## Abstract

We investigate the formation of transverse patterns in a doubly resonant degenerate optical parametric oscillator. Extending previous work, we treat the more realistic case of a spherical mirror cavity with a finite–sized input pump field. Using numerical simulations in real space, we determine the conditions on the cavity geometry, pump size and detunings for which pattern formation occurs; we find multistability of different types of optical patterns. Below threshold, we analyze the dependence of the quantum image on the width of the input field, in the near and in the far field.

© Optical Society of America

## Corrections

M. Marte, H. Ritsch, K. Petsas, A. Gatti, L. Lugiato, C. Fabre, and D. Leduc, "Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects: errata," Opt. Express**3**, 476-476 (1998)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-3-11-476

## 1. Introduction

Nowadays the optical parametric oscillator (OPO) has become a device with a broad range of applications [1]. The standard descriptions of the OPO are based on a small number of modes for the involved fields, which are characterized by their carrier frequency. In the standard configuration one uses a single mode for each of the involved fields (pump, signal and idler fields), which are coupled by some effective nonlinear coupling constant. All the other modes in the cavity are usually neglected, as they are assumed to be far off resonance or not coupled by the dynamics. In general this assumption is not valid and several transverse or longitudinal mode pairs may contribute to the dynamics. Recently models involving several modes have been developed [2–4]. In some cases a large number of modes contribute, leading to new dynamical phenomena. For the case of a resonator with planar mirrors, it has been demonstrated [5] that in general one has a multimode situation with spatial instabilities leading to the spontaneous formation of various types of transverse optical patterns. Interestingly, below threshold these patterns are somehow still present, but hidden in the correlations of the quantum noise (quantum images [6,7,8]).

In recent papers [2,4] we have shown that when using a resonator with nonplanar mirrors and a finite–sized pump field, a new nonlinear coupling between different signal and idler mode pairs exists. This introduces a strikingly different physical behaviour of the system and leads to new and interesting phenomena such as the combined oscillation of various modes above threshold with fixed relative phases. Even multistability between different such solutions can be obtained and the squeezing properties of the emitted light are modified [9]. In contrast to this treatment, which is restricted to a small number of effectively contributing modes, we will consider here the case of quasidegenerate cavity configuration with very small transverse mode splitting. In this limit a very large number of coupled modes participate in the dynamics. This strongly influences the pattern formation, both above threshold on a classical level and below threshold on a quantum level.

Apart from the possibility to have a prototype of a well–controlled, easy accessible and variable nonlinear dynamical system, there might also be some practical applications in image processing, storage and amplification [10,11] or ultrafast optical (de-)multiplexing. Let us finally remark that the basic structure of the underlying mathematical equations is very similiar to the Gross-Pitajewski equation governing the (almost zero-temperature) dynamics of a Bose-condensate in an external trap, where similiar features might be observable.

In this paper we analyze the case of a degenerate OPO with spherical mirrors and finite–size input beams. On a classical level, we analyze pattern formation above threshold, and show under which conditions some of the results of [5] extend to the case of spherical mirrors. On a quantum level, we consider the case of the OPO below threshold and study the variation of the quantum images in the near [12] and in the far [13] field, when the width of the input beam is gradually reduced.

## 2. Model

Let us consider a cavity with spherical mirrors and a *thin* nonlinear
*χ*
^{(2)}-crystal with an effective nonlinear
coupling strength *χ*. The cavity is coherently pumped by
an input field *E*_{p}
at a frequency
*ω*_{p}
from the outside and the single
input/output mirror is assumed highly reflective at the pump frequency as well as at
the signal frequency *ω*_{s}
=
*ω*_{p}
/2. We consider the doubly resonant
case of an OPO with a common cavity for the two fields, which have a common Rayleigh
length *z*_{r}
. For the overall geometry we restrict
ourselves to a quasi–planar or a quasi–confocal geometry. In
this case, the effective transverse mode spacing *ξ* is on
the order of, or even less than, the cavity linewidths
*k*_{s}
and *k*_{p}
for the signal and
pump field, respectively, and many transverse modes can contribute to the dynamics [4]. We will concentrate on the classical aspects of the field
dynamics first. Eliminating the longitudinal dependence by using the paraxial and
the mean field approximations, we find the following equations for the transverse
dynamics of the slowly varying pump field amplitude
*A*_{p}
(*r, ϕ, t*) and the
signal field *A*_{s}
(*r, ϕ, t*) [7]:

where the transverse variables *r* and *ϕ*
denote the distance from the axis of the system and the angular variable,
respectively. The effect of diffraction and spherical mirrors is contained in the
differential operator [12]:

Here *δ*_{k}
= ${\omega}_{k}^{00}$ - *ω*_{k}
(*k* ∈
{*p, s*}) is the detuning between the chosen carrier frequency of
the fields and the eigenfrequency of the *TEM*
_{00}-mode
closest to resonance. *E*_{p}
(*r,
ϕ*) represents an externally applied pump amplitude and ${w}_{k}=\sqrt{\frac{\left(2{z}_{r}c\right)}{{\omega}_{k}}}$ is the minimum waist of the intracavity fields. The additive noise
terms *W*_{k}
have been introduced to model fluctuations of
the pump field or other noise sources. On one hand such noise are helpful to speed
up the convergence of the numerical solutions; on the other hand, for a proper
choice of the time-correlation functions of *W*_{p}
and
*W*_{s}
Eqs. (1,2) are Langevin equations which govern the dynamics of the
system on a quantum level, in the Wigner representation [7]. Without such noise sources, we have a trivial solution of
these equations given by *A*_{s}
= 0,
*A*_{p}
= *E*_{p}
/
(*k*_{p}
+
*iδ*_{p}
). This solution is stable only below
threshold.

One notices that the eigenfunctions of each operator *L*_{k}
are just the usual transverse Gauss-Laguerre functions. Inserting the corresponding
mode expansions for all fields leads to the standard coupled mode equations [4]. Unfortunately, any analytical treatment of these equations
seems impossible at present. Let us, however, emphasize here that for the ideal
degenerate confocal cavity, *i.e*. for *ξ*
= 0, the spatial differential operator disappears from the above equations and we
get independent OPO’s at each spatial point. Such a system has quite
intriguing physical properties, as e.g. localized squeezing [14]. Note that on the other hand, in the limit in which a
quasi–planar cavity becomes an ideal planar cavity one has that
*ξ* → 0 but, simultaneously,
*w*_{k}
→ ∞ in such a way that
${\xi w}_{k}^{2}$ converges to a finite value, so that the effects of ${\nabla}_{T}^{2}$ (diffraction in the paraxial approximation) do not vanish.

The case of finite mirror size can be treated by confining the action of the
operators *L*_{k}
to the mirror surface, and by avoiding
periodic boundary conditions. Although in practice one usually works in a regime
where the mirror boundaries seem to play no essential role, the spontaneous
formation of optical patterns with broken rotational symmetry could be very
sensitive to even small boundary effects.

We will solve Eqs. (1,2) by numerical integration starting from zero for all three fields, using a combination of a split-step and a modified mid-point method [15]. Special care has to be taken concerning the noise part [7] as e.g. outlined in a recent work by M. San Miguel and R. Toral [16]. In order to test the integration procedure, we have used various boundary shapes (round, square) and sizes, as well as different noise intensities, grid sizes and time steps to check that the physical properties are independent of the numerical details of calculation.

## 3. Spatial patterns in a doubly resonant degenerate quasi–confocal OPO

We assume that the pump field has a simple Gaussian configuration with width
*w*_{e}
, *i.e*.
*E*_{p}
(*r, ϕ*) =
*E*
_{0} exp
(-*r*
^{2}/${w}_{e}^{2}$). In addition, we will assume a quasi-confocal cavity setup, which implies
inversion symmetry *A*_{k}
(*r, ϕ*)
= *A*_{k}
(*r, ϕ* +
*π*) [or alternatively
*A*_{k}
(*r, ϕ*) =
-*A*_{k}
(*r, ϕ*
+ *π*)] of all intracavity fields; this cuts
the number of points needed for the numerical solution in half.

Depending on the pump–size, pump–strength, detunings and on the
effective transverse mode spacing (including the crystal) we find a great diversity
in the physical behaviour of the system. Varying the mode spacing (e.g. by changing
the cavity geometry), we can control the number of effectively participating modes
in the dynamics from a nearly single mode case if *ξ*
≫ *k*_{s}
to a large number of contributing modes
for *ξ* ≪ *k*_{s}
. By
varying the detunings we can choose the modes out of the transverse manifold which
are dominantly excited. Similarly, by changing the shape (e.g. the size) of the pump
field we can select which modes are effectively excited. In addition, via the
pump–*size* we can also control the nonlinear
intermode coupling [2,4] from independent excitation (large pump
*w*_{e}
≫ *w*_{p}
) to a
strong mode coupling (*w*_{e}
≈
*w*_{s}
). Finally, we can influence the dynamics by
changing the pump *strength*.

Let us first consider the simple case of a perfect confocal cavity, where
*ξ* = 0. One easily finds that for points (*r,
ϕ*) where the pump intensity is large enough so that
|*E*_{p}
(*r,
ϕ*)|^{2} (${k}_{p}^{2}$ + ${\delta}_{p}^{2}$) (${k}_{s}^{2}$ + ${\delta}_{s}^{2}$)/*χ*
^{2}, the OPO is above threshold. For
instance, for *δ*_{s}
=
*δ*_{p}
= 0 and by choosing
*E*_{p}
real, one has in steady-state:

The intracavity pump field is clamped to a fixed value, and the intracavity signal field intensity at the location of the nonlinear crystal reproduces the variation of the pump field amplitude, reduced by a fixed quantity.

In contrast, for points where |*E*_{p}
(*r,
ϕ*)|^{2} <
(${k}_{p}^{2}$ + ${\delta}_{p}^{2}$) (${k}_{s}^{2}$ + ${\delta}_{s}^{2}$)/*χ*
^{2}, the OPO is below threshold. For
*δ*_{s}
=
*δ*_{p}
= 0 and *E*_{p}
real one has, e.g.:

For a Gaussian pump, one therefore finds there exists a circular area around the origin inside which the signal beam has the shape of the central part of Gaussian, and outside which the signal field is zero, whereas the intracavity pump field intensity has the shape of the outer part of a Gaussian outside the circle, and has a flat top inside the circle.

When *ξ* ≠ 0, the treshold has to be determined
numerically. In the following, we will focus on a number of selected examples. In
all calculations of this section we have *χ* =
0.1*k*_{s}
and *k*_{p}
=
3*k*_{s}
. For such values, the plane wave threshold
for a planar cavity corresponds to *E*_{p}
=
30*k*_{s}
for
*δ*_{s}
=
*δ*_{p}
= 0.

For *ξ* ≫ *k*_{s}
, the
results are qualitatively similar to those of the two-mode treatment of the
degenerate OPO. For a nonzero detuning of pump and signal fields, situations arise
that display spatially oscillating patterns, which might be related to the
self-pulsing found in [17]. The rotational symmetry is still preserved.

#### 3.1 Resonant multimode case

We proceed directly to a more complex situation and consider a small
(*ξ* ≈ *k*_{s}
),
or even very small (*ξ* ≪
*k*_{s}
) transverse mode spacing. As we can see from
Eqs. (1,2), this decreases the influence of diffraction, which
mediates spatial cross-coupling, and leads eventually to individual spatial
points oscillating independently. Let us for the moment assume
*δ*_{p}
=
*δ*_{s}
= 0, the corresponding stationary
signal field intensity distribution is shown in Fig. 1.

For a small transverse mode spacing *ξ* =
0.05*k*_{s}
, we see that inside a central region,
where the pump field is locally above threshold, the intracavity pump field is
clamped by the dynamics to its threshold value, yielding a flat top. This
behaviour is therefore very close to the one encountered in the perfect confocal
configuration. In the outside region, where the pump is below threshold, the
intracavity pump field is merely proportional to the input. This behaviour gets
more pronounced for smaller *ξ* and larger
*w*_{e}
, where more and more modes contribute to
the intracavity fields. The signal field is strongly confined to the above
threshold region, where its shape roughly corresponds to the input pump field.
Outside this region it is almost zero. The strong directional confinement of the
far field shows the transverse phase coherence of the total signal field, which
is still present despite the weak cross-coupling.

#### 3.2 Detuned multimode operation: transverse patterns

In this section we discuss the most complex situation and allow for multimode
operation as well as a detuning of the signal carrier field with respect to the
corresponding *TEM*
_{00} mode. As it is known this leads
to spatial instabilities and optical pattern formation [5] in the case of flat mirrors and plane wave input field.

Let us first look at the case of a *TEM*
_{00} input field.
Many modes are excited and we have a fairly strong cross-coupling between the
different modes. This leads to a fixed relative phase operation of several
modes, yielding various rather smooth spatial distributions, which can in some
sense be interpreted as an effective oscillating mode. This notion is, however,
somewhat artificial as the effective mode depends on the pump strength and pump
shape. An example is shown in Fig. 2, where we plot the stationary intracavity pump and
signal fields for a small transverse mode spacing.

The signal far field exhibits a ring-shaped distribution, which closely reminds
the below threshold results found in [7], while the near field configuration is Bessel-like. Only
in the case of a very small mode coupling *ξ* =
0.075*k*_{s}
, we find a pattern with broken
rotational symmetry, similar to the roll patterns discussed just below.

For a wider input pump field (*w*_{e}
≫
*w*_{p}
) again many modes are possibly excited,
but the relative phase coupling is less strong and there is more room for
dynamical adjustment of the fields. This allows for breaking rotational symmetry
and for the spontaneous dynamical formation of various optical patterns. The
situation is richer than in the plane wave case. In the following we will show
this on some specific examples.

Close to threshold one finds a roll (stripe) pattern as shown in Fig. 3, where we have small mode coupling, broad resonant pump and negative signal detuning.

Once such stripes in some direction have formed, they only very slowly change
their direction, driven by the input noise. However, comparing many different
runs, starting each time at zero field with a small amount of
“white” noise the stripes are randomly oriented, but show
a well defined modulation length depending on
*δ*_{s}
and *E*_{p}
. In
some rare cases a ring shape pattern appears. Once such a pattern has formed, it
seems to be stable against the formation of stripes even for a fairly large
amount of noise and for slow changes of the system parameters. As one might
expect, the stripe pattern leads to a two-peaked intensity distribution in the
far field.

For an increased pump strength other types of patterns appear. Roughly at 1.75
times threshold, spiral shaped patterns form frequently, as shown in Fig. 4. Once formed they show a slow rotation,
*i.e*. the end of the arms grow. Again, these turn out to be
quite stable against noise and slow changes of the system parameters. The
rotation speed depends on *E*_{p}
and
*w*_{p}
and is much slower than the other time scales
in the system. For these parameter ranges stripes and spirals are stable at the
same time. Even seeding a stripe pattern into the input for an existing spiral
has no effect, as the modification of the pump field induced by the spiral
prevents gain for the seeded stripes, and vice versa. Hence there is
multistability of various patterns induced *via* the backaction
of the existing pattern on the pump field.

Even further above threshold new patterns, cointaining a number of point–like structures, appear. Typical examples in the near field are shown in Fig. 5.

In all this, the shape of the mirror seems to have little influence; it is the finiteness and size of the pump, which plays the most important role.

## 4. Quantum images below threshold

In this section we consider a quasi–planar cavity, instead of
quasi–confocal. Below threshold, and not too close to threshold, the pump
depletion can be neglected. If moreover, as in [12], [13] we assume that the pump field is not reflected at all by
the cavity mirrors, the quantity *A*_{p}
in Eq. (2) can be expressed as a given function of *r,
ϕ* equal to the input field *E*_{in}
.
Hence, Eq. (2) becomes self-contained for the signal field
*A*_{s}
(*r,ϕ,t*), and Eq. (1) can be dropped altogether.

If we consider any circle 𝑐 centered on the system axis, at steady-state
below threshold, any observable 𝓞(*r, ϕ, t*)
is uniform on average over the circle because of the cylindrical symmetry and is
constant in time. However, if we consider the spatial correlation function at
steady-state *F*(*r*,
Δ*ϕ, t*) = ⟨𝓞
(*r, ϕ, t*) 𝓞 (*r,
ϕ′*, 0)⟩ where *r* is
the radius of circle 𝑐, we obtain a function of
Δ*ϕ* = *ϕ* -
*ϕ′* which exhibits a spatial modulation.
This structure, visible in the spatial correlation function, has been called
“*quantum image*” [6,12]. The same is true for the spectrum associated with the
correlation

As in [12,13], we will consider as observable a quadrature of the signal
field, 𝓞 = *A*_{s}
exp(-*iφ*_{L}
) +
${A}_{s}^{*}$ exp(*iφ*_{L}
), for an appropriate phase
*φ*_{L}
of the local oscillator field used
to detect the quadrature. While in [12,13] the input field was a plane wave, we assume here that it
corresponds to a gaussian profile of width *w*_{e}
, and
analyze how the result changes when we decrease gradually
*w*_{e}
from infinity (plane wave case) to values on the
order of *w*_{s}
. In particular, we want to see whether the
main phenomena identified in [12,13] persist, or are washed out by the finite size of the pump.

The calculation of the correlation function is carried out with the help of an
expansion of the signal field in terms of Gauss-Laguerre modes [12,13]. In the present case, however, the finiteness of
*w*_{e}
couples the modes [2,4]; a complete description of the calculation will be given in
a future paper [18]. Figures 6 and 7 show the spectral density *F̃*
(*r*, Δ*ϕ*,
*ω*) [normalized to
*F̃*(*r*,
Δ*ϕ* = 0, *ω*)]
for *ω* = 0. In the near field [*i.e*. in Figs. 6(a) and 7(a)] we take *r* =
*w*_{s}
and *φ*_{L}
= 0. In
the far field [Figs. 6(b) and 7(b)], on the other hand, we evaluate the correlation
function at a distance *z* = 20*z*_{r}
from
cavity center, and we take $r={w}_{s}{\left[1+{\left(\frac{z}{{z}_{r}}\right)}^{2}\right]}^{\frac{1}{2}}$ and [19]

where *q*_{c}
is the order of a frequency-degenerate family of
Gauss-Laguerre modes which is exactly resonant with the signal frequency
*ω*_{s}
[12] and *q* = 2*p* +
*l*, where *p* and *l* are the
radial and angular indices of the Gauss-Laguerre functions, respectively. In both figures 6 and 7 we have *q*_{c}
= 3. The value of
the input field is measured as a dimensioneless parameter
*Ē*_{in}
=
*E*_{in}*χ*/*k*_{s}
; in
the plane wave limit *w*_{e}
→ ∞, the
threshold value is *Ē*_{in}
= 1.

When the OPO is below but close to threshold the main feature is the presence of a
regular modulation in the correlation as a function of
Δ*ϕ*. For the value of
*Ē*_{in}
considered in Fig. 6, the system is 10% below threshold in the plane wave
limit *w*_{e}
→ ∞. As shown in [12] for *w*_{e}
= ∞, when
approaching the threshold the contribution of the resonant family becomes dominant
and, because the circle 𝑐 is chosen in such a way that the modes
*p* = 1, *l* = 1 vanish, the correlation function
*F̃* (*r*,
Δ*ϕ*, *ω* =
0)/*F̃*(*r*,
Δ*ϕ* = 0, *ω* =
0) arises only from the contribution of the two modes *p* = 0,
*l* = 3, so that getting close to threshold the curve approaches
the function cos (3Δ*ϕ*), as shown by the red
line in Fig. 6(a). Reducing *w*_{e}
, the
modulation becomes less regular and less pronounced, but it is still quite
remarkable for *w*_{e}
= 14*w*_{s}
. It
must be taken into account, in addition, that decreasing
*w*_{e}
the threshold value for
*Ē*_{in}
increases.

On the other hand, well below threshold the interesting feature arises in the far
field [13]. Precisely, as shown by Fig. 7(b), there is a peak for
Δ*ϕ* = *π* higher
than the peak in Δ*ϕ* = 0. As shown in [19,20] this is a purely quantum effect which provides evidence
from a spatial viewpoint of the twin photon emission in the OPO. Figure 7 shows that this feature (which becomes more and more
pronounced as *ξ* is decreased) is quite robust with
respect to the reduction of *w*_{e}
, up to when
*w*_{e}
becomes on the order of
*w*_{s}
. The correlation as a function of
Δ*ϕ* becomes broader as
*w*_{e}
is decreased.

## Acknowledgements

This research was carried out in the framework of Network QSTRUCT
(“*Quantum Structures*”) of the TMR program
of the EU, and supported by the Austrian Science Foundation FWF Project No. S6506.
M. M. was supported by an *APART* fellowship of the Austrian Academy
of Sciences.

## References and links

**1. ** Special issue on *χ*^{2} second order nonlinear optics, from fundamentals to applications, edited by C. Fabre and J.-P. Pocholle, in JEOS B: J. Quantum and Semiclassical Opt. 9, (2) (1997). Special issue on Optical Parametric Oscillators, in Applied Physics B, May 1998.

**2. **M.A.M. Marte, H. Ritsch, L. A. Lugiato, and C. Fabre, “Simultaneous multimode optical parametric oscillations in a triply resonant cavity,” Acta Physica Slovaca **47**, 233 (1997).

**3. **G. S. Agarwal and S. D. Das Gupta, “Model for mode hopping in optical parametric oscillators,” J. Opt. Soc. Am. B **14**, 2174 (1997). [CrossRef]

**4. **C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode couplings in optical parametric oscillators,” submitted to Appl. Phys. B, Special Issue on Optical Parametric Oscillators, edited by. J. Mlynek and S. 1Schiller.

**5. **G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A **49**, 2028 (1994). [CrossRef] [PubMed]

**6. **A. Gatti and L.A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A **52**, 1675 (1995). [CrossRef] [PubMed]

**7. **A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A **56**, 877 (1997). [CrossRef]

**8. **A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” Opt. Express **1**, 21 (1997), http://epubs.osa.org/oearchive/source/1968.htm. [CrossRef] [PubMed]

**9. **C. Schwob and C. Fabre, “Squeezing and quantum correlations in multimode optical parametric oscillators,” preprint, to be submitted to JEOS B: J. Quantum and Semiclassical Opt.

**10. **E. Lantz and F. Devaux, “Parametric amplification of images,” JEOS B: J. Quantum and Semi-classical Opt. **9**, 279 (1997).

**11. **M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A **52**, 4930 (1995). [CrossRef] [PubMed]

**12. **L. A. Lugiato and I. Marzoli, “Quantum spatial correlations in the optical parametric oscillator with spherical mirrors,” Phys. Rev. A **52**, 4886 (1995). [CrossRef] [PubMed]

**13. **L. A. Lugiato, S. M. Barnett, A. Gatti, I. Marzoli, G.-L. Oppo, and H. Wiedemann, “Quantum Images,” J. of Nonlinear Optical Phys. and Materials **5**, 809 (1996). [CrossRef]

**14. **L. A. Lugiato and P. Grangier, “Improving quantum-noise reduction with spatially multimode squeezed light,” J. Opt. Soc. Am. B **14**, 225 (1997). [CrossRef]

**15. **W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, *Numerical Recipes; the art of scientific computing*, Cambridge University Press, Cambridge (1986).

**16. **M. SanMiguel and R. Toral, *Instabilities and Nonequlibrium structures, VI*, Kluwer Academic Pub. (1997).

**17. **L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing snd chaos in optical parametric oscillators,” Il Nuovo Cimento 10 D, 959 (1988).

**18. **K. I. Petsas, A. Gatti, and L. A. Lugiato, “Quantum images in optical parametric oscillators with spherical mirrors and gaussian pump,” submitted to JEOS B: J. Quantum and Semiclassical Opt.

**19. **L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli, and G.-L. Oppo, “Quantum images in nonlinear optics,” J. Mod. Opt. **44**, 1899 (1997). [CrossRef]

**20. **I. Marzoli, A. Gatti, and L. A. Lugiato, “Spatial quantum signatures in parametric down-conversion,” Phys. Rev. Lett. **78**, 2092 (1997). [CrossRef]