Coherence properties of high-gain twin beams

Twin-beam coherence properties are analyzed in both spatial and spectral domains at high gain, including also the regime of pump depletion. We show an increase in the size of the intensity autocorrelation and cross-correlation areas at increasing pump power, replaced by a decrease in the pump depletion regime. This effect is interpreted as a progressive loss in the mode selection occurring at high-gain amplification. The experimental determination of the number of spatiospectral modes from the measurements of the $g^{\left(2\right)}$-intensity autocorrelation coefficient confirms this explanation.

Figure 1

Experimental setup used for the spatiospectral measurements of the twin beam: HWP, half-wave plate; PBS, polarizing cube beam splitter; BBO, nonlinear crystal; L, lens, with 60-mm focal length; ${\mathrm{M}}_j$, spherical mirrors; G, grating; and EMCCD, electron-multiplying camera.

Figure 2

(a) Single-shot image recorded by the EMCCD camera, in which the typical specklelike pattern of PDC in the spatiospectral domain is clearly evident. (b) Typical example of the intensity correlation coefficient ${\Gamma }_{k,l}^{(i,j)}$, in which the intensity autocorrelation and cross-correlation areas are clearly evident. In this particular case, as the pixel at coordinates $(i,j)$ was chosen on the left-hand side, the intensity autocorrelation area is on the same side, whereas the cross-correlation area is on the right-hand side of the image.

Figure 3

Evolutions of the (a) spectral and (b) spatial FWHM size of the intensity autocorrelation (red circles) and cross-correlation (black circles) areas measured from the $(\theta ,\lambda )$ spectra of the twin beam as functions of the pump mean power. A fourth-root function, as expected from the theory of the PDC structure under the assumption of an undepleted pump beam, is used to fit the first part of each data set. (c) Ratio $P_{\mathrm{out}}/P_{\mathrm{in}}$ of the pump power measured at the exit of the crystal to the pump power measured at the entrance, normalized to the power value before depletion, as a function of the pump mean power.

Figure 4

Simulations of the evolution of the (a) spectral and (b) spatial FWHM size of the intensity autocorrelation (line with red circles) and cross-correlation (line with black circles) areas of the twin beam as functions of the pump mean power. (c) Simulation of the ratio $P_{\mathrm{out}}/P_{\mathrm{in}}$ of the pump power at the exit of the crystal to the pump power at the entrance, normalized to the power value before depletion, as a function of the pump mean power.

Figure 5

Spectral and spatial pump beam profiles for different values of the pump mean power: (a) sections of the spectral profile at different pump powers (green, 15 mW; magenta, 35 mW; blue, 55 mW; and black, 99 mW), (b)–(d) maps of the spectral distributions upon subtraction of the distribution of the least intense measurement [(b) 35 mW, (c) 55 mW, and (d) 99 mW], (e) sections of the spatial profile at different pump powers (green, 15 mW; magenta, 35 mW; blue, 55 mW; and black, 99 mW), and (f)–(h) maps of the spatial distributions upon subtraction of the distribution of the least intense measurement.

Figure 6

Simulation of the spectral and spatial pump beam profiles for different values of the pump mean power: (a) sections of the spectral profile at different pump powers (green, 12 mW; magenta, 30 mW; blue, 55 mW; and black, 100 mW), (b)–(d) maps of the spectral distributions upon subtraction of the distribution at the lowest power [(b) 30 mW, (c) 55 mW, and (d) 100 mW], (e) sections of the spatial profile at different pump powers (green, 12 mW; magenta, 30 mW; blue, 55 mW; and black, 100 mW), and (f)–(h) maps of the spatial distributions upon subtraction of the distribution at the lowest power.

Figure 7

Evolution of the number of photons generated by PDC as a function of the square root of the pump peak power $P^{\prime }$ per pulse. The theoretical curve (red line) holding under the assumption of an undepleted pump beam is also shown. The inset shows the first part of the data shown in the main figure (black circles) together with the fitting curve predicted by the theory (red line).

Figure 8

(a) Intensity autocorrelation coefficient (red circles) dependent on the pump mean power and (b) number $K$ of spatiospectral modes determined from the $g^{\left(2\right)}$-intensity autocorrelation coefficient. In both panels the data shown as closed (open) circles characterize the signal (idler) arm.