# A classical model of spontaneous parametric down-conversion

Girish Kulkarni<EMAIL_ADDRESS>Department of Physics,
University of Ottawa, Ottawa, Ontario K1N 6N5, Canada  Jeremy Rioux
Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
Boris Braverman  Present address: QuEra Computing, Boston, MA 02135, USA
Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
Maria V. Chekhova  Max Planck Institute for the Science of Light, Staudtstr.
2, 91058 Erlangen, Germany  University of Erlangen-Nuremberg, Staudtstr. 7/B2,
91058 Erlangen, Germany  Robert W. Boyd  Department of Physics, University of
Ottawa, Ottawa, Ontario K1N 6N5, Canada  The Institute of Optics, University
of Rochester, Rochester, New York 14627, USA

###### Abstract

We model spontaneous parametric down-conversion (SPDC) as classical difference
frequency generation (DFG) of the pump field and a hypothetical stochastic
“vacuum” seed field. We analytically show that the second-order spatiotemporal
correlations of the field generated from the DFG process replicate those of
the signal field from SPDC. Specifically, for low gain, the model is
consistent with the quantum calculation of the signal photon’s reduced density
matrix; and for high gain, the model’s predictions are in good agreement with
our experimental measurements of the far-field intensity profile, orbital
angular momentum spectrum, and wavelength spectrum of the SPDC field for
increasing pump strengths. We further theoretically show that the model
successfully captures second-order SU(1,1) interference and induced coherence
effects in both gain regimes. Intriguingly, the model also correctly predicts
the linear scaling of the interference visibility with object transmittance in
the low-gain regime – a feature that is often regarded as a quintessential
signature of the nonclassicality of induced coherence. Our model may not only
lead to novel fundamental insights into the classical-quantum divide in the
context of SPDC and induced coherence, but can also be a useful theoretical
tool for numerous experiments and applications based on SPDC.

##  I Introduction

Spontaneous parametric down-conversion (SPDC) is a nonlinear optical
phenomenon in which an incident field known as pump interacts with a non-
centrosymmetric medium to produce a pair of fields known as signal and idler
[1, 2, 3]. In the quantum paradigm, the interaction can be modeled as a series
of infinitely-many contributing processes, where the $n$’th order contribution
physically corresponds to $n$ pump photons getting annihilated to produce $n$
signal photons and $n$ idler photons simultaneously [4]. The strength of the
interaction, referred to as gain, is directly proportional to the pump
amplitude [5]. For typical continuous-wave and low-power pulsed pump lasers,
the interaction is weak and can be approximated by the dominant first-order
contribution alone, which yields an entangled two-photon signal-idler state
[6, 7, 8]. In this low-gain regime, which is characterized by a linear growth
of the generated fields with respect to the pump amplitude, SPDC sources are
ubiquitously employed in fundamental quantum optics experiments [9, 10, 11,
12] and optical quantum technologies [13, 14, 15]. On the other hand, for
high-power pulsed pump lasers, the interaction can be strong and may comprise
of significant higher-order contributions, thereby producing a bright
multiphoton entangled state of the signal and idler fields [16, 17]. In this
high-gain regime, which is characterized by an exponential growth of the
generated fields with respect to the pump amplitude, SPDC has significant
potential for applications such as sub-shot noise imaging [18, 19] and
generation of nonclassical states of light [20, 21].

While the precise form of two-photon state produced in the perturbative low-
gain regime has long been analytically derived for a general pump profile [6,
7, 8], the theoretical characterization of the non-perturbative high-gain
regime has proved to be much more difficult. In order to correctly evaluate
higher-order contributions, one must account for the fact that the interaction
Hamiltonian may not commute with itself at different times, and consequently,
the Schrodinger time-evolution operator must strictly be expanded in terms of
the time-ordered Dyson series instead of the usual Taylor series [22].
However, it was shown that any finite-order truncation of the Dyson series
destroys the Gaussian character of the output state [23], whereas the exact
solution can be proved to be a Gaussian state solely from the quadratic form
of the interaction Hamiltonian [24]. Such issues can be avoided by turning to
the Heisenberg picture, where one instead obtains a set of coupled
differential equations for the evolution of the signal and idler mode
operators [5, 25, 26, 27]. However, these equations have so far been exactly
solved only for the special case of a monochromatic plane-wave pump [5], and
approximately solved for the cases of a broadband pump with a single spatial
mode [25], and a pump with narrow spectral and angular bandwidth [16]. For
more general cases, numerical solutions that are often computationally
intensive have to be sought. Some studies have explored analytical ansatz
solutions that assume gain-independent Schmidt modes for the output state [28,
29]. However, these solutions have limited applicability as they are unable to
account for experimental observations such as broadening of the angular
spectrum and wavelength spectrum of the SPDC field with increasing gain [30,
31]. Thus, a consistent analytical quantum description for high-gain SPDC has
still not been achieved for a general pump field.

In view of the aforementioned challenges encountered in performing a general
quantum calculation, one can instead ask: is there a classical analytic
description that can consistently explain at least a restricted class of
correlations in high-gain SPDC? For instance, in the low-gain regime, it is
known that second-order (in fields) autocorrelations of the individual fields
can be modeled classically [32, 33, 34, 35, 36]. A simple and elegant approach
in this context was developed in Ref. [32], where SPDC is modeled as the
paraxial propagation of a classical partially-coherent source created in the
nonlinear medium by the coupling of the pump field to a classical noise field
that simulates the vacuum fluctuations. The second-order correlations derived
using this approach were demonstrated to be consistent with the quantum
calculation of the reduced one-photon density matrix in the low-gain regime.
However, this approach does not appear to admit a straightforward
generalization to the high-gain regime as the generated field is a priori
assumed to scale linearly with the pump amplitude. Another approach, which has
also been neatly summarized in Ref. [32], is to model SPDC as a three-wave
mixing process between the pump and the vacuum fluctuations [33, 34, 35]. In
particular, the spatial and spectral profiles of the SPDC field can be derived
by solving differential equations involving a stochastic term that represents
the vacuum fluctuations [34, 35], although the origin of coherence is less
clear and has been attributed to various specific experimental conditions [37,
38]. However, to our knowledge, this latter approach has not been taken beyond
the monochromatic plane-wave pump approximation in the high-gain regime to
derive analytic solutions for realistic pump fields with finite spectral and
angular bandwidth. Moreover, this latter approach has also not yet been
employed for analyzing two-crystal settings such as SU(1,1) interferometers
[39, 40] and induced coherence experiments [11, 41].

In this work, we adopt the aforementioned latter approach and model SPDC as
classical difference frequency generation (DFG) of the pump field and a
classical stochastic seed field that simulates the zero-point vacuum
fluctuations. In contrast with previous studies, we derive analytical
expressions for the second-order spatiotemporal correlations of the generated
signal field for a narrow-but-finite-bandwidth pump in low and high gain
regimes. The paper is organized as follows: In Section II, we set up the model
and derive the second-order spatiotemporal correlation function of the
generated field in low and high gain regimes. In Section III, we perform
experimental measurements of the far-field intensity profile, orbital angular
momentum spectrum, and wavelength spectrum for increasing pump strengths and
demonstrate their agreement with the model’s predictions. In Section IV, we
extend the model to analyze SU(1,1) interference and induced coherence effects
in both low and high gain regimes. In Section V, we conclude with a summary
and outlook.

##  II Theoretical Model

Figure 1: Modeling SPDC as DFG of the pump with a hypothetical classical
“vacuum” seed field.

As depicted in Fig. 1, we model SPDC as DFG of the pump field $A_{p}$ with a
hypothetical classical seed field $A_{i}$ that mimics the effect of zero-point
vacuum fluctuations. Our goal is to compute the second-order spatiotemporal
correlations of the generated field $A_{s}$. We approximate the fields to be
paraxially propagating along the longitudinal $z$-axis. A crystal of length
$L$ and second-order nonlinear susceptibility $\chi^{(2)}$ is placed
perpendicular to the $z$-axis with its input face defining $z=0$. The Fourier
amplitudes (not including propagation phases) for the pump field, classical
“vacuum”, and the generated fields are denoted by stochastic functions
$A_{j}({\bm{q}_{j}},\omega_{j},z)$, where ${\bm{q}_{j}}$ and $\omega_{j}$
represent the transverse wavevector and frequency coordinates, for $j=p,i,$
and $s$, respectively. In terms of the position vector
${\bm{r}}\equiv({\bm{\rho}},z)$, where ${\bm{\rho}}$ is transverse position,
and wavevectors ${\bm{k}_{j}}\equiv({\bm{q}_{j}},k_{jz})$, we define the
electric field mode functions

$\displaystyle\bm{E}_{j}({\bm{r}},\omega_{j})$
$\displaystyle=\int\mathrm{d}{\bm{q}_{j}}A_{j}({\bm{q}_{j}},\omega_{j},z)\,e^{i({\bm{q}_{j}}\cdot{\bm{\rho}}+k_{jz}z)}.$
(1)

We now assume that the energy depletion of the pump due to the nonlinear
interaction is negligible, and consequently, the pump amplitude
$A_{p}({\bm{q}_{p}},\omega_{p})$ is independent of $z$. We emphasize that the
pump field can be pulsed or continuous-wave with any arbitrary spectral and
spatial profile. The evolution of the other two fields is governed by the wave
equation of nonlinear optics (see Eq. (2.1.23) of Ref. [3])

$\displaystyle\nabla^{2}\bm{E}_{j}({\bm{r}},\omega_{j})+\frac{n^{2}_{j}\omega^{2}_{j}}{c^{2}}\bm{E}_{j}({\bm{r}},\omega_{j})=-\frac{\omega^{2}_{j}}{\epsilon_{0}c^{2}}\bm{P}_{NL}({\bm{r}},\omega_{j}),$
(2)

where $j=s,i$ and $n_{j}\equiv n_{j}(\omega_{j})$ is the refractive index
inside the crystal. Assuming energy conservation, i.e,
$\omega_{p}=\omega_{s}+\omega_{i}$, the nonlinear polarization
$\bm{P}_{NL}({\bm{r}},\omega_{j})$ can be written as (see Eq. (1.5.30) of Ref.
[3])

$\displaystyle\bm{P}_{NL}({\bm{r}},\omega_{j})=4\epsilon_{0}d_{\rm
eff}\int\mathrm{d}\omega_{p}\,\bm{E}_{p}({\bm{r}},\omega_{p})\bm{E}_{l}^{*}({\bm{r}},\omega_{l}),$
(3)

where $j=s(i)$ and $l=i(s)$. In our scalar treatment, the quantity $d_{\rm
eff}=\chi^{(2)}/2$ is the nonlinear coupling coefficient of the crystal
medium, and its dependence on the frequencies of the coupling waves has been
ignored. Substituting equations (1) and (3) in (2) and making the slowly-
varying envelope approximation $\frac{\partial^{2}}{\partial
z^{2}}A_{j}({\bm{q}_{j}},\omega_{j},z)\ll k_{jz}\frac{\partial}{\partial
z}A_{j}({\bm{q}_{j}},\omega_{j},z)$ for $j=s,i$, we obtain (see Appendix A for
the detailed calculation)

$\displaystyle\frac{\partial A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial z}$
$\displaystyle=\frac{2id_{\rm
eff}\omega^{2}_{s}}{k_{sz}c^{2}}\iint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,A_{p}({\bm{q}_{p}},\omega_{p})$
$\displaystyle\hskip 56.9055pt\times
A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)e^{i\Delta k_{z}z},$ (4a)
$\displaystyle\frac{\partial A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)}{\partial
z}$ $\displaystyle=\frac{-2id_{\rm
eff}\omega^{2}_{i}}{k_{iz}c^{2}}\iint\mathrm{d}\omega_{s}\,\mathrm{d}{\bm{q}_{s}}\,A^{*}_{p}({\bm{q}_{p}},\omega_{p})$
$\displaystyle\hskip 42.67912pt\times
A_{s}({\bm{q}_{s}},\omega_{s},z)e^{-i\Delta k_{z}z},$ (4b)

where $\Delta k_{z}=k_{pz}-k_{sz}-k_{iz}$ is the longitudinal phase mismatch,
and its dependence on ${\bm{q}_{s}},{\bm{q}_{i}},\omega_{s},$ and $\omega_{i}$
has been suppressed for brevity. Also, we have assumed that the transverse
dimensions of the crystal are much larger than those of the pump beam, which
leads to conservation of transverse momentum, i.e,
${\bm{q}_{p}}={\bm{q}_{s}}+{\bm{q}_{i}}$. The above coupled differential
equations (4) describe the nonlinear interaction for arbitrary strengths of
the pump field.

###  II.1 Low-gain regime

When the pump power is low, the nonlinear interaction is weak. In the context
of DFG with monochromatic plane waves, it is known that under weak
interaction, the longitudinal growth of the seed wave is negligible (see Sec.
2.8 of Ref. [3]). Hence, we approximate
$A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)\approx
A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)$ and simply integrate Eq. (4a) from $z=0$
to $z=L$ to obtain

$\displaystyle A_{s}({\bm{q}_{s}},\omega_{s},L)=\frac{2id_{\rm
eff}\omega^{2}_{s}}{k_{sz}c^{2}}\iint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,A_{p}({\bm{q}_{p}},\omega_{p})$
$\displaystyle\hskip 85.35826pt\times
A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\int_{0}^{L}\mathrm{d}z\,e^{i\Delta
k_{z}z}$ $\displaystyle=\frac{d_{\rm
eff}L\omega^{2}_{s}}{k_{sz}c^{2}}\iint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,A_{p}({\bm{q}_{p}},\omega_{p})A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)$
$\displaystyle\hskip 42.67912pt\times\mathrm{sinc}\left(\Delta
k_{z}L/2\right)\,\mathrm{exp}\left(i\Delta k_{z}L/2\right),$ (5)

where $\mathrm{sinc}(x)\equiv(\sin x)/x$. Using the above equation, we can
evaluate the second-order correlation function [42]

$\displaystyle\langle
A_{s}({\bm{q}_{s}},\omega_{s},L)A^{*}_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s},L)\rangle=\frac{d_{\rm
eff}^{2}L^{2}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{k_{sz}k^{\prime}_{sz}c^{4}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}$
$\displaystyle\hskip
8.53581pt\times\mathrm{d}\omega^{\prime}_{i}\,\mathrm{d}{\bm{q}^{\prime}_{i}}\,\langle
A_{p}({\bm{q}_{s}}+{\bm{q}_{i}},\omega_{s}+\omega_{i})A^{*}_{p}({\bm{q}^{\prime}_{s}}+{\bm{q}^{\prime}_{i}},\omega^{\prime}_{s}+\omega^{\prime}_{i})\rangle$
$\displaystyle\hskip 36.98857pt\times\langle
A_{i}({\bm{q}^{\prime}_{i}},\omega^{\prime}_{i},0)A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\rangle\,\mathrm{sinc}\left(\Delta
k_{z}L/2\right)$ $\displaystyle\hskip
65.44133pt\times\mathrm{sinc}\left(\Delta
k^{\prime}_{z}L/2\right)\,e^{i\left(\Delta k_{z}-\Delta
k^{\prime}_{z}\right)L/2},$ (6)

where $\langle\cdots\rangle$ denotes an ensemble average and $\Delta
k^{\prime}_{z}$ is the phase mismatch evaluated for the primed variables. We
have assumed that the pump and classical “vacuum” fields have no mutual
correlation, and as a result, their individual correlation functions factor
out separately.

In order to evaluate the above equation (6), we must now substitute for
$\langle
A_{i}({\bm{q}^{\prime}_{i}},\omega^{\prime}_{i},0)A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\rangle$.
While it may seem reasonable to assume that different modes of the classical
“vacuum” are completely uncorrelated, i.e, $\langle
A_{i}({\bm{q}^{\prime}_{i}},\omega^{\prime}_{i},0)A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\rangle=C\,\delta({\bm{q}_{i}}-{\bm{q}^{\prime}_{i}})\delta(\omega_{i}-\omega^{\prime}_{i})$,
it is possible to explicitly derive this relation along with the value of the
scaling factor $C$ if we make a short detour to quantum theory. We note from
the quantized theory of the electromagnetic field that (see Sec. 2.3.3 of Ref.
[43])

$\displaystyle\hat{\bm{E}_{i}}({\bm{r}},\omega_{i})=i\int\mathrm{d}{\bm{q}_{i}}\,\sqrt{\frac{\hbar\omega_{i}}{4\pi\epsilon_{0}}}\,\hat{a}({\bm{q}_{i}},\omega_{i})\,e^{i({\bm{q}_{i}}\cdot{\bm{\rho}}+k_{iz}z)},$
(7)

where $\hat{\bm{E}_{i}}({\bm{r}},\omega_{i})$ is the quantized operator
corresponding to the classical amplitude $\bm{E}_{i}({\bm{r}},\omega_{i})$
defined in Eq. (1) and $\hat{a}({\bm{q}_{i}},\omega_{i})$ is the annihilation
operator for the mode specified by its arguments. Upon comparing equations (1)
and (7), it is evident that $A_{i}({\bm{q}_{i}},\omega_{i},0)$ corresponds to
$i\sqrt{\frac{\hbar\omega_{i}}{4\pi\epsilon_{0}}}\,\hat{a}({\bm{q}_{i}},\omega_{i})$.
Now for $A_{i}({\bm{q}_{i}},\omega_{i},0)$ to effectively simulate the zero-
point vacuum fluctuations, we must have $\langle
A_{i}({\bm{q}^{\prime}_{i}},\omega^{\prime}_{i},0)A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\rangle=\frac{\hbar\sqrt{\omega_{i}\omega^{\prime}_{i}}}{4\pi\epsilon_{0}}\langle\mathrm{vac}|\hat{a}_{i}({\bm{q}^{\prime}_{i}},\omega^{\prime}_{i})\hat{a}^{\dagger}_{i}({\bm{q}_{i}},\omega_{i})|\mathrm{vac}\rangle$,
where $|\mathrm{vac}\rangle$ is the quantum state of the vacuum. Using the
relation
$\langle\mathrm{vac}|\hat{a}_{i}({\bm{q}^{\prime}_{i}},\omega^{\prime}_{i})\hat{a}^{\dagger}_{i}({\bm{q}_{i}},\omega_{i})|\mathrm{vac}\rangle=\delta({\bm{q}^{\prime}_{i}}-{\bm{q}_{i}})\delta(\omega_{i}-\omega^{\prime}_{i})$
from quantum theory [44], we obtain

$\displaystyle\langle
A_{i}({\bm{q}^{\prime}_{i}},\omega^{\prime}_{i},0)A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\rangle=\frac{\hbar\omega_{i}}{4\pi\epsilon_{0}}\delta({\bm{q}_{i}}-{\bm{q}^{\prime}_{i}})\delta(\omega_{i}-\omega^{\prime}_{i}).$
(8)

Thus, we find that the different modes of the classical “vacuum” are indeed
completely uncorrelated. Moreover, the scaling factor
$\hbar\omega_{i}/4\pi\epsilon_{0}$ ensures consistency with the zero-point
energy of the vacuum fluctuations. The above derivation of Eq. (8) is the only
calculation in this paper that explicitly appeals to quantum theory [45].

We now substitute Eq. (8) in Eq. (6), replace the slowly-varying term
$\omega_{i}$ by its central value $\omega_{i0}$, and take it outside the
integral to obtain

$\displaystyle\langle
A_{s}({\bm{q}_{s}},\omega_{s},L)A^{*}_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s},L)\rangle=\frac{\hbar\omega_{i0}d_{\rm
eff}^{2}L^{2}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{4\pi\epsilon_{0}k_{sz}k^{\prime}_{sz}c^{4}}$
$\displaystyle\times\iint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\langle
A_{p}({\bm{q}_{s}}+{\bm{q}_{i}},\omega_{s}+\omega_{i})A^{*}_{p}({\bm{q}^{\prime}_{s}}+{\bm{q}_{i}},\omega^{\prime}_{s}+\omega_{i})\rangle$
$\displaystyle\hskip 8.53581pt\times\mathrm{sinc}\left(\Delta
k_{z}L/2\right)\mathrm{sinc}\left(\Delta
k^{\prime}_{z}L/2\right)\,e^{i\left(\Delta k_{z}-\Delta
k^{\prime}_{z}\right)L/2}.$ (9)

The above expression is consistent with the reduced density matrix of the
signal photon obtained from the full quantum-mechanical treatment of low-gain
SPDC [6, 8, 7]. The integration over the classical “vacuum” modes on the
right-hand side of Eq. (9) is formally equivalent to performing a partial
trace of the two-photon density matrix over the idler photon. Thus, the
generated field from DFG in the low-gain regime replicates the signal field
from SPDC with respect to second-order spatiotemporal correlations, and in
this context the classical “vacuum” field in DFG plays a role equivalent to
that of the idler field from SPDC. However, we emphasize that the latter
equivalence is valid only for the specific purpose of deriving the second-
order correlations of the signal field because the correlations of the
classical “vacuum” do not reflect those of the idler field in reality.

###  II.2 High-gain regime

When the pump power is high and consequently, the nonlinear interaction is
strong, we can no longer ignore the longitudinal growth of the seeding
classical “vacuum”. As a result, the derivation of the signal field
correlations is slightly more involved and proceeds a bit differently. In
particular, we will now use the position-time representation
$V_{p}({\bm{\rho}},t)$ of the pump amplitude, which is related to the Fourier
amplitude $A_{p}({\bm{q}_{p}},\omega_{p})$ as

$\displaystyle
V_{p}({\bm{\rho}},t)=\iint\mathrm{d}{\bm{q}_{p}}\mathrm{d}\omega_{p}\,A_{p}({\bm{q}_{p}},\omega_{p})e^{i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}.$
(10)

We invert the Fourier relation in Eq. (10), substitute it in Eq. (4), and
differentiate with respect to $z$ to obtain

$\displaystyle\frac{\partial^{2}A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial
z^{2}}=\frac{2id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t$
$\displaystyle\hskip 28.45274pt\times
V_{p}({\bm{\rho}},t)e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}\,e^{i\Delta
k_{z}z}\Bigg{[}\frac{\partial A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)}{\partial
z}\Bigg{.}$ $\displaystyle\hskip 113.81102pt\Bigg{.}+i\Delta
k_{z}A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)\Bigg{]}.$ (11)

In most experiments, the angular and frequency bandwidths $\delta{\bm{q}_{p}}$
and $\delta\omega_{p}$ of the pump are much smaller than the corresponding
bandwidths $\delta{\bm{q}_{s}}$ and $\delta\omega_{s}$ of the generated SPDC
field, respectively. Under this condition of a “narrow-band pump”, the
variation of $\Delta k_{z}$ with respect to ${\bm{q}_{p}}$ and $\omega_{p}$ is
much slower than that of $A_{p}({\bm{q}_{p}},\omega_{p})$, which is usually
sharply-peaked around the central wavevector ${\bm{q}_{p0}}=0$ and the central
pump frequency $\omega_{p0}$. As a result, one can define $\Delta\bar{k}_{z}$
as the central value of $\Delta k_{z}$ evaluated for the conditions
${\bm{q}_{s}}+{\bm{q}_{i}}=0$ and $\omega_{s}+\omega_{i}=\omega_{p0}$,
approximate $\Delta k_{z}\approx\Delta\bar{k}_{z}$ (this approximation is
exactly true only for the monochromatic plane-wave pump case) in the second
term on the right hand side, and take it outside the integral. We note that
any quantity appearing with an overbar notation in the rest of this paper must
be understood as the central value of that quantity evaluated for the
aforementioned conditions. Using equations (4) and (10) in Eq. (11) (see
Appendix B.I for details), we obtain

$\displaystyle\frac{\partial^{2}A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial
z^{2}}-i\Delta\bar{k}_{z}\frac{\partial
A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial z}$ $\displaystyle\hskip
85.35826pt-\bar{G}^{2}({\bm{\rho}},t)\,A_{s}({\bm{q}_{s}},\omega_{s},z)=0,$
(12)

where we have defined

$\displaystyle G^{2}({\bm{\rho}},t)=\frac{4d_{\rm
eff}^{2}\omega^{2}_{s}\omega^{2}_{i}}{k_{sz}k_{iz}c^{4}}|V_{p}({\bm{\rho}},t)|^{2}.$
(13)

Upon solving Eq. (12) subject to the initial conditions
$A_{s}({\bm{q}_{s}},\omega_{s},z=0)=0$ and $\partial
A_{s}({\bm{q}_{s}},\omega_{s},z)/\partial z|_{z=0}$ evaluated using equations
(4a) and (10), we obtain (see Appendix B.II for details)

$\displaystyle A_{s}({\bm{q}_{s}},\omega_{s},L)=\frac{2id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V_{p}({\bm{\rho}},t)$
$\displaystyle\hskip 28.45274pt\times
e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}\,A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right]$
$\displaystyle\hskip 113.81102pte^{i(\Delta k_{z}-\Delta\bar{k}_{z}/2)L},$
(14)

where

$\displaystyle\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)\equiv\sqrt{\bar{G}^{2}({\bm{\rho}},t)-\left(\frac{\Delta\bar{k}_{z}}{2}\right)^{2}}.$
(15)

Using equations (8) and (14), and taking $\omega_{i}\approx\omega_{i0}$
outside the integral, we find (see Appendix B.III for the detailed
calculation)

$\displaystyle\langle
A_{s}({\bm{q}_{s}},\omega_{s},L)A^{*}_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s},L)\rangle=\frac{2\hbar\omega_{i0}d^{2}_{\rm
eff}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{(2\pi)^{4}\epsilon_{0}k_{sz}k^{\prime}_{sz}c^{4}}$
$\displaystyle\hskip
14.22636pt\times\iint\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,\langle|V_{p}({\bm{\rho}},t)|^{2}\rangle\,e^{-i[({\bm{q}_{s}}-{\bm{q}^{\prime}_{s}}){\bm{\rho}}-(\omega_{s}-\omega^{\prime}_{s})t]}$
$\displaystyle\hskip
28.45274pt\times\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right]\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}},t)}\right]$
$\displaystyle\hskip 113.81102pt\times
e^{i(\Delta\bar{k}_{z}-\Delta\bar{k}^{\prime}_{z})L/2}.$ (16)

The above expression quantifies the second-order spatiotemporal correlations
of the generated field for a narrow-band pump in the high-gain regime. It may
be verified that Eq. (16) is consistent with previous quantum derivations of
the high-gain SPDC field correlations for a monochromatic plane-wave pump [5]
and a narrow-band pump [16] (also, see Eq. B.1 in Appendix B of Ref. [46],
where the expression derived in Ref. [16] has been used for quantifying
spatial correlations in degenerate SPDC with a quasimonochromatic pump with
narrow angular bandwidth). We however note that owing to the “narrow-band”
approximation involved in the above derivation, in the low-gain limit
$G({\bm{\rho}},t)\to 0$ wherein $\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)\to
i\Delta\bar{k}_{z}/2$, Eq. (16) is in general only approximately equivalent to
Eq. (9). Nevertheless, it can be verified that the equivalence is exact for
the case of a monochromatic plane-wave pump, for which the “narrow-band”
approximation is exactly true. In the following section, we experimentally
test the predictions of Eq. (16) by measuring the far-field intensity profile,
OAM spectrum, and frequency spectrum in high-gain SPDC for increasing gain
with a pulsed Gaussian pump.

##  III Single-crystal setup: Experiments and theory

In our experimental setup depicted in Fig. 2, a $355$ nm vertically-polarized
$30$-ps $50$-Hz pulsed Gaussian Nd:YAG laser (EKSPLA PL2231) is spatially-
filtered and used to pump a $3$-mm type-I $\beta$-barium borate SPDC crystal
(cut to produce horizontally-polarized collinear degenerate emission for
perpendicular incidence of a vertically-polarized pump) placed at the pump’s
waist plane. The combination of the half-wave plate (HWP) and polarizing beam-
splitter (PBS) is used to control the pump amplitude reaching the crystal.
This pump amplitude is inferred up to an overall scaling factor from energy
measurements using the energy meter (Coherent EnergyMax USB-J-10MB-HE). The
beam-waist size $w_{p}$, defined as the $1/e^{2}$ half-width of the intensity
profile at the waist plane, was measured using a Gentec Beamage-3.0 beam
profiler to be $w_{p}=185\,\mu$m. The residual pump after the crystal is
removed by means of two dichroic mirrors (DMs), and the generated SPDC field
is guided towards different parts of the setup for measurements as described
in the following subsections.

Figure 2: Schematic of the experimental setup used for measuring the far-field
intensity profile, OAM spectrum, and wavelength spectrum in high-gain SPDC for
increasing pump strengths. SF: spatial filter, HWP+PBS: half-wave plate +
polarizing beam-splitter combination for pump amplitude control, BBO:
$\beta$-barium borate nonlinear crystal, DM: dichroic mirror, BD: beam dump,
BS: beam-splitter, FM: flip mirror, T: translation stage to control the
interferometric phase $\delta$, IF: interference filter centered at $710$ nm.

###  III.1 Far-field intensity

We place a lens with its focal plane coinciding with the crystal output face,
and directly image the SPDC far-field on an Andor Ixon-897 EMCCD camera (pixel
size $16\,\mu$m $\times 16\,\mu$m) that is placed at the plane indicated by
the dashed line after the flip mirror (FM) location depicted in Fig. 2. The
crystal orientation was approximately set to satisfy the collinear emission,
and a $10$ nm interference filter (IF) centered at the degenerate wavelength
$\lambda_{s}=2\lambda_{p0}=710$ nm was placed at the camera entrance. We
rotate the HWP across different angles and acquire five sets of ten images
each of the SPDC far-field for each corresponding pump amplitude, which is
simultaneously measured using the energy meter up to an overall scaling
factor. We then take diametric slices along the horizontal and vertical axes
of each SPDC profile and compute the full-widths-at-half-maximum (FWHMs). We
observe a slight asymmetry between the widths in the two directions due to
spatial walk-off. However, we ignore walk-off effects in our study and
consider the mean of the two widths as the width of the far-field profile. The
FWHM width is converted from number of pixels to angle in milliradians by
using the camera pixel-size and the focal length of the collimating lens.

For obtaining the theoretical prediction, we fix the variables in the temporal
degree of freedom by assuming degenerate SPDC $\lambda_{s}=710$ nm with a
quasimonochromatic pump of wavelength $\lambda_{p0}=355$ nm. We verify that
the pump angular bandwidth $\delta{\bm{q}_{p}}=2/w^{2}_{p}\approx
58\,\mathrm{mm}^{-2}$ is indeed much smaller than the SPDC angular bandwidth
$\delta{\bm{q}_{s}}=|{\bm{k}_{s}}|/L\approx 4910\,\mathrm{mm}^{-2}$ [16]. We
then use Eq. (16) to compute the far-field intensity as

$\displaystyle\langle|A_{s}({\bm{q}_{s}})|^{2}\rangle$
$\displaystyle=\frac{K_{\rm
arb}}{k^{2}_{sz}}\int\mathrm{d}{\bm{\rho}}\,\langle|V_{p}({\bm{\rho}})|^{2}\rangle$
$\displaystyle\hskip
28.45274pt\times\left|\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}})L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}})}\right|^{2},$
(17)

where $K_{\rm arb}$ is an overall scaling constant. Throughout this paper, the
quantity $K_{\rm arb}$ must be interpreted as an arbitrary scaling factor
whose value is irrelevant for our purposes. The frequency-time coordinates
have been suppressed for brevity.

For our experiments, the pump field profile and phase-mismatch are given by

$\displaystyle V_{p}({\bm{\rho}})$
$\displaystyle=g\exp\\{-|{\bm{\rho}}|^{2}/w^{2}_{p}\\},$ (18a)
$\displaystyle\Delta\bar{k}_{z}$
$\displaystyle=|{\bm{k}_{p}}|-2\sqrt{|{\bm{k}_{s}}|^{2}-|{\bm{q}_{s}}|^{2}},$
(18b)

where $g$ is a pump amplitude scaling factor, $|{\bm{k}_{s}}|=2\pi
n_{so}/\lambda_{s}$, $|{\bm{k}_{p}}|=2\pi\eta_{p}(\theta_{p})/\lambda_{p0}$,
$\theta_{p}$ is the angle between the pump propagation direction and the optic
axis inside the crystal, and
$\eta_{p}(\theta_{p})=n_{pe}n_{po}/\sqrt{n_{po}^{2}\sin^{2}\theta_{p}+n_{pe}^{2}\cos^{2}\theta_{p}}$
is the effective refractive index of the extraordinary-polarized pump inside
the crystal [47]. The values $n_{p(e)o}$ and $n_{so}$ of the (extra)ordinary
and ordinary refractive indices of BBO for the pump and signal wavelengths,
respectively, can be obtained using the Sellmeier relations [48]

$\displaystyle n_{e}^{2}(\lambda)$
$\displaystyle=2.7405+\frac{0.0184}{\lambda^{2}-0.0179}-0.0155\lambda^{2},$
(19a) $\displaystyle n_{o}^{2}(\lambda)$
$\displaystyle=2.3730+\frac{0.0128}{\lambda^{2}-0.0156}-0.0044\lambda^{2},$
(19b)

where $\lambda$ is the corresponding wavelength in microns. Using these
parameters with Eq. (17), we perform a least-squares fit between theory and
experiment of the logarithmic total far-field intensity with $g$ as the fit
parameter. The value of $\theta_{p}$ for the simulations is allowed to have a
small offset from the collinear emission condition
$\theta^{\rm(coll)}_{p}=32.914^{\circ}$ because in experiments, $\theta_{p}$
is set by hand to approximately satisfy collinear emission. We ignore walk-off
effects and exploit the rotational symmetry of the pump and the SPDC field to
significantly speed up our simulations.

Figure 3: Broadening of the far-field intensity profile of the SPDC field with
increasing gain. (a) and (b) depict the experimental and theoretical images
for $g=3.5$. The experimental and theoretical total intensities and full-
widths-at-half-maximum (FWHMs) of diametric slices of such profiles for
increasing pump amplitudes are depicted in (c).

We depict our experimental and theoretical results in Fig. 3. In our
simulations, we chose $\theta_{p}=32.9105^{\circ}$, which is less than
$0.004^{\circ}$ away from $\theta^{\rm(coll)}_{p}=32.914^{\circ}$. The
experimental and theoretical far-field intensity profiles for $g=3.5$ (in
arbitrary units) are shown in Fig. 3(a) and 3(b), respectively. Apart from
some visible aberrations in the experimental image Fig. 3(a) that appear due
to burnt spots and dust on a dichroic mirror, the profiles match closely. In
Fig. 3(c), we show the trends of the full widths at half maximum (FWHMs) of
diametric slices and total intensities of the far-field profiles with
increasing pump amplitudes. The observed increasing trend of the FWHMs
indicating the broadening of the angular spectrum with gain has also been
reported previously [31]. Here, we find good agreement of our classical
model’s predictions with experiment.

###  III.2 OAM spectrum

In SPDC with a zero-OAM Gaussian pump field, OAM conservation constrains the
form of the generated state to a high-dimensional superposition involving
signal and idler modes with opposite anti-correlated OAM values. In the low-
gain regime, these modes are populated by one photon each [49], whereas in the
high-gain regime, the same modes can be populated by higher but correlated
numbers of photons [50]. In both cases, the second-order spatial correlations
of the signal field are equivalent to those of a field that is an incoherent
mixture of different OAM states, and the distribution of the weights of the
OAM states is referred to as the OAM spectrum. We now study the behavior of
the OAM spectrum in high-gain SPDC for increasing pump strengths.

We define the polar coordinates $(q_{s},\phi_{s})$ of ${\bm{q}_{s}}$ to be
related to the Cartesian coordinates as
$(q_{s}\sin\phi_{s},q_{s}\cos\phi_{s})=(q_{sx},q_{sy})$. The generated field
of Eq. (14) can be written as [51]

$\displaystyle
A_{s}(q_{s},\phi_{s})=\sum_{l=-\infty}^{+\infty}\sum_{p=0}^{\infty}\alpha_{lp}LG^{l}_{p}(q_{s})\,e^{il\phi_{s}},$
(20)

where $l$ and $p$ are the azimuthal and radial indices of the Laguerre-Gauss
mode functions $LG^{l}_{p}(q_{s})\,e^{il\phi_{s}}$, and $\alpha_{lp}$ are
stochastic coefficients that satisfy
$\sum_{p}\langle\alpha_{lp}\alpha^{*}_{l^{\prime}p}\rangle=S_{l}\delta_{ll^{\prime}}$,
where $\delta_{ll^{\prime}}$ is the Kronecker delta symbol. The quantities
$S_{l}$, which include contributions from all the constituent radial modes,
are collectively referred to as the OAM spectrum. The angular coherence
function $W(\phi_{s1},\phi_{s2})=W(\phi_{s1}-\phi_{s2})$ defined as [52, 51]

$\displaystyle W(\phi_{s1}-\phi_{s2})=\int\mathrm{d}q_{s}\,q_{s}\,\langle
A_{s}(q_{s},\phi_{s1})A^{*}_{s}(q_{s},\phi_{s2})\rangle,$ (21)

has been shown to be related to the OAM spectrum $S_{l}$ by the Fourier
transform relation [52, 51]

$\displaystyle
S_{l}=\int_{-\pi}^{\pi}\mathrm{d}\Delta\phi\,W(\Delta\phi)\,e^{il\Delta\phi}.$
(22)

Thus, the angular coherence function $W(\phi_{s},-\phi_{s})=W(2\phi_{s})$ can
be computed using equations (16) and (21), following which Eq. (22) can be
used to obtain the OAM spectrum.

Figure 4: Narrowing of the OAM spectrum of the SPDC signal field with
increasing gain. (a) and (b) are the constructive and destructive
interferograms, and (c) depicts the measured OAM spectrum alongwith the
theoretical prediction for $g=3.0$. (d) depicts the total intensity and
Schmidt number $K_{a}$ as a function of pump amplitude $g$.

In our experiments, we employ the interferometric technique reported in Ref.
[51] to measure the OAM spectrum of the high-gain SPDC field. As shown in Fig.
2, the SPDC field is incident into a Mach-Zehnder interferometer with an odd
and even number of mirrors in the two arms, and its far-field interferograms
are imaged using the EMCCD camera. The constructive and destructive
interferograms are acquired by changing the interferometric phase $\delta$
using the translation stage, and their difference image is used to compute the
OAM spectrum [51]. However, this direct procedure results in large errors in
the value of $S_{0}$ due to shot-to-shot fluctuations in the pump energy. We
partially address this problem by averaging the interferograms over two
hundred pulses, but interferometric phase drifts prevent us from increasing
the acquisition time further. Moreover, the larger acquisition times also
introduce errors due to systematic energy drifts of the laser. Therefore, we
normalize the constructive and destructive interferograms such that the
intensities in a $20\times 20$ pixels region away from the central
interference fringe for both images are the same. Such a normalization
effectively simulates the situation where the two interferograms were obtained
for identical pump energies and almost entirely eliminates the error in
$S_{0}$. In addition, the sum image of the two interferograms after this
normalization effectively approximates the intensity profile of the SPDC
field, which is then used for fitting the experimental results to the model’s
predictions. In this way, we experimentally obtain the OAM spectra and the
total intensities for different pump amplitudes.

We compute the theoretical predictions using the same parameters that were
used in the previous subsection. We perform a fit between theory and
experiment of the logarithmic total intensities for different pump amplitudes
with $g$ as the fit parameter. Using equations (16) and (21), we then compute
$W(\phi_{s},-\phi_{s})$, and subsequently use Eq. (22) to compute the OAM
spectrum for each pump amplitude. For our simulations, we choose
$\theta_{p}=32.894^{\circ}$, which is only $0.02^{\circ}$ away from
$\theta^{\rm(coll)}_{p}=32.914^{\circ}$. The rotational and reflection
symmetries of the pump and SPDC fields can be used to significantly speed up
the computations.

We depict our experimental and theoretical results in Fig. 4. The constructive
and destructive experimentally-measured interferograms for $g=3.0$ (in arb
units) are depicted in Fig. 4(a) and Fig. 4(b). The measured OAM spectrum with
the theoretically-predicted spectrum are shown in Fig. 4(c). The width of the
OAM spectrum can be quantified using the angular Schmidt number $K_{a}\equiv
1/\sum_{l}S^{2}_{l}$, where the spectrum is normalized such that
$\sum_{l}S_{l}=1$. In Fig. 4(d), we depict the trends of the total intensity
and the Schmidt number for increasing pump amplitudes. The decreasing Schmidt
number implies that the OAM spectrum narrows with increasing gain. Again, we
find that our experimental measurements are in good agreement with the
predictions of our classical model.

###  III.3 Wavelength spectrum

We raise the flip mirror depicted in Fig. 2 and guide the SPDC field into a
spectrometer to measure its wavelength spectrum. The spectrometer is an
imaging spectrograph (Princeton Instruments Acton Series SP2558 500 mm triple-
grating) with a CCD camera (PIXIS:100BR eXcelon, pixel size 20 $\mu$m $\times$
20 $\mu$m). An aperture of 1 mm radius is used to select only the central
portion of the collinear emission far-field corresponding to
${\bm{q}_{s}}\approx 0$, and a lens with focal length $500$ mm is used to
focus the light from the aperture onto the entrance slit of the spectrometer.
We enable vertical binning so that the signal at a certain wavelength is the
sum of the photoelectron counts over all the pixels that correspond to that
wavelength. The integration time for each of the spectra is 4000 ms. We record
ten spectra for each pump amplitude over the range from 610 nm to 810 nm, and
to cover this range, we repeat the acquisition for different angular positions
of the grating (1200 grooves per mm, 750 nm blaze). We apply a median filter
to the recorded spectra to eliminate outlier peaks that appear at random
locations.

For evaluating the theoretically predicted spectrum, we set ${\bm{q}_{s}}=0$
in collinear SPDC, and obtain the frequency spectrum from Eq. (16) as

$\displaystyle\langle|A_{s}(\omega_{s})|^{2}\rangle=K_{\rm
arb}\,\omega^{2}_{s}\iint\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,\langle|V_{p}({\bm{\rho}},t)|^{2}\rangle\,$
$\displaystyle\hskip
85.35826pt\times\left|\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right|^{2},$
(23)

The spatial coordinates of the signal field have been suppressed for brevity.
The wavelength spectrum $S(\lambda_{s})$ can be computed using the relation
$S(\lambda_{s})=\langle|A_{s}(\omega_{s})|^{2}\rangle|\mathrm{d}\omega_{s}/\mathrm{d}\lambda_{s}|$
that upon using $\omega_{s}=2\pi c/\lambda_{s}$ yields

$\displaystyle S(\lambda_{s})=K_{\rm arb}\langle|A_{s}(2\pi
c/\lambda_{s})|^{2}\rangle/\lambda^{2}_{s},$ (24)

We now compute then $S(\lambda_{s})$ using the relations,

$\displaystyle V_{p}({\bm{\rho}},t)$
$\displaystyle=g\,\exp\left\\{-\frac{t^{2}}{(2\Delta
t)^{2}}\right\\}\,\exp\left\\{-\frac{|{\bm{\rho}}|^{2}}{w_{p}^{2}}\right\\},$
(25a) $\displaystyle\Delta\bar{k}_{z}$
$\displaystyle=\frac{2\pi\eta_{p}(\theta_{p})}{\lambda_{p0}}-\frac{2\pi
n_{o}(\lambda_{s})}{\lambda_{s}}-\frac{2\pi
n_{o}(\bar{\lambda}_{i})}{\bar{\lambda}_{i}},$ (25b)

where $g$ is a pump amplitude scaling factor,
$\bar{\lambda}_{i}=1/(1/\lambda_{p0}-1/\lambda_{s})$, and $\Delta
t=30/2.355=12.738$ ps, where we have converted the FWHM pulse width of the
power to the standard deviation width [53]. Using these parameters and the
dispersion relations (19), we perform a least-squares fit of the total
intensities of the recorded spectra to the total intensities predicted by
equation (23) with $g$ as the fit parameter. We then use the fit value of $g$
to compute the theoretically-predicted spectra for different pump amplitudes.
The spatial rotational symmetry of the pump can be exploited to significantly
reduce computation time.

Figure 5: Broadening of the wavelength spectrum of the SPDC signal field with
increasing gain. (a) depicts the wavelength spectrum for $g=2.8$. (b) depicts
the total intensity and FWHM widths of the spectra for different pump
amplitudes.

In Fig. 5, we depict our experimental and theoretical results. In Fig. 5(a),
we depict the experimental and theoretical wavelength spectrum for $g=2.8$ (in
arb units), and in Fig. 5(b), we depict the total intensities and the FWHM
widths of the spectra for different pump strengths. The increasing FWHM width
implies that the wavelength spectrum gets broader with increasing pump
strength in high-gain SPDC [30]. Our experimental results are in good
agreement with the theoretical predictions of our classical model.

##  IV Two-crystal setups: Theory

###  IV.1 SU(1,1) interference

Figure 6: Conceptual schematic depiction of SU(1,1) interference as
interpreted in our classical model.

SU(1,1) interferometers are nonlinear interferometers that can be created by
essentially replacing the beam-splitters in linear SU(2) interferometers, such
as Mach-Zehnder interferometers, by nonlinear crystals [39, 40, 54]. It is
known that such interferometers can achieve a phase sensitivity approaching
the Heisenberg limit with the important advantage that they have a better loss
tolerance than linear SU(2) interferometers employing squeezed states of
light. In addition to phase metrology [55], SU(1,1) interferometers are also
useful for applications such as radiation shaping [56], microscopy [57],
optical coherence tomography [58], and sensing [59].

We consider the SU(1,1) setup shown in Fig. 6 that involves two crystals 1 and
2, each of length $L$, separated by an air gap of length $D$. We define the
longitudinal coordinate $z^{\prime}$ such that $z^{\prime}=0$ corresponds to
the input face of crystal 2. We assume that there is no postselection taking
place between the crystals, and that $D$ is small enough such that the far-
fields of both the crystals can be assumed to overlap perfectly. As the field
$A_{j}$ for $j=p,s,i$ propagates across crystal 1 and the air gap to reach
crystal 2, it acquires a phase $k_{jz}L+k^{\rm(air)}_{jz}D$, where
$k^{\rm(air)}_{jz}$ represents the longitudinal wavevector component of the
corresponding field in air. As a result, the field generated in crystal 2
follows the relation

$\displaystyle\frac{\partial
A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime})}{\partial z^{\prime}}=\frac{2id_{\rm
eff}\omega^{2}_{s}}{k_{sz}c^{2}}\iint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,A_{p}({\bm{q}_{p}},\omega_{p})$
$\displaystyle\hskip 28.45274pt\times e^{i[\Delta k^{\rm(air)}_{z}D+\Delta
k_{z}L]}A^{*}_{i}({\bm{q}_{i}},\omega_{i},z^{\prime})e^{i\Delta
k_{z}z^{\prime}},$ (26)

where $\Delta
k^{\rm(air)}_{z}=k^{\rm(air)}_{pz}-k^{\rm(air)}_{sz}-k^{\rm(air)}_{iz}$ is the
longitudinal phase-mismatch in air. Using equations (4a) and (26), the net
field $A_{s}({\bm{q}_{s}},\omega_{s})$ at the output of the interferometer can
be written as

$\displaystyle A_{s}({\bm{q}_{s}},\omega_{s})=\frac{2id_{\rm
eff}\omega^{2}_{s}}{k_{sz}c^{2}}\iint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,A_{p}({\bm{q}_{p}},\omega_{p})$
$\displaystyle\times\Big{[}\int_{0}^{L}\mathrm{d}z\,A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)e^{i\Delta
k_{z}z}+\,e^{i[\Delta k^{\rm(air)}_{z}D+\Delta k_{z}L]}\Big{.}$
$\displaystyle\Big{.}\hskip
56.9055pt\times\int_{0}^{L}\mathrm{d}z^{\prime}\,A^{*}_{i}({\bm{q}_{i}},\omega_{i},z^{\prime})e^{i\Delta
k_{z}z^{\prime}}\Big{]}.$ (27)

In what follows, we derive the second-order spatiotemporal correlations of the
output field in low and high gain regimes, and illustrate the interference in
the spatial domain through numerical simulations.

#### Low-gain regime

In the low-gain regime, as the interaction is weak, the longitudinal growth of
the classical “vacuum” across both the crystals is negligible. Therefore, we
approximate $A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)\approx
A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)$ in both terms of the right-hand side of
Eq. (27), and perform the integrations over $z$ and $z^{\prime}$ to obtain

$\displaystyle A_{s}({\bm{q}_{s}},\omega_{s})=\frac{d_{\rm
eff}L\omega^{2}_{s}}{k_{sz}c^{2}}\iint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,A_{p}({\bm{q}_{p}},\omega_{p})A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)$
$\displaystyle\hskip 56.9055pt\times\mathrm{sinc}\left(\Delta
k_{z}L/2\right)\,e^{i\Delta k_{z}L/2}$ $\displaystyle\hskip
85.35826pt\times\left\\{1+e^{i[\Delta k^{\rm(air)}_{z}D+\Delta
k_{z}L]}\right\\}.$ (28)

Using the above equation alongwith Eq. (8), we obtain

$\displaystyle\langle
A_{s}({\bm{q}_{s}},\omega_{s})A^{*}_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s})\rangle=\frac{\hbar\omega_{i0}d_{\rm
eff}^{2}L^{2}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{4\pi\epsilon_{0}k_{sz}k^{\prime}_{sz}c^{4}}\iint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}$
$\displaystyle\times\langle
A_{p}({\bm{q}_{s}}+{\bm{q}_{i}},\omega_{s}+\omega_{i})A^{*}_{p}({\bm{q}^{\prime}_{s}}+{\bm{q}_{i}},\omega^{\prime}_{s}+\omega_{i})\rangle$
$\displaystyle\hskip 8.53581pt\times\mathrm{sinc}\left(\Delta
k_{z}L/2\right)\mathrm{sinc}\left(\Delta
k^{\prime}_{z}L/2\right)\,e^{i\left(\Delta k_{z}-\Delta
k^{\prime}_{z}\right)L/2}$ $\displaystyle\hskip
42.67912pt\times\left\\{1+e^{i[\Delta k^{\rm(air)}_{z}D+\Delta
k_{z}L]}\right\\}$ $\displaystyle\hskip 85.35826pt\times\left\\{1+e^{-i[\Delta
k^{{}^{\prime}\rm(air)}_{z}D+\Delta k^{\prime}_{z}L]}\right\\}.$ (29)

The above expression quantifies the second-order spatiotemporal correlations
of the output field of the SU(1,1) interferometer in the low-gain regime.

#### High-gain regime

In the high-gain regime, the initial classical “vacuum”
$A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)$ will first be amplified over the length
$L$ of crystal 1 to $A^{*}_{i}({\bm{q}_{i}},\omega_{i},L)$, which then seeds
the DFG process in crystal 2. In order to quantify the growth of the classical
“vacuum”, we define
$A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)=f(z)A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)$,
differentiate Eq. (4b) with respect to $z$, and make the “narrow-band pump”
approximation to obtain

$\displaystyle\frac{\partial^{2}f(z)}{\partial
z^{2}}+i\Delta\bar{k}_{z}\frac{\partial f(z)}{\partial
z}-\bar{G}^{2}({\bm{\rho}},t)\,f(z)=0.$ (30)

Solving the above Eq. (30) subject to the initial conditions $f(z=0)=1$ and
$\partial f(z)/\partial z|_{z=0}=0$, we obtain

$\displaystyle
f(z)=e^{-i\Delta\bar{k}_{z}z/2}\Bigg{\\{}\cosh\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)z\Bigg{.}$
$\displaystyle\hskip
56.9055pt\Bigg{.}+\frac{i\Delta\bar{k}_{z}}{2\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\sinh\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)z\Bigg{\\}}.$
(31)

The above relation quantifies the growth of the classical “vacuum” inside
crystal 1.

We now note that the generated field in crystal 2 will be governed by Eq.
(12). Moreover, the field $A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime}=0)$ at the
entrance of crystal 2 is given by Eq. (14) as the field remains constant
through the air gap. In addition, $\partial
A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime})/\partial z^{\prime}|_{z^{\prime}=0}$
can be evaluated using Eq. (26). Solving Eq. (12) for the above initial
conditions, we obtain (see Appendix B. IV for details)

$\displaystyle A_{s}({\bm{q}_{s}},\omega_{s})=\frac{4id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V_{p}({\bm{\rho}},t)$
$\displaystyle\hskip 28.45274pt\times
e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}\,A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\left[\frac{\sinh\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right]$
$\displaystyle\hskip 56.9055pt\times e^{i(\Delta
k_{z}-\Delta\bar{k}_{z}/2)L}\,h(\Delta k^{\rm(air)}_{z},D),$ (32)

where we have defined

$\displaystyle h(\Delta k^{\rm(air)}_{z},D)\equiv e^{i\Delta
k^{\rm(air)}_{z}D/2}\Big{[}\cosh\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L\Big{.}$
$\displaystyle\Big{.}\times\cos\left\\{\Delta
k^{\rm(air)}_{z}D/2\right\\}-\frac{\Delta\bar{k}_{z}}{2\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\sinh\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L\Big{.}$
$\displaystyle\hskip 99.58464pt\Big{.}\times\sin\left\\{\Delta
k^{\rm(air)}_{z}D/2\right\\}\Big{]}.$ (33)

Using the above relations and Eq. (8), we obtain

$\displaystyle\langle
A_{s}({\bm{q}_{s}},\omega_{s})A^{*}_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s})\rangle=\frac{4\hbar\omega_{i0}d^{2}_{\rm
eff}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{(2\pi)^{4}\epsilon_{0}k_{sz}k^{\prime}_{sz}c^{4}}\iint\mathrm{d}{\bm{\rho}}\,\mathrm{d}t$
$\displaystyle\times\langle|V_{p}({\bm{\rho}},t)|^{2}\rangle\,e^{-i[({\bm{q}_{s}}-{\bm{q}^{\prime}_{s}}){\bm{\rho}}-(\omega_{s}-\omega^{\prime}_{s})t]}$
$\displaystyle\times\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right]\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}},t)}\right]$
$\displaystyle\times\bar{h}(\Delta k^{\rm(air)}_{z},D)\,\bar{h}^{*}(\Delta
k^{{}^{\prime}\rm(air)}_{z},D)\,e^{i(\Delta\bar{k}_{z}-\Delta\bar{k}^{\prime}_{z})L/2}.$
(34)

The above equation quantifies the second-order spatiotemporal correlations of
the output field of the SU(1,1) interferometer in the high-gain regime.

#### Numerical simulations in the spatial domain

Figure 7: SU(1,1) far-field interference patterns in the low and high gain
regimes for increasing air gap lengths.

We fix the frequency variables by restricting our attention to degenerate SPDC
with a monochromatic pump, and suppress those variables for brevity. This
restriction physically corresponds to placing a narrowband filter centered at
the degenerate emission wavelength. We numerically simulate collinear
degenerate SPDC for the same parameters used in Sec. IIIA. For computing the
phase mismatch $\Delta k^{\rm air}_{z}$, we use the dispersion relation for
air from Ref. [60], according to which the refractive index of air for the
pump wavelength $355$ nm is $n^{\rm air}_{p}=1.00028571$, and that for the
signal (idler) wavelength $710$ nm is $n^{\rm air}_{s(i)}=1.00027571$. We use
Eq. (29) and $\Delta k_{z}=|{\bm{q}_{s}}-{\bm{q}_{i}}|^{2}/4k_{s}$ to compute
the low-gain interference patterns, and use Eq. (34) to compute the high-gain
interference patterns for different air gap lengths and depict them in Fig. 7.
It is evident that in both gain regimes, the interference exhibits high
visibility. Moreover, with increasing air gap, the fringe separation reduces,
and owing to the finite dispersion of air, the intensity at the center also
varies. These predictions qualitatively agree with previous experimental
observations [61, 31].

###  IV.2 Induced coherence

In their seminal 1991 experiment [11], Zou, Wang and Mandel built a modified
SU(1,1) interferometric setup in which the signal fields from the two crystals
were separated and superposed on a beam-splitter. Ordinarily, the signal
fields from two distinct SPDC processes do not interfere with each other.
However, Zou et al. [11] observed that when the idler paths from the two
processes are aligned, the signal fields become mutually coherent and exhibit
interference. This intriguing effect, termed as “induced coherence”, has not
only revealed fundamental insights about interference and indistinguishability
[11, 62], but has also been harnessed in a variety of applications such as
imaging [41] and spectroscopy [63].

The physical origin of induced coherence has been extensively debated over the
years since its inception [11, 62, 64, 65, 66, 67]. In particular, the
possibility of the coherence originating from “induced emission”, i.e, the
idler photons from the first crystal stimulating emission in the second
crystal, was recognized [11, 62]. Zou et al. ruled out this possibility by
performing their experiment at weak pump power where induced emission could be
shown to be negligible. They argued that the induced coherence is a
consequence of the signal photon paths being rendered indistinguishable by the
alignment of the idler paths. Their theoretical analysis also supported this
quantum interpretation by correctly predicting the experimentally-observed
linear scaling of the interference visibility with respect to the
transmittance of an object placed in the idler path between the crystals.
Subsequently, Wiseman and Mølmer [65] performed a four-mode quantum
calculation for arbitrary pump strengths and showed that the linear scaling of
the visibility is the true signature of induced emission being negligible, and
that the scaling is not linear when induced emission is significant. Moreover,
they termed the former regime as “quantum”, and the latter regime as
“classical”. Thus, it was implied that induced coherence without induced
emission is an intrinsically quantum-mechanical effect with no classical
explanation.

Figure 8: Conceptual schematic of the induced coherence experiment [11]
interpreted in classical SPDC. Here, the partially-transmissive object O is
modeled as a beam-splitter.

In more recent years, with the series of intriguing experimental
demonstrations of “quantum imaging with undetected photons” [41], “quantifying
the momentum correlation between two light beams by detecting one” [68], and
“interference fringes controlled by noninterfering photons” [69], the question
of whether induced coherence admits a classical explanation has gained renewed
interest. While imaging with undetected light has been demonstrated in
classical settings where an auxiliary external field stimulates the emission
in both crystals [70, 71], it was explicitly stated that the case of
spontaneous low-gain emission cannot be explained within classical physics
[71]. Subsequently, a theoretical study showed that induced coherence persists
even when the pump field is a single photon Fock state, where the occurrence
of stimulated emission is strictly impossible [67]. Based on this result, the
authors concluded that any classical or semiclassical explanation of induced
coherence is effectively ruled out. Nevertheless, we submit that their study
does not preclude the existence of a classical model that successfully
captures the various induced coherence-related experiments that have been
performed so far in which the pump can be treated classically. In what
follows, we show that the classical SPDC model of the present paper captures
several important experimentally observed features of induced coherence in low
and high gain regimes, including the linear scaling of the visibility for low
gain that is often regarded as the signature quantum feature of induced
coherence.

In Fig. 8, we depict the conceptual schematic of a prototypical induced
coherence experiment involving two identical crystals as interpreted in the
classical SPDC model of the present paper. The pump field is split into two
equal parts: one part pumps crystal 1; the other part acquires a uniform phase
$\phi_{0}$ and pumps crystal 2. A classical “vacuum” field $A_{i}$ seeds the
process in crystal 1 and then encounters the object O that is assumed to have
a complex field transmittance $Te^{i\gamma}$. The object O is modeled as a
beam-splitter of splitting ratio $T:\sqrt{1-T^{2}}$ with a second classical
“vacuum” $B_{i}$ that is completely uncorrelated with the first classical
“vacuum” $A_{i}$ entering its other port. The output superposition of the two
“vacua” from this hypothetical beam-splitter then seeds the process in crystal
2. The fields $A_{s1}$ and $A_{s2}$ generated from the two crystals 1 and 2
are superposed on a perfectly symmetric 50:50 beam-splitter and the
interference is recorded on a CCD camera. We emphasize that Fig. 8 must be
viewed as a conceptual depiction because the actual geometry of the setup can
be entirely different. For instance, the experiment can also be implemented
using collinear phase-matching [41], but the underlying concept remains the
same. In what follows, we fix the temporal variables by assuming degenerate
SPDC with a quasimonochromatic pump, and derive the output spatial
interference patterns recorded in the induced coherence experiment in low and
high gain regimes.

#### Low-gain regime

Figure 9: Numerical simulations of the case similar to the one studied by
Lemos et al. [41] where the object O has a binary structured transmission
profile $T$. As shown in (a), we assume $T({\bm{q}_{i}})$ shaped in the Greek
symbol $\Psi$. (b) and (c) are interferograms obtained for $\phi_{0}=0$ and
$\phi_{0}=\pi$, respectively. (d) and (e) are the sum and difference images,
respectively.

In the low-gain regime, the growth of the “vacuum” $A^{*}_{i}({\bm{q}_{i}},0)$
can be neglected. Using Eq. (5), the generated fields from the two crystals
can then be written as

$\displaystyle A_{s1}({\bm{q}_{s}})=\frac{K_{\rm
arb}}{k_{sz}}\int\mathrm{d}{\bm{q}_{i}}\,A_{p}({\bm{q}_{p}})A^{*}_{i}({\bm{q}_{i}},0)\,\mathrm{sinc}\left(\Delta
k_{z}L/2\right)$ $\displaystyle\hskip 113.81102pt\times e^{i\Delta k_{z}L/2},$
(35a) $\displaystyle A_{s2}({\bm{q}_{s}})=\frac{K_{\rm
arb}}{k_{sz}}\int\mathrm{d}{\bm{q}_{i}}\,A_{p}({\bm{q}_{p}})\,e^{i\phi_{0}}\,\mathrm{sinc}\left(\Delta
k_{z}L/2\right)$ $\displaystyle\times e^{i\Delta
k_{z}L/2}\Big{[}T\,e^{i\gamma}A^{*}_{i}({\bm{q}_{i}},0)+\sqrt{1-T^{2}}\,B^{*}_{i}({\bm{q}_{i}},0)\Big{]},$
(35b)

The above fields overlap at the beam-splitter to yield the field
$A_{s}({\bm{q}_{s}})=\left\\{A_{s1}({\bm{q}_{s}})+A_{s2}({\bm{q}_{s}})\right\\}/\sqrt{2}$
at the CCD camera. We now note that $B_{i}$ has the same autocorrelations as
that of $A_{i}$, i.e, $\langle
B_{i}({\bm{q}^{\prime}_{i}},0)B^{*}_{i}({\bm{q}_{i}},0)\rangle=\langle
A_{i}({\bm{q}^{\prime}_{i}},0)A^{*}_{i}({\bm{q}_{i}},0)\rangle=C\delta({\bm{q}_{i}}-{\bm{q}^{\prime}_{i}})$,
where $C$ is a scaling factor. However, $B_{i}$ has no mutual correlations
with $A_{i}$, i.e, $\langle
A_{i}({\bm{q}^{\prime}_{i}},0)B^{*}_{i}({\bm{q}_{i}},0)\rangle=0$. Using these
relations, the measured intensity takes the form

$\displaystyle I=\langle|A_{s}({\bm{q}_{s}})|^{2}\rangle$
$\displaystyle=\frac{K_{\rm
arb}}{k^{2}_{sz}}\int\mathrm{d}{\bm{q}_{i}}|A_{p}({\bm{q}_{p}})|^{2}\,\mathrm{sinc}^{2}\left(\Delta
k_{z}L/2\right)$ $\displaystyle\hskip
28.45274pt\times\Big{\\{}1+T\cos\left(\gamma+\phi_{0}\right)\Big{\\}},$ (36)

In general, the transmittance $T$ and phase $\gamma$ could be spatially-
structured, in which case they would explicitly depend on $\bm{q}_{i}$.
However, when $T$ and $\gamma$ are uniform, the visibility
$V({\bm{q}_{s}})=(I_{\rm max}-I_{\rm min})/(I_{\rm max}+I_{\rm min})$ scales
linearly with $T$, in agreement with the quantum-mechanical prediction and
experimental observations [11, 62]. Thus, our classical model is able to
correctly predict the linear scaling of the visibility with object
transmittance in the low-gain regime – a feature that is often regarded as the
quintessential signature of nonclassicality. We will now theoretically analyze
two intriguing cases of the object O within our classical model.

We first consider the case where the object O has a binary structured
transmission profile $T({\bm{q}_{i}})$ and constant phase $\gamma=0$ similar
to the one studied by Lemos et al. [41]. Specifically, as shown in Fig. 9 (a),
we assume that $T({\bm{q}_{i}})$ has the shape of the Greek symbol $\Psi$. We
assume collinear degenerate SPDC with the same parameters as chosen in Sec.
IIIA. Using Eq. (36), we numerically compute the interferograms for
$\phi_{0}=0$ and $\phi_{0}=\pi$ and depict them in Fig. 9(b) and Fig. 9(c),
respectively. The sum and difference images of these interferograms are
depicted in Fig. 9(d) and Fig. 9(e), respectively. We note that the
interferograms in Fig. 9(b) and Fig. 9(c) display the object O even though the
interfering signal fields themselves have not directly interacted with the
object, whereas the classical “vacuum” that has directly interacted with the
object is never detected. This intriguing phenomenon was termed “quantum
imaging with undetected photons” in Lemos et al. [41], and was explained in
terms of the quantum-mechanical interpretation of induced coherence. Here, it
is evident that the same feature is also captured within our classical model.
We emphasize that while other studies have previously demonstrated this
feature in classical settings involving a real external stimulating field [70,
71], to our knowledge, this is the first classical treatment applied to the
spontaneous low-gain limit in which Zou et al. [11, 62] and Lemos et al. [41]
performed their experiments.

Figure 10: Numerical low-gain simulations of the case considered in A.
Hochrainer et al. [68] where object O is a lens with $T=1$. (a) depicts the
phase $\gamma\sim|{\bm{q}_{i}}|^{2}$ modulo $2\pi$. (b), (c), (d), and (e)
depict interference patterns for increasing pump beam-waist $w_{p}$. (f)
depicts the visibility computed along a horizontal slice through the center of
the interferograms by varying the phase $\phi_{0}$ from $0$ to $2\pi$, and (g)
depicts the scaling of the FWHM of the visibility profiles with $w_{p}$.

Next, we consider the case studied by Hochrainer et al. [68] in which the
object O is a lens – a pure phase object with $T=1$ and
$\gamma=\lambda_{i}d\,|{\bm{q}_{i}}|^{2}/(4\pi)$ as depicted in Fig. 10a. We
set the equivalent free-space propagation distance $d=22.6$ mm in our
analysis, and study the interference for different values of the pump beam-
waist $w_{p}$. The other parameters are chosen to be the same as those used in
Sec. IIIA. In Fig. 10, we depict our numerical simulations of Eq. (36)
corresponding to the low-gain regime. As shown in (b), (c), (d), and (e) for
small values of $w_{p}$, the interference is blurred out away from the center,
whereas for large values of $w_{p}$, the interference exhibits sharp contrast
even for large radial distances. This effect can be interpreted in our
classical model as follows: for small $w_{p}$, the pump has a large angular
bandwidth, implying a large spread in ${\bm{q}_{p}}$. Consequently, the far-
field intensity at ${\bm{q}_{s}}$ is a superposition of a large number of
interference patterns corresponding to several different ${\bm{q}_{i}}$ that
satisfy ${\bm{q}_{i}}={\bm{q}_{p}}-{\bm{q}_{s}}$. As a result, the
interference pattern is blurred. On the other hand, for large $w_{p}$, the
pump’s angular bandwidth is small, and consequently, the averaging effect is
less pronounced leading to high interference visibility. For different values
of $w_{p}$, we then compute the visibility $V({\bm{q}_{s}})$ along a
horizontal slice through the center by numerically varying the phase
$\phi_{0}$ from $0$ to $2\pi$. As shown in Fig. 10(f), for small $w_{p}$, the
visibility decays rapidly in the radial directions, whereas the decay is
slower for larger $w_{p}$. In Fig. 10(f), we plot the FWHM of the visibility
curves for increasing $w_{p}$ alongwith a linear fit. This implies that the
interference visibility can be used to infer the pump’s angular bandwidth
which, in the usual quantum mechanical picture of SPDC, also determines the
transverse momentum correlations between the signal and idler photons.
Therefore, the visibility can be used for “quantifying the momentum
correlations between two photons by detecting one”, as experimentally
demonstrated by Hochrainer et al. [68]. Here, we find that our above results
qualitatively agree with those depicted in Fig. 2 of Ref. [68], but the
interpretation is classical within our model.

#### High-gain regime

In the high-gain regime, the initial “vacuum” $A^{*}_{i}({\bm{q}_{i}},0)$ is
amplified in crystal 1 to
$A^{*}_{i}({\bm{q}_{i}},L)=A^{*}_{i}({\bm{q}_{i}},0)f(L)$, which then
superposes with the “vacuum” $B^{*}_{i}({\bm{q}_{i}},0)$ from the other port
of O to seed crystal 2. Using equations (14) and (31), the fields from the two
crystals can be written as,

$\displaystyle A_{s1}({\bm{q}_{s}})=\frac{K_{\rm
arb}}{k_{sz}}\iint\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,V_{p}({\bm{\rho}})\,e^{-i{\bm{q}_{p}}\cdot{\bm{\rho}}}\,e^{i(\Delta
k_{z}-\Delta\bar{k}_{z}/2)L}$ $\displaystyle\hskip 56.9055pt\times
A^{*}_{i}({\bm{q}_{i}},0)\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}})L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}})}\right],$
(37a) $\displaystyle A_{s2}({\bm{q}_{s}})=\frac{K_{\rm
arb}}{k_{sz}}\iint\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,V_{p}({\bm{\rho}})\,e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\phi_{0})}$
$\displaystyle\hskip 28.45274pt\times e^{i(\Delta
k_{z}-\Delta\bar{k}_{z}/2)L}\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}})L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}})}\right]$
$\displaystyle\hskip
14.22636pt\times\Big{[}T\,e^{i\gamma}A^{*}_{i}({\bm{q}_{i}},0)f(L)+\sqrt{1-T^{2}}\,B^{*}_{i}({\bm{q}_{i}},0)\Big{]}.$
(37b)

The above fields overlap at the beam-splitter to yield the field
$A_{s}({\bm{q}_{s}})=\left\\{A_{s1}({\bm{q}_{s}})+A_{s2}({\bm{q}_{s}})\right\\}/\sqrt{2}$,
and the measured intensity at the CCD camera takes the form

$\displaystyle\langle|A_{s}({\bm{q}_{s}})|^{2}\rangle=\frac{K_{\rm
arb}}{k^{2}_{sz}}\int\mathrm{d}{\bm{\rho}}\,\langle|V_{p}({\bm{\rho}})|^{2}\rangle\left|\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}})L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}})}\right|^{2}$
$\displaystyle\hskip
14.22636pt\times\left[1+\bar{T}^{2}|f(L)|^{2}+2\,\bar{T}\,|f(L)|\right.$
$\displaystyle\left.\hskip
28.45274pt\times\cos\\{\mathrm{arg}\,f(L)+\bar{\gamma}+\phi_{0}\\}+(1-\bar{T}^{2})\right],$
(38)

As seen in the above expression, when $|f(L)|\gg 1$, the terms quadratic in
$\bar{T}$ don’t cancel, and as a result, the visibility $V({\bm{q}_{s}})$ is
not linear in $\bar{T}$.

We numerically study the effect of increasing pump strength on the spatial
interference. For simplicity, we now assume the object is no longer
structured, but has a uniform transmittance $T$, and compute the behavior of
the visibility for increasing pump strengths. We assume collinear degenerate
SPDC with the same parameters chosen in Sec. IIIA. In Fig. 11(a), we depict
the visibility with respect to transmittance for different pump strengths $g$
(in arb units) computed by numerically varying $\phi_{0}$ from $0$ to $2\pi$
in Eq. (38). While the visibility has a linear dependence on transmittance for
$g=0.01$ in the low-gain regime, the dependence is in general neither linear
nor monotonic for $g=1.01$ and $g=2.01$ that correspond to the high-gain
regime. The latter fact may be understood as follows: for $T=0$, the fields
$A_{s1}({\bm{q}_{s}})$ and $A_{s2}({\bm{q}_{s}})$ are equal in magnitude, but
have no mutual coherence, and consequently, the visibility vanishes. However,
for $T>0$, the visibility increases with respect to $T$ until some critical
value, but when the classical “vacuum” seeding the second crystal is stronger
than that seeding the first crystal, the field $A_{s2}({\bm{q}_{s}})$ is
stronger than the field $A_{s1}({\bm{q}_{s}})$, which causes the visibility to
decrease. This behavior of the visibility in the high-gain regime has also
been predicted in existing quantum calculations for a monochromatic plane-wave
pump [66, 65, 64].

Figure 11: (a) depicts the interference visibility, and (b) depicts the degree
of coherence $|\mu({\bm{q}_{s}})|$ as a function of object transmittance $T$
for different pump amplitudes $g$ (in arb units).

At this juncture, one can ask whether the decline in visibility is only due to
the disproportion of the overlapping fields $A_{s1}({\bm{q}_{s}})$ and
$A_{s2}({\bm{q}_{s}})$ or whether there is also a simultaneous reduction in
the degree of their mutual coherence. This question is important from a
practical standpoint in the context of imaging because in the former case, the
fields $A_{s1}({\bm{q}_{s}})$ and $A_{s2}({\bm{q}_{s}})$ can be equalized in
magnitude by suitably attenuating $A_{s2}({\bm{q}_{s}})$ to realize perfect
visibility [64, 66], whereas in the latter case, the visibility cannot be
optimized beyond $|\mu({\bm{q}_{s}})|$, where the mutual degree of coherence
$\mu({\bm{q}_{s}})$ is defined as $\mu({\bm{q}_{s}})\equiv\langle
A^{*}_{s1}({\bm{q}_{s}})A_{s2}({\bm{q}_{s}})\rangle/\sqrt{\langle
A_{s1}({\bm{q}_{s}})A^{*}_{s1}({\bm{q}_{s}})\rangle\langle
A_{s2}({\bm{q}_{s}})A^{*}_{s2}({\bm{q}_{s}})\rangle}$. Using equations (37),
we obtain

$\displaystyle\mu({\bm{q}_{s}})=\frac{\int\mathrm{d}{\bm{\rho}}\,\langle|V_{p}({\bm{\rho}})|^{2}\rangle|\,\Phi(\Delta\bar{k}_{z},{\bm{\rho}},L)|^{2}\,f(L)\,\bar{T}\,e^{i\bar{\gamma}}}{\sqrt{\int\mathrm{d}{\bm{\rho}}\,\langle|V_{p}({\bm{\rho}})|^{2}\rangle|\,\Phi(\Delta\bar{k}_{z},{\bm{\rho}},L)|^{2}}\sqrt{\int\mathrm{d}{\bm{\rho}}\,\langle|V_{p}({\bm{\rho}})|^{2}\rangle|\,\Phi(\Delta\bar{k}_{z},{\bm{\rho}},L)|^{2}\left\\{1+\bar{T}^{2}(|f(L)|^{2}-1)\right\\}}},$
(39)

where
$\Phi(\Delta\bar{k}_{z},{\bm{\rho}},L)\equiv\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},L)/\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},L)$.
It may be verified that Eq. (39) is consistent with previous theoretical
calculations for a monochromatic plane-wave pump [66, 65]. We compute and plot
the behavior of $|\mu({\bm{q}_{s}})|$ with respect to transmittance $T$ in
Fig. 11(b) for different pump amplitudes. In much similarity with the
monochromatic plane-wave pump case [66, 65], $|\mu({\bm{q}_{s}})|$ scales
linearly with $T$ for low-gain, but the scaling does not remain linear for
increasing gain. Nevertheless, $\mu({\bm{q}_{s}})$ remains very high even for
large gain, therefore allowing the possibility to realize imaging with high
visibility by means of attenuation. There is a small discrepancy from perfect
coherence in the high-gain regime for unity transmittance, i.e,
$\mu({\bm{q}_{s}})<1$ for $T=1$. In fact, it is evident from Eq. (39) that
owing to the expicit dependence of $f(L)$ on ${\bm{\rho}}$, in general
$|\mu({\bm{q}_{s}})|\,\neq 1$ for $T=1$. However, we are unable to ascertain
if this discrepancy is a genuine physical effect, or an artifact of the
“narrow-band” approximation because the discrepancy vanishes for the
monochromatic plane-wave pump case.

##  V Summary and Outlook

We describe a classical model that simulates SPDC as DFG of the pump field
with a hypothetical stochastic field that mimics the effect of vacuum
fluctuations. We show that the second-order spatiotemporal correlations of the
field generated from DFG replicate those of the signal field from SPDC. In
particular, for low gain, the second-order correlations predicted by the model
are identical to those predicted by the quantum calculation of the reduced
density matrix of the signal photon, whereas for high gain, we performed
experimental measurements of the far-field intensity profile, OAM spectrum,
and the wavelength spectrum for increasing pump strengths and demonstrated
their agreement with the model’s predictions. The far-field intensity profile
and wavelength spectrum exhibit broadening, whereas the OAM spectrum exhibits
narrowing with increasing gain.

Next, we use the model to theoretically analyse second-order interference in
SU(1,1) interferometers and induced coherence experiments. We derive
analytical expressions for the second-order spatiotemporal correlations of the
output field of a prototypical SU(1,1) interferometer, and illustrate some
salient features of the interference in the spatial domain through numerical
simulations. We then apply the model to the induced coherence experiment, and
derive expressions for spatial interference in the low and high gain regimes.
Interestingly, the model correctly predicts the experimentally-observed linear
scaling of the visibility with object transmittance in the spontaneous low-
gain limit – a feature that is often regarded as the quintessential signature
of the nonclassicality of induced coherence. We then apply the model to
numerically analyze the intriguing induced coherence-related phenomena known
as “quantum imaging with undetected photons” [41] and “quantifying the
momentum correlation between two light beams by detecting one” [68]. Finally,
we analyze the behavior of visibility and degree of mutual coherence of the
interfering fields in the high-gain regime for increasing pump strengths. We
find the behavior to be consistent with previous studies that assumed a
monochromatic plane-wave pump [65, 66].

In future, our work may potentially pave the way towards a better
understanding of the classical-quantum divide in the context of SPDC and
induced coherence. The model itself could be viewed as a specific application
of stochastic electrodynamics – a broader research program that attempts to
explain a variety of quantum phenomena by positing the existence of a
classical stochastic background radiation field that statistically mimics the
zero-point vacuum fluctuations [72, 73, 74, 75, 76]. It may be possible to
push the model further to investigate higher-order correlations and other
intriguing fundamental effects related to SPDC and induced coherence, which
could potentially shed light on the limits to which classical physics can be
used to approximate the quantum world.

In addition to fundamental implications, our work may also have significant
practical applicability. Our model can be a useful theoretical tool for
analysing high-gain SPDC [30, 56, 50, 31] and induced coherence experiments
[41, 68, 69]. Moreover, using the connection between Schmidt decomposition in
quantum theory and the coherent-mode decomposition in classical coherence
theory [32, 51], our model can be used to extract important properties of the
global quantum state such as the Schmidt spectrum and Schmidt modes [77]. A
precise knowledge of these properties may not only lead to a better
quantitative understanding of high-dimensional multiphoton entanglement of the
high-gain SPDC field [78], but may also inform experiments aimed at harnessing
the underlying correlations for applications in imaging [18, 19], quantum
state preparation [20], phase metrology [55], radiation shaping [56],
microscopy [57], sensing [59], and spectroscopy [63].

## Acknowledgments

We acknowledge useful discussions with Samuel Lemieux, Nicolás Quesada, Jeff
Lundeen, and Polina Sharapova. B.B acknowledges support from the Banting
Postdoctoral Fellowship.

## APPENDIX A: DERIVATION OF THE DFG EQUATIONS

Upon substituting equations (1) and (3) in (2), we obtain

$\displaystyle\left[\nabla^{2}+\frac{n^{2}_{j}\omega^{2}_{j}}{c^{2}}\right]\int\mathrm{d}{\bm{q}_{j}}A_{j}({\bm{q}_{j}},\omega_{j},z)\,e^{i({\bm{q}_{j}}\cdot{\bm{\rho}}+k_{jz}z)}$
$\displaystyle=-\frac{4d_{\rm
eff}\omega^{2}_{j}}{c^{2}}\iiint\mathrm{d}\omega_{p}\,\mathrm{d}{\bm{q}_{p}}\,\mathrm{d}{\bm{q}_{l}}\,A_{p}({\bm{q}_{p}},\omega_{p})\,A^{*}_{l}({\bm{q}_{l}},\omega_{l},z)$
$\displaystyle\hskip
113.81102pt\times\,e^{i\left[({\bm{q}_{p}}-{\bm{q}_{l}})\cdot{\bm{\rho}}+(k_{pz}-k_{lz})z\right]},$
(40)

for $j=s(i)$ and $l=i(s)$. We first focus only on the left hand side of the
above equation. Using $\nabla\equiv(\nabla_{\perp},\partial/\partial z)$ and
$|{\bm{k}_{j}}|=n_{j}\omega_{j}/c=\sqrt{|{\bm{q}_{j}}|^{2}+k^{2}_{jz}}$ and
simplifying, we obtain

$\displaystyle\int\mathrm{d}{\bm{q}_{j}}\left(|{\bm{k}_{j}}|^{2}-q^{2}_{j}\right)\,A_{j}({\bm{q}_{j}},\omega_{j},z)\,e^{i({\bm{q}_{j}}\cdot{\bm{\rho}}+k_{jz}z)}+\frac{\partial}{\partial
z}\int\mathrm{d}{\bm{q}_{j}}\left[\frac{\partial
A_{j}({\bm{q}_{j}},\omega_{j},z)}{\partial
z}+ik_{jz}A_{j}({\bm{q}_{j}},\omega_{j},z)\right]\,e^{i({\bm{q}_{j}}\cdot{\bm{\rho}}+k_{jz}z)}$
$\displaystyle=\int\mathrm{d}{\bm{q}_{j}}\left\\{k^{2}_{jz}\,A_{j}({\bm{q}_{j}},\omega_{j},z)\,e^{i({\bm{q}_{j}}\cdot{\bm{\rho}}+k_{jz}z)}+\left[\frac{\partial^{2}A_{j}({\bm{q}_{j}},\omega_{j},z)}{\partial
z^{2}}+2ik_{jz}\frac{\partial A_{j}({\bm{q}_{j}},\omega_{j},z)}{\partial
z}-k^{2}_{jz}A_{j}({\bm{q}_{j}},\omega_{j},z)\right]e^{i({\bm{q}_{j}}\cdot{\bm{\rho}}+k_{jz}z)}\right\\}.$

We make the slowly-varying envelope approximation
$\frac{\partial^{2}}{\partial z^{2}}A_{j}({\bm{q}_{j}},\omega_{j},z)\ll
k_{jz}\frac{\partial}{\partial z}A_{j}({\bm{q}_{j}},\omega_{j},z)$, use a
different dummy variable ${\bm{q}^{\prime}_{j}}$ instead of ${\bm{q}_{j}}$,
and equate to the right hand side of Eq. (40) to obtain

$\displaystyle\int\mathrm{d}{\bm{q}^{\prime}_{j}}\,2ik^{\prime}_{jz}\frac{\partial
A_{j}({\bm{q}^{\prime}_{j}},\omega_{j},z)}{\partial
z}e^{i({\bm{q}^{\prime}_{j}}\cdot{\bm{\rho}}+k^{\prime}_{jz}z)}$
$\displaystyle=-\frac{4d_{\rm
eff}\omega^{2}_{j}}{c^{2}}\iiint\mathrm{d}\omega_{p}\,\mathrm{d}{\bm{q}_{p}}\,\mathrm{d}{\bm{q}_{l}}\,A_{p}({\bm{q}_{p}},\omega_{p})\,A^{*}_{l}({\bm{q}_{l}},\omega_{l},z)\,e^{i\left[({\bm{q}_{p}}-{\bm{q}_{l}})\cdot{\bm{\rho}}+(k_{pz}-k_{lz})z\right]}.$

Multiplying both sides by $e^{-i{\bm{q}_{j}}\cdot{\bm{\rho}}}$ and integrating
with respect to ${\bm{\rho}}$ over the transverse extent of the crystal, we
obtain

$\displaystyle\iint\mathrm{d}{\bm{\rho}}\,\mathrm{d}{\bm{q}^{\prime}_{j}}\,2ik^{\prime}_{jz}\frac{\partial
A_{j}({\bm{q}^{\prime}_{j}},\omega_{j},z)}{\partial
z}e^{i\left[({\bm{q}^{\prime}_{j}}-{\bm{q}_{j}})\cdot{\bm{\rho}}+k^{\prime}_{jz}z\right]}$
$\displaystyle=-\frac{4d_{\rm
eff}\omega^{2}_{j}}{c^{2}}\int\mathrm{d}{\bm{\rho}}\iiint\mathrm{d}\omega_{p}\,\mathrm{d}{\bm{q}_{p}}\,\mathrm{d}{\bm{q}_{l}}\,A_{p}({\bm{q}_{p}},\omega_{p})\,A^{*}_{l}({\bm{q}_{l}},\omega_{l},z)$
$\displaystyle\hskip 170.71652pt\,\times
e^{i\left[({\bm{q}_{p}}-{\bm{q}_{l}}-{\bm{q}_{j}})\cdot{\bm{\rho}}+(k_{pz}-k_{lz})z\right]}.$

We now assume that the transverse extent of the crystal is much larger than
the pump spot-size and therefore, the integration over ${\bm{\rho}}$ can be
performed over the entire infinite range of ${\bm{\rho}}$ to yield

$\displaystyle\int\,\mathrm{d}{\bm{q}^{\prime}_{j}}\,2ik^{\prime}_{jz}\frac{\partial
A_{j}({\bm{q}^{\prime}_{j}},\omega_{j},z)}{\partial
z}e^{ik^{\prime}_{jz}z}\delta({\bm{q}^{\prime}_{j}}-{\bm{q}_{j}})$
$\displaystyle=-\frac{4d_{\rm
eff}\omega^{2}_{j}}{c^{2}}\iiint\mathrm{d}\omega_{p}\,\mathrm{d}{\bm{q}_{p}}\,\mathrm{d}{\bm{q}_{l}}\,A_{p}({\bm{q}_{p}},\omega_{p})\,A^{*}_{l}({\bm{q}_{l}},\omega_{l},z)\,e^{i(k_{pz}-k_{lz})z}$
$\displaystyle\hskip
227.62204pt\times\delta({\bm{q}_{p}}-{\bm{q}_{l}}-{\bm{q}_{j}}),$

where the Dirac delta relation on the right-hand side expresses conservation
of transverse momentum, i.e, ${\bm{q}_{p}}={\bm{q}_{s}}+{\bm{q}_{i}}$. Using
$\omega_{p}=\omega_{s}+\omega_{i}$ and $\Delta k_{z}=k_{pz}-k_{sz}-k_{iz}$,
the above equation can be simplified to yield

$\displaystyle\frac{\partial A_{j}({\bm{q}_{j}},\omega_{j},z)}{\partial z}$
$\displaystyle=\frac{2id_{\rm
eff}\omega^{2}_{j}}{k_{jz}c^{2}}\iint\mathrm{d}\omega_{l}\,\mathrm{d}{\bm{q}_{l}}\,A_{p}({\bm{q}_{p}},\omega_{p})A^{*}_{l}({\bm{q}_{l}},\omega_{l},z)e^{i\Delta
k_{z}z},$

which represents Eq. (4a) for $(j,l)=(s,i)$, and Eq. (4b) for $(j,l)=(i,s)$
upon complex conjugation.

## APPENDIX B: DETAILED CALCULATIONS FOR THE HIGH-GAIN REGIME:

### I. Derivation of the second-order differential equation (12) for the
signal field

Upon substituting Eq. (10) in equations (4), we obtain

$\displaystyle\frac{\partial A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial
z}=\frac{2id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V_{p}({\bm{\rho}},t)\,e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)e^{i\Delta
k_{z}z},$ (41a) $\displaystyle\frac{\partial
A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)}{\partial z}=\frac{-2id_{\rm
eff}\omega^{2}_{i}}{(2\pi)^{3}k_{iz}c^{2}}\iiiint\mathrm{d}\omega_{s}\,\mathrm{d}{\bm{q}_{s}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V^{*}_{p}({\bm{\rho}},t)\,e^{i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}A_{s}({\bm{q}_{s}},\omega_{s},z)e^{-i\Delta
k_{z}z}.$ (41b)

We now consider the right-hand side of Eq. (11)

$\displaystyle\frac{2id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}tV_{p}({\bm{\rho}},t)e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}\,e^{i\Delta
k_{z}z}\Bigg{[}\frac{\partial A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)}{\partial
z}+i\Delta k_{z}A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)\Bigg{]}$
$\displaystyle=\frac{4d^{2}_{\rm
eff}\omega^{2}_{s}\bar{\omega}^{2}_{i}}{(2\pi)^{6}k_{sz}\bar{k}_{iz}c^{4}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\iiiint\mathrm{d}\omega^{\prime}_{s}\,\mathrm{d}{\bm{q}^{\prime}_{s}}\,\mathrm{d}{\bm{\rho}^{\prime}}\,\mathrm{d}t^{\prime}V_{p}({\bm{\rho}},t)V^{*}_{p}({\bm{\rho}^{\prime}},t^{\prime})e^{-i[({\bm{q}_{p}}\cdot{\bm{\rho}}-{\bm{q}^{\prime}_{p}}\cdot{\bm{\rho}^{\prime}})-(\omega_{p}t-\omega^{\prime}_{p}t^{\prime})]}A_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s},z)$
$\displaystyle\hskip 56.9055pt\times e^{i(\Delta k_{z}-\Delta
k^{\prime}_{z})z}+i\Delta\bar{k}_{z}\frac{2id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V_{p}({\bm{\rho}},t)\,e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}A^{*}_{i}({\bm{q}_{i}},\omega_{i},z)e^{i\Delta
k_{z}z}$ $\displaystyle=\frac{4d^{2}_{\rm
eff}\omega^{2}_{s}\bar{\omega}^{2}_{i}}{(2\pi)^{6}k_{sz}\bar{k}_{iz}c^{4}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\iiiint\mathrm{d}\omega^{\prime}_{s}\,\mathrm{d}{\bm{q}^{\prime}_{s}}\,\mathrm{d}{\bm{\rho}^{\prime}}\,\mathrm{d}t^{\prime}V_{p}({\bm{\rho}},t)V^{*}_{p}({\bm{\rho}^{\prime}},t^{\prime})e^{-i[({\bm{q}_{s}}\cdot{\bm{\rho}}-{\bm{q}^{\prime}_{s}}\cdot{\bm{\rho}^{\prime}})-(\omega_{s}t-\omega^{\prime}_{s}t^{\prime})+{\bm{q}_{i}}({\bm{\rho}}-{\bm{\rho}^{\prime}})-\omega_{i}(t-t^{\prime})]}$
$\displaystyle\hskip 284.52756pt\times
A_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s},z)e^{i(\Delta k_{z}-\Delta
k^{\prime}_{z})z}+i\Delta\bar{k}_{z}\frac{\partial
A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial z}$
$\displaystyle=\frac{4d^{2}_{\rm
eff}\omega^{2}_{s}\bar{\omega}^{2}_{i}}{(2\pi)^{3}k_{sz}\bar{k}_{iz}c^{4}}\iint\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\iiiint\mathrm{d}\omega^{\prime}_{s}\,\mathrm{d}{\bm{q}^{\prime}_{s}}\,\mathrm{d}{\bm{\rho}^{\prime}}\,\mathrm{d}t^{\prime}V_{p}({\bm{\rho}},t)V^{*}_{p}({\bm{\rho}^{\prime}},t^{\prime})e^{-i[({\bm{q}_{s}}\cdot{\bm{\rho}}-{\bm{q}^{\prime}_{s}}\cdot{\bm{\rho}^{\prime}})-(\omega_{s}t-\omega^{\prime}_{s}t^{\prime})]}\delta({\bm{\rho}}-{\bm{\rho}^{\prime}})\delta(t-t^{\prime})A_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s},z)$
$\displaystyle\hskip 341.43306pt\times e^{i(\Delta k_{z}-\Delta
k^{\prime}_{z})z}+i\Delta\bar{k}_{z}\frac{\partial
A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial z}$
$\displaystyle=\frac{4d^{2}_{\rm
eff}\omega^{2}_{s}\bar{\omega}^{2}_{i}}{(2\pi)^{3}k_{sz}\bar{k}_{iz}c^{4}}\iint\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\iint\mathrm{d}\omega^{\prime}_{s}\,\mathrm{d}{\bm{q}^{\prime}_{s}}\,|V_{p}({\bm{\rho}},t)|^{2}e^{-i[({\bm{q}_{s}}-{\bm{q}^{\prime}_{s}})\cdot{\bm{\rho}}-(\omega_{s}-\omega_{s})t]}\,A_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s},z)\,e^{i(\Delta
k_{z}-\Delta k^{\prime}_{z})z}+i\Delta\bar{k}_{z}\frac{\partial
A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial z}$
$\displaystyle=\frac{4d^{2}_{\rm
eff}\omega^{2}_{s}\bar{\omega}^{2}_{i}}{k_{sz}\bar{k}_{iz}c^{4}}|V_{p}({\bm{\rho}},t)|^{2}\iint\mathrm{d}\omega^{\prime}_{s}\,\mathrm{d}{\bm{q}^{\prime}_{s}}\,\delta({\bm{q}_{s}}-{\bm{q}^{\prime}_{s}})\,\delta(\omega_{s}-\omega_{s})\,A_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s},z)\,e^{i(\Delta
k_{z}-\Delta k^{\prime}_{z})z}+i\Delta\bar{k}_{z}\frac{\partial
A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial z}$
$\displaystyle=\frac{4d^{2}_{\rm
eff}\omega^{2}_{s}\bar{\omega}^{2}_{i}}{k_{sz}\bar{k}_{iz}c^{4}}|V_{p}({\bm{\rho}},t)|^{2}\,A_{s}({\bm{q}_{s}},\omega_{s},z)+i\Delta\bar{k}_{z}\frac{\partial
A_{s}({\bm{q}_{s}},\omega_{s},z)}{\partial z}.$

We note that in the second-last step of the above calculation, we have used
the fact that the pump intensity profile $|V_{p}({\bm{\rho}},t)|^{2}$ has a
very slow variation with respect to ${\bm{\rho}}$ and $t$ due to our
assumption that the frequency and angular bandwidth of the pump is much
smaller than that of the generated field. Consequently, the function
$|V_{p}({\bm{\rho}},t)|^{2}$ could be taken out of the integral. Upon shifting
the above result to the left-hand side of Eq. (11), we obtain Eq. (12).

### II. Solution of the second-order differential equation (12) for the signal
field

Substituting an ansatz solution of the form
$A_{s}({\bm{q}_{s}},\omega_{s},z)=e^{rz}$ in Eq. (12), we obtain
$r^{2}-i\Delta\bar{k}_{z}r-\bar{G}^{2}({\bm{\rho}},t)=0$, which has the roots
$r_{\pm}=i\Delta\bar{k}_{z}/2\pm\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)$,
where $\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)$ is defined by Eq. (15). The
general solution then takes the form
$A_{s}({\bm{q}_{s}},\omega_{s},z)=Ce^{r_{+}z}+De^{r_{-}z}$, where $C$ and $D$
are scaling factors to be determined by initial conditions. The first
condition $A_{s}({\bm{q}_{s}},\omega_{s},z=0)=0$ implies $C+D=0$, and the
value of $\partial A_{s}({\bm{q}_{s}},\omega_{s},z)/\partial z|_{z=0}$
obtained by evaluating Eq. (41a) for $z=0$ yields

$\displaystyle C=\frac{id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,\frac{V_{p}({\bm{\rho}},t)}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\,e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}A^{*}_{i}({\bm{q}_{i}},\omega_{i},0).$

We again use the approximation that the function
$\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)$ from its dependence on
$|V_{p}({\bm{\rho}},t)|^{2}$ has a very slow variation with respect to
${\bm{\rho}}$ and $t$. The above relation then implies
$A_{s}({\bm{q}_{s}},\omega_{s},z)=2C\,e^{i\Delta\bar{k}_{z}z/2}\,\mathrm{sinh}\left[\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)z\right]$,
which yields Eq. (14) for $z=L$.

### III. Evaluating the high-gain spatiotemporal correlation function of the
signal field:

Using Eq. (14), we evaluate

$\displaystyle\langle
A_{s}({\bm{q}_{s}},\omega_{s},z)A^{*}_{s}({\bm{q}^{\prime}_{s}},\omega^{\prime}_{s},z)\rangle=\frac{4d^{2}_{\rm
eff}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{(2\pi)^{6}k_{sz}k^{\prime}_{sz}c^{4}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\iiiint\mathrm{d}\omega^{\prime}_{i}\,\mathrm{d}{\bm{q}^{\prime}_{i}}\,\mathrm{d}{\bm{\rho}^{\prime}}\,\mathrm{d}t^{\prime}\langle
V_{p}({\bm{\rho}},t)V_{p}({\bm{\rho}^{\prime}},t^{\prime})\rangle$
$\displaystyle\times
e^{-i[({\bm{q}_{p}}\cdot{\bm{\rho}}-{\bm{q}^{\prime}_{p}}\cdot{\bm{\rho}^{\prime}})-(\omega_{p}t-\omega^{\prime}_{p}t^{\prime})]}\,\langle
A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)A_{i}({\bm{q}^{\prime}_{i}},\omega^{\prime}_{i},0)\rangle\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right]\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}^{\prime}},t^{\prime})L}{\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}^{\prime}},t^{\prime})}\right]e^{i[(\Delta
k_{z}-\Delta
k^{\prime}_{z})+(\Delta\bar{k}_{z}-\Delta\bar{k}^{\prime}_{z})/2]L}$
$\displaystyle=\frac{4d^{2}_{\rm
eff}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{(2\pi)^{6}k_{sz}k^{\prime}_{sz}c^{4}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\iiiint\mathrm{d}\omega^{\prime}_{i}\,\mathrm{d}{\bm{q}^{\prime}_{i}}\,\mathrm{d}{\bm{\rho}^{\prime}}\,\mathrm{d}t^{\prime}\langle
V_{p}({\bm{\rho}},t)V_{p}({\bm{\rho}^{\prime}},t^{\prime})\rangle\,e^{-i[({\bm{q}_{p}}\cdot{\bm{\rho}}-{\bm{q}^{\prime}_{p}}\cdot{\bm{\rho}^{\prime}})-(\omega_{p}t-\omega^{\prime}_{p}t^{\prime})]}$
$\displaystyle\hskip
56.9055pt\times\frac{\hbar\omega_{i}}{4\pi\epsilon_{0}}\delta({\bm{q}_{i}}-{\bm{q}^{\prime}_{i}})\delta(\omega_{i}-\omega^{\prime}_{i})\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right]\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}^{\prime}},t^{\prime})L}{\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}^{\prime}},t^{\prime})}\right]e^{i[(\Delta
k_{z}-\Delta
k^{\prime}_{z})+(\Delta\bar{k}_{z}-\Delta\bar{k}^{\prime}_{z})/2]L}$
$\displaystyle=\frac{4d^{2}_{\rm
eff}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{(2\pi)^{6}k_{sz}k^{\prime}_{sz}c^{4}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\frac{\hbar\omega_{i}}{4\pi\epsilon_{0}}\iint\mathrm{d}{\bm{\rho}^{\prime}}\,\mathrm{d}t^{\prime}\langle
V_{p}({\bm{\rho}},t)V_{p}({\bm{\rho}^{\prime}},t^{\prime})\rangle\,e^{-i[({\bm{q}_{s}}\cdot{\bm{\rho}}-{\bm{q}^{\prime}_{s}}\cdot{\bm{\rho}^{\prime}})-(\omega_{s}t-\omega^{\prime}_{s}t^{\prime})]}e^{i[{\bm{q}_{i}}\cdot({\bm{\rho}}-{\bm{\rho}^{\prime}})-\omega_{i}(t-t^{\prime})]}$
$\displaystyle\hskip
170.71652pt\times\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right]\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}^{\prime}},t^{\prime})L}{\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}^{\prime}},t^{\prime})}\right]e^{i[(\Delta
k_{z}-\Delta
k^{\prime}_{z})+(\Delta\bar{k}_{z}-\Delta\bar{k}^{\prime}_{z})/2]L}$
$\displaystyle=\frac{2\hbar\omega_{i0}d^{2}_{\rm
eff}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{(2\pi)^{4}\epsilon_{0}k_{sz}k^{\prime}_{sz}c^{4}}\iint\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\iint\mathrm{d}{\bm{\rho}^{\prime}}\,\mathrm{d}t^{\prime}\langle
V_{p}({\bm{\rho}},t)V_{p}({\bm{\rho}^{\prime}},t^{\prime})\rangle\,e^{-i[({\bm{q}_{s}}\cdot{\bm{\rho}}-{\bm{q}^{\prime}_{s}}\cdot{\bm{\rho}^{\prime}})-(\omega_{s}t-\omega^{\prime}_{s}t^{\prime})]}\delta({\bm{\rho}}-{\bm{\rho}^{\prime}})\delta(t-t^{\prime})$
$\displaystyle\hskip
227.62204pt\times\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right]\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}^{\prime}},t^{\prime})L}{\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}^{\prime}},t^{\prime})}\right]e^{i(\Delta\bar{k}_{z}-\Delta\bar{k}^{\prime}_{z})L/2}$
$\displaystyle=\frac{2\hbar\omega_{i0}d^{2}_{\rm
eff}\omega^{2}_{s}{\omega^{\prime}_{s}}^{2}}{(2\pi)^{4}\epsilon_{0}k_{sz}k^{\prime}_{sz}c^{4}}\iint\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\langle|V_{p}({\bm{\rho}},t)|^{2}\rangle\,e^{-i[({\bm{q}_{s}}-{\bm{q}^{\prime}_{s}})\cdot{\bm{\rho}}-(\omega_{s}-\omega^{\prime}_{s})t]}\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)}\right]\left[\frac{\mathrm{sinh}\,\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}},t)L}{\Gamma(\Delta\bar{k^{\prime}}_{z},{\bm{\rho}},t)}\right]e^{i(\Delta\bar{k}_{z}-\Delta\bar{k}^{\prime}_{z})L/2},$

which is Eq. (16).

### IV. Evaluating the output field from the SU(1,1) setup:

As the field $A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime})$ satisfies Eq. (12),
we can assume an ansatz solution of the form
$A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime})=Ce^{r_{+}z^{\prime}}+De^{r_{-}z^{\prime}}$,
where
$r_{\pm}=i\Delta\bar{k}_{z}/2\pm\Gamma(\Delta\bar{k}_{z},{\bm{\rho}},t)$.
Thus, we have $C+D=A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime}=0)$ and
$Cr_{+}+Dr_{-}=\dot{A}_{s}({\bm{q}_{s}},\omega_{s},z^{\prime}=0)=\partial
A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime})/\partial z|_{z^{\prime}=0}$, which
implies
$C=\left\\{\dot{A}_{s}({\bm{q}_{s}},\omega_{s},z^{\prime}=0)-r_{-}A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime}=0)\right\\}/(r_{+}-r_{-})$
and
$D=\left\\{r_{+}A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime}=0)-\dot{A}_{s}({\bm{q}_{s}},\omega_{s},z^{\prime}=0)\right\\}/(r_{+}-r_{-})$.
Substituting from equations (14) and (26), we obtain

$\displaystyle A_{s}({\bm{q}_{s}},\omega_{s},z^{\prime}=L)=\frac{2id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V_{p}({\bm{\rho}},t)\,e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}\,A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\,e^{i(\Delta
k_{z}-\Delta\bar{k}_{z}/2)L}$ $\displaystyle\hskip
56.9055pt\times\left\\{e^{\Gamma L}\left[\frac{e^{i\Delta
k^{\rm(air)}_{z}D}}{2\Gamma}\Bigg{\\{}\cosh\Gamma
L+\frac{i\Delta\bar{k}_{z}}{2\Gamma}\sinh\Gamma
L\Bigg{\\}}+\left\\{\Gamma-i\Delta\bar{k}_{z}/2\right\\}\frac{\mathrm{sinh}\,\Gamma
L}{2\Gamma^{2}}\right]\right.$ $\displaystyle\hskip
113.81102pt\left.+e^{-\Gamma
L}\left[\left\\{i\Delta\bar{k}_{z}/2+\Gamma\right\\}\frac{\mathrm{sinh}\,\Gamma
L}{2\Gamma^{2}}-\frac{e^{i\Delta
k^{\rm(air)}_{z}D}}{2\Gamma}\Bigg{\\{}\cosh\Gamma
L+\frac{i\Delta\bar{k}_{z}}{2\Gamma}\sinh\Gamma L\Bigg{\\}}\right]\right\\}$
$\displaystyle=\frac{2id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V_{p}({\bm{\rho}},t)\,e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}\,A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\,e^{i(\Delta
k_{z}-\Delta\bar{k}_{z}/2)L}$ $\displaystyle\hskip
56.9055pt\times\left\\{e^{i\Delta k^{\rm(air)}_{z}D}\Bigg{\\{}\frac{\sinh
2\Gamma L}{2\Gamma}+\frac{i\Delta\bar{k}_{z}}{2\Gamma^{2}}\sinh^{2}\Gamma
L\Bigg{\\}}+\frac{\mathrm{sinh}\,\Gamma
L}{2\Gamma^{2}}\left(2\Gamma\cosh\Gamma L-i\Delta\bar{k}_{z}\sinh\Gamma
L\right)\right\\}$ $\displaystyle=\frac{2id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V_{p}({\bm{\rho}},t)\,e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}\,A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\,e^{i(\Delta
k_{z}-\Delta\bar{k}_{z}/2)L}$ $\displaystyle\hskip
113.81102pt\times\left\\{\frac{\sinh 2\Gamma L}{2\Gamma}\left(1+e^{i\Delta
k^{\rm(air)}_{z}D}\right)-\frac{i\Delta\bar{k}_{z}}{2}\left(\frac{\sinh\Gamma
L}{\Gamma}\right)^{2}\left(1-e^{i\Delta k^{\rm(air)}_{z}D}\right)\right\\}$
$\displaystyle=\frac{2id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V_{p}({\bm{\rho}},t)\,e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}\,A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\,e^{i(\Delta
k_{z}-\Delta\bar{k}_{z}/2)L}\,e^{i\Delta k^{\rm(air)}_{z}D/2}$
$\displaystyle\hskip 113.81102pt\times\left\\{\frac{\sinh 2\Gamma
L}{\Gamma}\cos\left\\{\Delta
k^{\rm(air)}_{z}D/2\right\\}-\Delta\bar{k}_{z}\left(\frac{\sinh\Gamma
L}{\Gamma}\right)^{2}\sin\left\\{\Delta k^{\rm(air)}_{z}D/2\right\\}\right\\}$
$\displaystyle=\frac{4id_{\rm
eff}\omega^{2}_{s}}{(2\pi)^{3}k_{sz}c^{2}}\iiiint\mathrm{d}\omega_{i}\,\mathrm{d}{\bm{q}_{i}}\,\mathrm{d}{\bm{\rho}}\,\mathrm{d}t\,V_{p}({\bm{\rho}},t)\,e^{-i({\bm{q}_{p}}\cdot{\bm{\rho}}-\omega_{p}t)}\,A^{*}_{i}({\bm{q}_{i}},\omega_{i},0)\,e^{i(\Delta
k_{z}-\Delta\bar{k}_{z}/2)L}\,e^{i\Delta
k^{\rm(air)}_{z}D/2}\left(\frac{\sinh\Gamma L}{\Gamma}\right)$
$\displaystyle\hskip 113.81102pt\times\left[\cosh\Gamma L\,\cos\left\\{\Delta
k^{\rm(air)}_{z}D/2\right\\}-\frac{\Delta\bar{k}_{z}}{2\Gamma}\sinh\Gamma
L\sin\left\\{\Delta k^{\rm(air)}_{z}D/2\right\\}\right],$

which is Eq. (32).

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