## Abstract

We display the results of the numerical simulations of a set of Langevin equations, which describe the dynamics of a degenerate optical parametric oscillator in the Wigner representation. The scan of the threshold region shows the gradual transformation of a quantum image into a classical roll pattern. An experiment on parametric down- conversion in lithium triborate shows strikingly similar results in both the near and the far field, displaying qualitatively the classical features of quantum images.

© Optical Society of America

## 1. Introduction

Most fascinating in Science is the general phenomenon of spontaneous formation of
spatial pattern from a homogeneous background, when a control parameter, say
*β*, is raised to the critical value
*β*_{c} , where the pattern formation
starts [1,2]. For a 2D system with translational and rotational symmetry, the analysis
is best carried out in terms of the spatial Fourier transform of the field in play. The
instability which leads to pattern formation develops over a critical circle of radius
*k*_{c} in the Fourier plane. When the
control parameter *β* is slightly larger than
*β*_{c} small perturbations of the
form exp (ik⃗ · x⃗), where *k*⃗ =
(*k*_{x} ,
*k*_{y} ), x⃗ =
(*x*, *y*), with |*k*⃗| =
*k*_{c} , do not decay in time, but grow and
give rise to a spatial structure. Usually the system breaks spontaneously also the
rotational symmetry and only a small set of points on the circle
|*k*⃗| = *k*_{c}
contribute to the final pattern. If, for example, only two vectors
*k*⃗ and -*k*⃗ contribute to the pattern
with equal weight, the combination of exp (ik⃗ · x⃗) and exp
(-ik⃗ · x⃗) gives rise to a roll structure. In general, the
selection of the final pattern is governed by the nonlinearity with the constraint of
conservation of the total wave vector.

When the control parameter *β* is smaller than but close enough
to *β*_{c} , the onset of a spatial
structure is heralded by the spatial correlation function, which exhibits a modulation
with a wavelength
2*π*/*k*_{c} [2].

In the case of nonlinear optical systems, the spatial Fourier transform corresponds to
the “far field” configuration of the field itself. The law of
conservation of total wave vector is what is usually called the phase matching condition
in nonlinear optics. ^{2} Most important of all,
nonlinear optical structures are capable of displaying relevant quantum features, even
at room temperature. The main reason for this is that the interaction processes which
give rise to pattern formation in optics, correspond to simultaneous destruction and
creation of photons, and this circumstance creates correlations of quantum nature.

If, for example, a degenerate optical parametric oscillator (OPO) below threshold is
considered, the control parameter is the input field amplitude, and the critical point
*β*_{c} corresponds to the threshold
for signal generation. Below threshold, according to the semiclassical picture, the
signal field is exactly zero. On the other hand, in a fully quantum description, due to
the presence of quantum noise, the signal field is zero only on average and on
approaching threshold it shows a spatial modulation, basically with a wavelength
2*π*/*k*_{c} , with a
noteworthy level of spatial order. We use the name “quantum image”
[4–5] to designate these noisy field
configurations generated by quantum fluctuations. The analysis of spatial correlations
in the far field shows clear signatures of the quantum nature of the fluctuations that
create the image [6, 5]. In this paper we show in detail the scan of the threshold region
of the OPO, with a gradual transformation of the quantum image to a classical image (the
roll pattern).

In addition we describe an experiment, performed in the laboratories of the Como branch
of Milano University, which studies the propagation of a pump field along a sample of
χ^{(2)} material. The pump field generates a signal and an idler
field at lower frequencies, via a process of parametric down conversion induced by
quantum fluctuations. The signal field is detected by means of a CCD camera, in the near
and in the far field; in both cases the field spatial distribution is closely similar to
that shown by our previous numerical results. Despite differences between the
experimental setting and the theoretical model studied here, the basic physical
mechanisms which give rise to the patterns are closely similar, and this can well
explain the impressive similarity of the pictures.

## 2. The Langevin equation model in the Wigner representation

Let us consider (Fig. 1) a cavity with plane
mirrors, containing a medium with a χ^{(2)} nonlinearity. Mirror
M_{1} has high reflectivity, while mirror M_{2} is completely
reflecting, so that the cavity is one-ended. A coherent and stationary field
*E*
_{in} of frequency 2*ω*_{s} and
with a plane wave configuration is injected into the cavity. The output is formed by a
field with the same frequency as the input field and, if the pump intensity overcomes a
suitable critical value which is the threshold for the optical parametric oscillator, by
another field, named the signal field, with frequency
*ω*_{s} . The generation of the
signal field occurs via a process of parametric down-conversion, in which photons of the
pump field are converted into pairs of photons of the signal field.

Our model is formulated in terms of Langevin equations for the field distributions
*A*
_{0}(*x*⃗,*t*) and *A*
_{1}(*x*⃗,*t*) associated with the pump of
frequency 2*ω*_{s} , and with the signal
of frequency *ω*_{s} , respectively.
More precisely, by *A*
_{0} and *A*
_{1} we indicate the slowly varying envelopes of the pump field and of the
signal field. They depend only on time and on the transverse coordinate
*x*⃗ = (*x*,*y*), while the
dependence on the longitudinal coordinate *z* has been eliminated by
assuming the uniform field approximation, so that only one longitudinal cavity mode is
relevant for both the pump and the signal field.

The starting point of the model is the master equation for the density operator of the
system, which generalizes the standard description of Drummond, McNeil and Walls [8] to include the spatial degrees of freedom. Using
the Wigner representation, this master equation is translated into a generalized
Fokker-Planck equation for the Wigner functional[5]. This equation includes also terms with third order functional
derivatives, which are neglected. Using Ito rules the Fokker-Planck equation obtained in
this way is finally translated into classical looking Langevin equations for the scaled
fields envelopes *A*
_{0}, *A*
_{1}, which read:

$$+\sqrt{\frac{2}{{\gamma}_{0}{n}_{\mathit{th}}}}{\xi}_{0}(\overrightarrow{x},t)],$$

$$+\frac{1}{\sqrt{{\gamma}_{0}{n}_{\mathit{th}}}}{\xi}_{1}(\overrightarrow{x},t)],$$

where:

- γ
_{0}and γ_{1}are the cavity damping rates for the pump and signal fields, respectively; - the detuning parameters are defined as$${\Delta}_{0}=\frac{{\omega}_{0}-2{\omega}_{s}}{{\gamma}_{0}}\phantom{\rule{1.2em}{0ex}}{\Delta}_{1}=\frac{{\omega}_{1}-{\omega}_{s}}{{\gamma}_{1}},$$
for the signal and pump field, respectively, where ω

_{0}and ω_{1}are the longitudinal cavity frequencies closest to 2*ω*_{s}and*ω*_{s}, respectively; - the transverse Laplacian$${\nabla}_{\perp}^{2}=\frac{{\partial}^{2}}{\partial {x}^{2}}+\frac{{\partial}^{2}}{\partial {y}^{2}}$$
describes diffraction in the paraxial approximation. In the case of optical systems, diffraction together with nonlinearities is the very origin of spontaneous pattern formation, and plays a role similar to diffusion in nonlinear chemical reactions. Here

*x*and*y*are dimensionless transverse coordinates, since we have scaled our space variables to the lengthwhich represents the characteristic length for pattern formation in a OPO cavity;

*E*is the scaled amplitude of the input field*E*_{in};*ξ*_{i}(*x*⃗,*t*),*ξ*_{0}(*x*⃗,*t*) are Langevin noise terms which describe quantum fluctuations. They have zero average, and the only nonvanishing correlation functions between them are:$$\u3008{\xi}_{1}^{*}(\overrightarrow{x},y){\xi}_{1}(\overrightarrow{x}\prime ,\mathit{t\prime})\u3009=\u3008{\xi}_{0}^{*}(\overrightarrow{x},y){\xi}_{0}(\overrightarrow{x}\prime ,\mathit{t\prime})\u3009=\frac{1}{2}\delta \left(\overrightarrow{x}-\overrightarrow{\mathit{x\prime}}\right)\delta \left(t-\mathit{t\prime}\right),$$so that noise is white both in space and in time;

- the parameter
*n*_{th}, whose inverse measures the level of quantum noise, is given by:where

*g*is the coupling constant of the interaction between the two fields; this parameter represents the number of pump photon in the characteristic area ${l}_{d}^{2}$ which are needed to trigger the thresholdfor signal generation.

The semiclassical picture is obtained by dropping the noise terms
*ξ*
_{0} and *ξ*
_{1} in Eqs. (1), (2). As shown in [8], in the case Δ_{1} < 0 the threshold of the
OPO corresponds to

and immediately above threshold the **OPO** emits a signal field in the form of
two plane waves with propagation direction symmetrically tilted with respect to the axis
of the cavity. These two waves interfere with “roll” pattern, shown in
Fig. 2a. The roll is the simplest spatial
pattern that can spontaneously form in nature. In the far field, on the other hand, the
two plane waves give rise to two localized spots (Fig.
2b). Although always symmetrical with respect to the origin, the position of
the two spots can arbitrarily rotate around the origin; correspondingly the orientation
of the rolls in the near field is arbitrary and depends only on initial conditions.

## 3. From below to above threshold

In the quantum description, as that given by Eqs. (1), (2) including the
Langevin force terms, there are photons in the signal field even below threshold. As a
matter of fact, it is well known that the degenerate **OPO** signal field below
threshold is in a squeezed vacuum state; its spatial structure was analyzed in [9,4]. This
means that below threshold the signal field is zero only on average. More important, as
we show here, the signal field displays a spatial modulation with a noteworthy level of
self-organisation. As a matter of fact, well below threshold the signal field is
completely noisy both in the far and in the near field, and corresponds basically to a
white noise configuration. However, as one gets closer to threshold it acquires an
increasing level of spatial order [5].

A short animation (Fig. 3) shows the dynamics of the quantum images as the input field amplitude is gradually varied from below to above threshold. The scan of the threshold region is built up in the following way: by fixing the input field value, we let the system evolve until it reaches the regime. Actually this stage is quite long due to critical slowing down of fluctuations close to threshold. At this point we record the dynamics of the system at regime, by taking a few snapshots of the field distribution in the transverse plane, at time intervals of 3${\mathrm{\gamma}}_{1}^{-1}$. Then, the input field amplitude is slightly increased, and the same procedure repeated. We do these steps for several increasing values of the input field, until the system is above the threshold.

The left panel in the animation shows the configuration of the most amplified quadrature
component of the signal field Re*A*
_{1}(*x*⃗,*t*) in the near field, while
the right panel displays the corresponding intensity distribution of the signal far
field. What we see here is a stochastic realization of the dynamics of the system,
obtained by numerically integrating Eqs. (1), (2) in a square with
periodic boundary conditions; the numerical procedure is outlined in [5]. The far field distribution is obtained by
Fourier transforming the near field. Parameters are chosen in such a way that the
threshold corresponds to *E* = √2.

Starting from values of the input field below the threshold, we observe in the animation an irregular spot pattern in the near field.

Due to the presence of noise, the spots perform a slow random walk in the transverse
plane; as a consequence the signal field mean value vanishes everywhere. However, in a
single snapshot, the distribution of spots over the transverse plane displays a clear
spatial order, and the probability of finding two spots at a distance *r*
apart has maxima when *r* is an integer multiple of the critical
wavelength 2*π*/*k*_{c} ,
which characterizes the modulation of the roll pattern above threshold. The far field
intensity distribution appears to be concentrated on the critical circle of radius
*k*_{c} . Over the far field circle, two
couples of bright spots, opposite to each other with respect to the center of the
circle, are clearly visible.

As the input field amplitude is increased across the threshold, the system undergoes a process of progressive coalescence of the spots into stripes, to approach a roll configuration as that shown by Fig. 2a. For quite a range of values of the input field the system is still “undecided” on which direction it should form them. Further on, beyond the threshold, the rotational symmetry is broken and rolls emerge in a definite direction; this is a priori arbitrary and, in the case of these numerical simulations, the system selects vertical rolls. Correspondingly, the two surviving spots in the far field pattern are aligned on the horizontal axis.

The animation shows the gradual transition from a quantum image (the spot pattern) below threshold, to a classical image (the roll pattern) above threshold. The far field intensity distribution shows correlated pair of maxima, in symmetrical position with respect to the center of the critical circle; this provides a spatial evidence of the mechanism of twin photon production. This feature, together with the analysis of spatial correlation between intensities fluctuations in the far field, gives evidence of the quantum nature of the fluctuations that originate the signal field [5,6].

## 4. An experimental approach to quantum images

The experiment, performed at the Optical Nonlinear Processes Laboratory in Como,
concerns the characterization of the transverse intensity distribution of the parametric
down-conversion generated by single pump pulses in a single nonlinear crystal,
*via* travelling-wave parametric amplification of the vacuum-state
fluctuation (the quantum noise).

The schematic lay out of the experimental set up is given in Fig. 4. The pump source is a frequency-doubled,
feedback-controlled mode-locked, chirped-pulse amplified Nd:glass laser (model TWINKLE,
produced by Light Conversion, Vilnius), delivering 1.2 ps single pulses at 527 nm, with
transform-limited spectral bandwidth, energy up to 2.5 mJ and repetition rate of 2.5 Hz.
The pulse energy and the beam diameter at the entrance of the non-linear crystal were
set to about 0.2 mJ and 2 mm, respectively. At the corresponding intensity (about
4*GW*/*cm*
^{2}) the signal-idler emission is generated in our crystal in regime of linear
amplification, i.e. with negligible pump depletion (below 1%).

The nonlinear crystal is a 15-mm long lithium triborate (**LBO**), cut for type
1 phase matching at *θ* = 90 deg. and Φ = 11.6 deg. For
this cut, the ordinary polarized signal and idler beams exhibit a lateral walkoff of
about 1 deg. respect to the extraordinary-polarized pump. Due to the relatively large
beam diameter (with respect to the crystal length) we do not expect any relevant
contribution of the lateral walkoff in our experiment. As far as the temporal walkoff
(the group-velocity mismatch) is concerned, our pulse duration is also sufficiently
large to make it ineffective in our experiment.

We detected the energy angular distribution of the parametric emission by means of a
silicon CCD camera placed in the focal plane of a positive lens, behind the crystal (far
field detection). Due to the cut off of the detector sensitivity at wavelengths larger
than 1 *μm* only the signal wave was detected (here and in the
following we arbitrary call signal and idler the shorter and longer wavelength
components of the amplified noise, respectively). The near field energy distribution of
the signal superfluorescence was obtained by imaging the exit face of the nonlinear
crystal onto the CCD camera by means of suitable magnifying objective lenses. In both
measurements the pump was eliminated by means of a low-band pass filter.

For a given crystal orientation, *θ* and Φ, with respect
to the incident pump, we expect the parametric emission to occur at all angles (and
wavelengths) which fulfill the phase-matching requirements (as previously mentioned, we
do not expect any filter due to the spatial [11]
or temporal [12] walkoff). The angular intensity
distribution, however, does not exhibit a flat dependence on the signal-to-pump angle of
non-collinear phase matching, *α* (see Fig. 4 for the definition): as it is evident from the single-shot
angular spectrum in Fig. 5a, the signal emission
is sharply peaked at a given a angle, whose value depends on the crystal orientation,
*θ* and Φ.

Preliminary measurements lead us to attribute this behaviour to the peculiar features of
the non-collinear tuning curves. In fact, for the given crystal orientation, the non
collinear phase-matching angle *α* initially increases at
shortening the signal wavelength from the collinear value (λ_{s} ≃ 1*μm*) toward the tuning edge (λ_{s} ≃ 0.7*μm*), it reaches a maximum value
*α*_{max} for λ_{s} ≃ 0.8*μm* and finally decreases back rapidly on
approaching the very edge of the tuning. Therefore, we obtained a very large energy
density just at *α* =
*α*_{max} , due to the large
number of wavelength which contribute to the emission in the same direction (the
measured spectral bandwidth at *α* =
*α*_{max} is Δλ_{s} ≃ 80*nm*). Moreover, the fast drop of the detector
sensitivity on going from 0.84*μm* to
1*μm* contributes in increasing the contrast between the ring
and the inner part of the circle in Fig. 5a.
Note that also the 0.76--0.7*μm* branch of the signal emission is
not detected in Fig. 5a. In fact, it is also
below the threshold of the detector sensitivity, being it distributed on a very broad
range of *α*’s.

The single-shot near-field intensity profile of the signal is presented in Fig. 5b, for the same crystal orientation and
pump-pulse parameters as in Fig. 5a. From the
figure it is evident the regular pattern structure (filamentation), whose details were
changing from shot to shot (patterns are cancelled if averaged acquisition is taken). In
agreement with our model, the characteristic spatial wavelength of this structure
decreases when the crystal is rotated in order to get larger rings in the angular
spectrum. In particular, from the 32 mrad half divergence shown in Fig. 5a one would expect a spatial wavelength of about
25*μm* in the pattern, in good agreement with the results
shown in Fig. 5b.

Clearly there are important differences between the experiment and the theoretical model
analysed in the previous sections. First of all, the experiment is performed in a pulsed
configuration, while the model assumes a cw input. Second, the experiment is cavityless.
Third the parametric down–conversion is nondegenerate, while the theoretical
model assumes degeneracy. However, in both cases the patterns arise from a gain
mechanism originated from parametric down-conversion. In the case of the OPO with
optical resonator, analysed in the theory, the gain originates from the instability at
the OPO threshold. In the experiment, the gain arises from the phase matching condition.
A theory, formulated for a cavityless configuration, which leads to results
qualitatively similar to those of the experiment, is the one which describes the onset
of filamentation from quantum noise in propagation along χ^{(3)} media
10].

Undoubtedly, there is a striking similarity between the experimental pictures and those obtained by numerical simulations of the quantum image dynamics, both in the near and in the far field. However, it must be noted, that the similarities concern only the classical aspects of the quantum image, in the sense that the same pictures could be generated by classical noise as well. In order to evidence unambiguous signatures of the quantum nature of fluctuations, as analysed in [6], one has to measure, in the far field, the spatial correlation function between intensity fluctuations of the signal and idler fields. This means that the experiment has to be performed in conditions such that both the idler field (red ring in Fig. 4) and the signal (blue ring in Fig. 4) are detected. The quantum nature of fluctuations is then shown by the fact the the maximum of correlation exists between opposite points on the two rings (the two black spots in Fig.4). Alternatively, the experiment can be realized in a degenerate configuration, as that assumed by the theory, in which only one ring is generated in the far field, and the maximal correlation arises between opposite points of the same ring [6]. These points are left for future investigations.

## Acknowledgments

P. Di Trapani and A. Berzanskis acknowledge the Deutscher Akademischer Austausch-dienst and the Vigoni-Programm 1997 for the financial support and give special thanks to Dr. W. Chinaglia for precious help in the experiment. G.-L. Oppo and R. Martin acknowledge EPSRC (grant K/70212) of the United Kingdom. This research was carried out in the framework of the activities of the Network QSTRUCT (Quantum Structures) of the TMR Program of the European Union.

## Footnotes

^{2} | For a general introduction on the field of optical pattern formation, see e.g. [3] and references quoted therein. |

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