These waves start to develop at certain twin-beam intensities and move from the signal- and idler-beam centers towards their tails. They manifest themselves via the change of coherence volume monitored in the far field by the measurement of local modified ${\overline{g}}^{\left(2\right)}$ function, which acts as a sensitive and stable tool for investigating field intensity coherence.

Figure 1

(a) Experimental setup used for the determination of spatiospectral intensity correlations in a TWB. Nd:YLF: pump laser; ATT.: varying pump-beam attenuation; BBO: nonlinear crystal; DM: dichroic mirror to deflect the pump beam; FL: Fourier lens with 100-mm focal length; A: aperture with vertical rectangular slit; SPEC.: imaging spectrometer; EMCCD: electron-multiplying CCD camera. (b) Ring formed by the signal-beam intensity in the far field for central frequency. (c) Output of the imaging spectrometer. From left to right: single-shot image of both beams showing the specklelike pattern with correlated grains, multiply exposed image forming two horizontal strips (upper strip: signal beam, lower strip: idler beam) and scheme for the determination (observation) of intensity auto- (AC) and cross- (XC) correlations; color bar below quantifies the intensities in arbitrary units. In (c), the frequency axis $\omega$ extends from 677.3 to 718.7 nm, the signal ($k_{\mathrm{s}}^{\perp }$) and idler ($k_{\mathrm{i}}^{\perp }$) radial wave-vector axes cover the range from 0 to 40 mrad. In (b) and (c), red arrows indicate the propagation of waves in TWB coherence at increasing TWB intensity. They start to develop above a certain threshold pump-power $P_{\mathrm{th}}$.

Figure 2

Functions (a) ${\overline{g}}_{1\mathrm{D}}^{\left(2\right)}\left(\mathrm{\Delta }a\right)$ from Eq. (4) of the Gaussian model and (b) ${\overline{g}}_{1\mathrm{D}}^{\left(2\right)}\left(\mathrm{\Delta }\tilde{a}\right)$ derived from the experimental data by grouping the pixels in the spectral (red $▵$) and radial wave-vector (black $\circ$) directions as they depend on normalized coherence lengths $\mathrm{\Delta }a=\mathrm{\Delta }A/d$ and $\mathrm{\Delta }\tilde{a}=\sqrt{ln\left(2\right)}\mathrm{\Delta }a$. In (b), solid curves connect the experimental points.

Figure 3

(a) Function ${\overline{g}}^{\left(2\right)}$ measured at individual pixels and widths (b) $\mathrm{\Delta }{A}_{\mathrm{s},\phi }$, (c) $\mathrm{\Delta }{A}_{\mathrm{s},\omega }$, and (d) $\mathrm{\Delta }{A}_{\mathrm{s},k}$ of, in turn, azimuthal wave-vector, spectral and radial wave-vector intensity autocorrelation functions (FWHM) for the signal beam at central frequency ${\omega }_{\mathrm{s}}^0$ and central radial wave vector $k_{\mathrm{s}}^{\perp 0}$ as they depend on pump power $P$. Experimental data are plotted as isolated symbols with error bars, and solid curves originate in the theoretical model [33].

Figure 4

(a) Experimental and (b) theoretical ${\overline{g}}^{\left(2\right)}$ function for individual pixels, (c) experimental and (d) theoretical number $M_g=1/{\overline{g}}^{\left(2\right)}$ of local modes and experimental widths (e) $\mathrm{\Delta }{A}_{\mathrm{s},\omega }$ and (f) $\mathrm{\Delta }{A}_{\mathrm{s},k}$ of frequency and radial wave vector, respectively, intensity autocorrelation functions [FWHM] as they depend on normalized radial wave vector $k_{\mathrm{s}}^{\perp }/k_{\mathrm{s}}^0$ and pump power $P$ for the central signal-beam frequency ${\omega }_{\mathrm{s}}^0$. Relative errors are better than 5% in (f), 11% in (e), and 12% in (a), (c).

Figure 5

Positions of maxima of ${\overline{g}}^{\left(2\right)}$ function drawn in Figs. 4 and 4 in signal-beam radial wave vector $k_{\mathrm{s}}^{\perp }$ as they depend on pump power $P$. Experimental (theoretical) data are plotted as isolated symbols (by solid curves).

Figure 6

(a) Experimental and (b) theoretical profiles of the signal-beam ${\overline{g}}^{\left(2\right)}$ function drawn in Fig. 4 [relative errors better than 12%] along normalized radial wave vector $k_{\mathrm{s}}^{\perp }/k_{\mathrm{s}}^0$ for powers $P$ equal to 23 mW [30 mW] (solid curves with black $\circ$), 43 mW [45 mW] (red $*$), 55 mW [60 mW] (green $▵$), 80 mW [90 mW] (blue $\square$), 115 mW [120 mW] (magenta $\diamond$), and 142 mW (150 mW) (cyan $+$) in (a) [(b)]. Signal-beam intensity profiles $I_{\mathrm{s},k}$ [relative error 1%] are drawn by dashed curves.

Figure 7

Functions ${\overline{g}}^{\left(2\right)}$ obtained by grouping 1 (black $▵$), 4 (red $*$), and 8 (blue $\circ$) pixels in (a) frequency and (b) radial wave-vector directions for the signal beam at central frequency ${\omega }_{\mathrm{s}}^0$ and central radial wave vector $k_{\mathrm{s}}^{\perp 0}$ as they depend on pump power $P$. Experimental data are plotted as isolated symbols with error bars; solid curves originate in the theoretical model.