# Partial coherence and coherence length in StimPDC

G. H. dos Santos<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Santa Catarina, CEP 88040-900, Florianóplis, SC, Brazil  R. C.
Souza Pimenta  Departamento de Física, Universidade Federal de Santa Catarina,
CEP 88040-900, Florianóplis, SC, Brazil  R. M. Gomes  Instituto de Física,
Universidade Federal de Goiás, CEP 74690-900, Goiânia, GO, Brazil  S. P.
Walborn<EMAIL_ADDRESS>Departamento de Física, Universidad de Concepción,
160-C Concepción, Chile  Millennium Institute for Research in Optics,
Universidad de Concepción, 160-C Concepción, Chile  P. H. Souto Ribeiro
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Santa
Catarina, CEP 88040-900, Florianóplis, SC, Brazil

###### Abstract

In parametric down conversion, a nonlinear crystal is pumped by a laser and
spontaneous emission takes place in signal and idler modes according to the
phase matching conditions. A seed laser can stimulate the emission in the
signal beam if there is mode overlap between them. This also enhances the
emission in the idler beam, affecting its coherence properties. While the
degree of coherence of the idler field as a function of the seed power has
already been studied, the transverse coherence length has not yet been
properly investigated. The transverse coherence length is a key parameter of
optical beams that determines the beam divergence, for example. Here, we
present a theoretical and experimental investigation of the transverse
coherence length in stimulated down conversion. In addition, we make a
connection between stimulated down conversion and partially coherent sources
like the Gaussian Schell model beams, and show that in general the idler field
cannot be described with this model.

###### pacs:

05.45.Yv, 03.75.Lm, 42.65.Tg

##  I Introduction

The coherence of a light field is a fundamental property that affects all
degrees of freedom. In several circumstances it is possible to treat temporal
coherence and transverse spatial coherence as independent properties, and here
we focus in the spatial coherence. Most natural light sources are spatially
incoherent in the source, but becomes partially coherent with
propagationborn_wolf99 . Therefore, partial coherence is a relevant issue in
most applications of natural light. A particularly relevant case of partially
coherent light is the so called Gaussian Schell Model beam (GSM)Schell56 ;
friberg94 , which finds applications in imaging, optical communication, light
scattering, nonlinear optics, and others Ma:17 ; ismail2017 ; Zhang:19 ;
ismail2020 ; Hutter20 ; Hutter21 ; Santos2022 ; Cai2022 .

On the other hand, laser fields are nearly perfect coherent. In this case the
light can be described by deterministic electric field functions, while the
partially coherent ones require other mathematical tools that include
stochastic fluctuations, as in the GSM model. We analyze here a different kind
of partially coherent light source: stimulated parametric down conversion
(StimPDC). In this nonlinear process, spontaneously emitted light is mixed
with stimulated emission in the so-called idler mode. The degree of coherence
of the light emitted depends on the stimulating power Ribeiro95  and so it is
expected that the coherence properties will also depend on this parameter. We
present here a theoretical and experimental investigation of the coherence
length of the light produced in StimPDC. To our knowledge, this coherence
property has not yet been investigated previously.

##  II Theory

The mutual coherence of the idler field in StimPDC with a coherent seed beam
was studied in Ref. Ribeiro95 , where it was shown that:

$\mu_{i}=\frac{\mu_{sp}I_{sp}+\mu_{c}I_{c}}{I_{sp}+I_{c}},$ (1)

where $\mu_{i}$ is the normalized mutual coherence between fields at points 1
and 2 in a plane transverse to the propagation direction of the idler beam
(the position dependence is not included explicitly in $\mu$ for brevity). For
StimPDC the mutual coherence is composed of two terms. The first, which we
denote as $\mu_{sp}$, is the mutual coherence between fields at the same
points due to the contribution of the spontaneously emitted light, and
$\mu_{c}$ is the contribution due to the coherent stimulation by the seed beam
and is ideally unit. The parameters $I_{sp}$ and $I_{c}$ describe the
intensities of the spontaneous emission and stimulated emission, respectively.

The normalized mutual coherence $\mu$ can be calculated using the cross
spectral density (CSD) $W(\mathbf{r}_{1},\mathbf{r}_{2})$:

$\mu\equiv\mu(\mathbf{r}_{1},\mathbf{r}_{2})=\frac{W(\mathbf{r}_{1},\mathbf{r}_{2})}{\sqrt{W(\mathbf{r}_{1},\mathbf{r}_{1})W(\mathbf{r}_{2},\mathbf{r}_{2})}}.$
(2)

As mentioned above, the output idler beam has components originating from the
spontaneous and stimulated emission at the source. It has been shown recently
that, in typical experimental situations, both of these contributions can be
described using the well-known GSM beams, which have a cross-spectral density
(CSD) given by

$W_{GSM}(\mathbf{r},\mathbf{r}^{\prime})=Ie^{-\frac{r^{2}+r^{\prime
2}}{4w^{2}}}e^{-\frac{(\mathbf{r}-\mathbf{r}^{\prime})^{2}}{2\delta^{2}}}e^{-ik\frac{(\mathbf{r}^{2}-\mathbf{r}^{\prime
2})}{2R}},$ (3)

where $I$ is the maximum beam intensity, $w$ is the beam waist, $\delta$ the
transverse coherence length, and $R$ the phase curvature.

In the case of the spontaneous emission term and nearly degenerate signal and
idler fields, Hutter et. al. Hutter20 ; Hutter21  showed that the idler beam
produced in SPDC with a GSM pump beam is described by a GSM beam. Thus, we can
define the spontaneous (“sp”) component with CSD $W_{sp}=W_{GSM}$ for some
specific parameters $I,w,\delta,R$. The stimulated (“stim”) component, on the
other hand, has been studied in Ref. Santos2022 . Considering both the seed
and pump as GSM beams, the CSD of the stimulated component can be written as
$W_{stim}=W_{GSM}$ for some intensity depending also on nonlinear coefficients
of the crystal, and the following idler beam parameters given as functions of
the pump and seed beam parameters:

$\frac{1}{w_{stim}^{2}}=\frac{1}{w_{seed}^{2}}+\frac{1}{w_{pump}^{2}},$ (4)

transverse coherence length

$\frac{1}{\delta_{stim}^{2}}=\frac{1}{\delta_{seed}^{2}}+\frac{1}{\delta_{pump}^{2}},$
(5)

phase curvature

$\frac{k_{i}}{R_{stim}}=\frac{k_{pump}}{R_{pump}}-\frac{k_{seed}}{R_{seed}},$
(6)

where $stim,seed,pump$ stand for stimulated idler, seed, and pump,
respectively.

Including the contribution coming from both the spontaneous and stimulated
components in StimPDC, the CSD for the idler field becomes:

$W_{i}(\mathbf{r}_{1},\mathbf{r}_{2})=W_{stim}(\mathbf{r}_{1},\mathbf{r}_{2})+W_{sp}(\mathbf{r}_{1},\mathbf{r}_{2}).$
(7)

Both $W_{stim}$ and $W_{sp}$ can be GSM-type, depending on the pump and seed
CSD and relative intensities. Therefore, $W_{i}$ will be GSM-type in several
practical conditions. However, in general it may not be a GSM beam.

Dividing Eq. (7) by the total intensity $I_{i}=I_{stim}+I_{sp}$ leads to a
relation for the normalized mutual coherence of the total idler field:

$\mu_{i}=(1-\beta)\mu_{sp}+\beta\mu_{stim},$ (8)

where $\beta=I_{stim}/(I_{sp}+I_{stim})$.

Let us consider the case where pump and seed beams are pure Gaussian beams, so
that $\delta_{pump}=\delta_{seed}\rightarrow\infty$, giving a stimulated
component which is completely coherent
$\delta_{stim}\rightarrow\delta_{c}=\infty$. In this case, the normalized
mutual coherence from Eq. (8) leads directly to (1) with $\mu_{stim}=\mu_{c}$,
as observed years ago. For the GSM fields, the normalized coherence is a
function of the relative distance $d=|\mathbf{r}_{1}-\mathbf{r}_{2}|$ between
points 1 and 2. However, if we assume that the stimulated component is
coherent, $\mu_{stim}=\mu_{c}$ is constant, ideally unity and independent of
$d$. Thus, we can rewrite the mutual coherence of StimPDC as

$\mu_{i}(d)=(1-\beta)\,\mu_{sp}(d)+\beta\,\mu_{c},$ (9)

where now $\beta=I_{c}/(I_{sp}+I_{c})$. Typically, the transverse coherence
length $l_{t}$ is defined as the value of $d$ for which the absolute value of
the normalized mutual coherence is $|\mu_{i}(d=l_{t})|=\epsilon$, where
$\epsilon$ is a small arbitrary constant.

The coherence length can be obtained by solving the following relation for
$l_{t}$ and some specific function $\mu_{sp}(x)$:

$|\mu_{i}(l_{t})|=|(1-\beta)\,\mu_{sp}(l_{t})+\beta\,\mu_{c}|=\epsilon.$ (10)

Before we further discuss the characterization of the coherence length, let us
present an experiment and results investigating the mutual coherence of
StimPDC.

##  III Experiment

The experimental setup is sketched in Fig. 1. A $405$ nm diode laser is used
to pump a 2 mm long Beta-Barium-Borate (BBO) nonlinear crystal (NLC), where
parametric down conversion takes place. The pump beam is linearly polarized
along the horizontal direction and the phase matching is type I. A second
diode laser called seed oscillating at $780$ nm is aligned with one of the
down converted modes, say the signal beam, and its emission is stimulated. A
scheme using a variable neutral density filter (VNDF), a beam splitter (BS1)
and a power meter is used to control the intensity of the seed beam. The
lenses L1 and L2 are used to control the size beam on the nonlinear crystal.
The corresponding idler beam at $840$ nm is indirectly stimulated, due to the
photon-pair emission in the process. In the idler beam, a lens L3 transposes
the image plane to a Michelson interferometer, which is 8 cm away from the
crystal. The strategy to measure the transverse coherence length consists in
misaligning the interfering beams from the interferometer shifting one path by
a tilt in the mirror M3 so that different points in the wavefront are
superposed with different separations as illustrated in Fig. 1.

Figure 1: Experimental setup for measuring the coherence length StimPDC light.
@405 is a blue laser at 405 nm, @780 is a red laser at 780 nm,
$\frac{\lambda}{2}$ is a half-wave plate, VNDF is a variable neutral density
filter, NLC is a nonlinear crystal, M1, M2, and M3 are mirrors, BS1 and BS2
are beam splitters, L1, L2, and L3 are lenses. A CMOS Camera measure the
interference pattern of the idler beam in a controlled misaligned Michelson
interferometer.

##  IV Results and Discussion of Coherence Length

Fig. 2 shows plots of the mutual coherence, identified as the visibility of
interference patterns for different displacements $d$. They were obtained
varying the separation between the interfering beams at the output of the
Michelson interferometer. The separations are on the order of tens of microns,
and are controlled by tilting one of the beams and measuring the displacement
between the peaks of the two interfering distributions taken separately. The
interference patterns are photographed with a CMOS camera. Measurements are
performed for several different separations and several different ratios
between the spontaneously emitted and the stimulated emitted light
contributions.

Figure 2: Mutual coherence as characterized by the visibility of the
interference pattern as a function of beams displacement $d$ in a misaligned
Michelson interferometer

Qualitatively, we observe that the visibilities of the interference patterns
decrease with increasing separation between interfering beams, as expected.
Moreover, the degree of coherence for displacement $d>0$ always increases when
the seed power increases.

Using Eq. (2) and the GSM for the spontaneous contribution, we can calculate
$\mu_{sp}(d)=e^{-\frac{-d^{2}}{2\delta_{sp}^{2}}}$. The normalized mutual
coherence of the idler field is then

$\displaystyle\mu_{i}(d)=(1-\beta)\,\mbox{e}^{-\frac{(\mathbf{r}-\mathbf{r}^{\prime})^{2}}{2\delta_{sp}^{2}}}+\beta.$
(11)

The absolute values in Eq. 10 were dropped, because all quantities are
positive. The data points were adjusted to Eq. (11). With $\beta=0$ we found
$\delta_{sp}=20.72\pm 0.59\mu\mathrm{m}$, and used this value for all the
curve fits so that only $\beta$ was the only free parameter, the values of
which are shown in Table 1. As seen in Fig. 2 and also in Eq. (11), one
signature of the light emitted in StimPDC process is that the degree of
coherence does not always decrease tending to zero. The minimum value of the
degree of coherence is given by $\beta$, the normalized intensity of the
stimulated component, since the stimulated light is nearly as coherent as the
pump and seed. Therefore, if one characterizes coherence length using a
minimum value of the mutual coherence as in Eq. (10), there may be partially
coherent beams (those with $\beta>\epsilon)$ with infinite coherence length,
in principle.

At the same time, the choice of parameter $\epsilon$ is arbitrary. To have
some operational meaning, it should correspond to the coherence requirements
for a particular application. For example, in quantum information, certain
conclusions about the quantum nature of the source can only be reached for a
minimum value of the visibility borges10 ; carvalho12 ; borges14 . Thus, in
order to characterize the coherence length of a StimPDC light source, we
introduce the term “$\epsilon$-coherence length”, where $\epsilon$ is the
minimum required interference visibility (mutual coherence) and can in
principle take on values between zero and one. In Fig. 2 thus plot the cutoff
for the $1/e$-coherence length, $1/2$-coherence length, and $7/8$ coherence
length. The coherence lengths obtained are summarized in table

$I_{seed}$ ($\mu$W) |  $\beta$ ($\%$) |  $\epsilon={1}/{e}$ ($\mu$m) |  $\epsilon=1/2$ ($\mu$m) |  $\epsilon=7/8$ ($\mu$m)
---|---|---|---|---
0 | 0 | 29.30 | 24.40 | 10.71
90 | $4.9\pm 3.5$ | 30.64 | 25.32 | 11.00
270 | $13.9\pm 2.7$ | 33.74 | 27.32 | 11.61
540 | $20.5\pm 2.5$ | 36.88 | 29.17 | 12.12
3000 | $44.9\pm 1.8$ | $\infty$ | 45.30 | 14.87
10700 | $76.0\pm 1.7$ | $\infty$ | $\infty$ | 25.12
Table 1: Normalized intensity of the stimulated component and
$\epsilon$-coherence lengths obtained for different seed beam intensities.

We might look to other parameters in Eq. (11) to characterize the coherence
length of a StimPDC light source. The width of the gaussian contribution
$\delta_{sp}$ is due to the spontaneous component alone, being the same for
all values of $\beta$, and thus not a good parameter (except perhaps when
$\beta$ is small, where the usual definition works fine). The parameter
$\beta$ gives the minimum coherence, and does not reveal information about the
distance $d$ at which it can be reached.

##  V Conclusion

We have studied the coherence properties of the idler beam in StimPDC, with a
view to measure the coherence length full idler beam, taking into account
spontaneous and stimulated emission of light. We showed theoretically that the
idler beam is a weighted combination of two GSM beams, and thus is not a GSM
beam in the general case. This leads to a normalized mutual coherence
function, that is not a simple gaussian distribution, which can render the
usual definition of the coherence length meaningless in some cases. We note
that other experiments have shown non-trival and even non-monotonic coherence
functions walborn11b ; almeida12 . Experimentally, we considered the case
where the pump and seed laser beams have infinite coherence length. This
configuration allows measuring and characterizing the effective coherence
length of the idler as a function of the coherence length arising from the
SPDC component and the normalized intensity of stimulation. We evaluated the
visibility pattern as a function of the seed beam intensity, obtained in a
Michelson interferometer in which one path is displaced horizontally by a
controlled length. Motivated by quantum information applications, we introduce
the notion of “$\epsilon$-coherence length”, defined as the largest distance
between two interfering points of the field for which the visibility is above
$\epsilon$, and use it to characterize the idler field. Our work contributes
to the development of new approaches for synthesizing partially coherent
optical beams and introduces the notion of coherence length in non-trivial
field distributions, according to its practical application.

###### Acknowledgements.

This work was funded by the the Brazilian agencies Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq - DOI 501100003593), Instituto
Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ
465469/2014-0), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
(CAPES DOI 501100002322), Fundação de Amparo à Pesquisa do Estado de Santa
Catarina (FAPESC - DOI 501100005667), Fundação de Amparo à Pesquisa do Estado
de Goiás (FAPEG - DOI 501100005285), and the Chilean agencies Fondo Nacional
de Desarrollo Científico y Tecnológico (FONDECYT - DOI 501100002850, regular
project 1200266); National Agency of Research and Development (ANID), and
Millennium Science Initiative Program—ICN17-012.

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