Figure 4: The graph adjacency matrix ${\bf V}$, characterizing the connections
between different nodes of DEGM after lifelong learning.

(a) The edges of DEGM after the MSFIRC lifelong learning. (b) The edges of
DEGM after the CCCOSCZC lifelong learning.

Figure 5: Edge information of DEGM after lifelong learning. The red and blue represent the basic and specific node, respectively. Criteria | Dataset | BE | LGM | DEGM | DEGM-2 | CN-DPM*
---|---|---|---|---|---|---
SL | MNIST | 26.3 | 685.3 | 22.3 | 22.3 | 21.9
SVHN | 47.0 | 941.7 | 30.1 | 29.0 | 39.3
Fashion | 43.8 | 663.4 | 37.7 | 27.4 | 36.6
IFashion | 45.9 | 1148.4 | 35.6 | 27.4 | 38.4
RMNIST | 27.9 | 704.2 | 20.2 | 22.1 | 25.3
Cifar10 | 994.4 | 1241.1 | 615.3 | 608.1 | 892.1
Average | 197.5 | 897.4 | 126.9 | 122.7 | 175.6
Table 4: The results under MSFIRC lifelong learning. Criteria | Dataset | BE | LGM | DEGM | DEGM-2 | CN-DPM*
---|---|---|---|---|---|---
SL | CelebA | 213.9 | 535.6 | 229.2 | 217.0 | 215.4
CACD | 414.9 | 814.3 | 368.3 | 281.95 | 347.3
3D-Chair | 649.1 | 2705.9 | 324.0 | 291.46 | 513.8
Omniglot | 875.1 | 5958.9 | 225.6 | 195.7 | 343.2
ImageNet* | 758.4 | 683.1 | 689.6 | 652.8 | 769.1
Car | 745.1 | 583.7 | 588.8 | 565.9 | 709.8
Zappos | 451.1 | 431.2 | 263.4 | 275.8 | 280.7
| CUB | 492.0 | 330.2 | 461.3 | 569.6 | 638.6
| Average | 575.0 | 1505.4 | 393.8 | 381.3 | 477.2
Table 5: The results under CCCOSCZC lifelong learning.

We show the adjacency matrix of the nodes from the graph ${\bf V}$ of DEGM
after MSFIRC lifelong learning in Fig. 4a where ”C1” represents the first
node. We can observe that DEGM creates three basic nodes and three specific
nodes, respectively. Three specific nodes have edges from the first and second
basic nodes. We also show ${\bf V}$ of DEGM after CCCOSCZC lifelong learning
in Fig. 4b. DEGM creates basic nodes when learning the first, third, fourth
and fifth task. We also show the edge information between members of
$\mathcal{S}$ and members of $\mathcal{G}$ in Fig. 5 where red colour
represents the basic nodes and blue colour represents the specific nodes.

###  L.2 Results for generalization bounds

To estimate the discrepancy, we train an auxiliary model on the distribution
$\mathbb{P}^{i}\otimes{\tilde{\mathcal{P}}_{(i+1)}}$ for each $(i+1)$-th task
learning. Then we calculate the discrepancy by using Definition 3 of the paper
for each training epoch.

In the following, we train a single model with GR under MNIST, Fashion and
IFashion (MFI) lifelong learning. The images from all databases have pixel
values within the range $[0,255]$. We implement the decoder by a neural
network that outputs the mean vector of a Gaussian distribution with the
diagonal covariance matrix (diagonal element is $1.0$). We estimate the
reconstruction error term by using :

$\displaystyle{\log{p_{\theta}}\left({{\bf x}\,|\,{\bf
z}}\right)}=-\frac{1}{{2\sigma^{2}}}{\left\|{{\bf{x}}-{\mu_{\theta}}\left({\bf{z}}\right)}\right\|^{2}}-\frac{1}{2}\log
2\pi\sigma^{2}$ (131)

It can be noted that we also normalize this reconstruction error by dividing
$28\times 28$, as done in Chen et al. (2016b). We evaluate the average risk
and the discrepancy distance for each training epoch (See details in Lemma 1
from the paper). To calculate $|KL_{1}-KL_{2}|$ and discrepancy distance, we
randomly choose 10000 number of samples from ${\mathcal{P}}_{(1:t)}$ and
${\mathbb{P}}^{t-1}\otimes{\tilde{\mathcal{P}}}_{t}$ at the $t$-th task
learning, respectively. The results are presented in Fig. 6a where the source
risk is continuously decreased while the discrepancy is increased as learning
more tasks. We also evaluate the risk, $|KL_{1}-KL_{2}|$ and discrepance
distance on MNIST, shown in Fig. 6b. In order to select the generated images
that are belonging to MNIST, we train a task-specific classifier that predicts
the task label for giving samples.

(a) Evaluation on all tasks. (b) Evaluation on MNIST

Figure 6:  The risk, $|KL_{1}-KL_{2}|$ and discrepancy distance estimated by a
single VAE model with GR under MFI lifelong learning.

In the following, we investigate the results for Lemma 2 of the paper. We
consider a sequence of MNIST, Fashion, IFashion (MFI) learning tasks. We
consider a mixture model ${\bf M}=\\{{\mathcal{M}}_{1},{\mathcal{M}}_{2}\\}$
consisting of two components after LLL in which ${\mathcal{M}}_{1}$ is fixed
after the first task learning while ${\mathcal{M}}_{2}$ is used to learn
Fashion and IFashion, respectively. We also consider to learn a single model
${\mathcal{M}}$ with GR under MFI lifelong learning. We evaluate the average
target risk (NLL estimated by ELBO) for each training epoch and the results
are reported in Fig. (7) where ”single” and ”mixture” represent
${\mathcal{M}}$ and ${\bf M}$, respectively. As shown from the results, the
expansion mechanism can achieve a tight GB, as discussed in Lemma 2 of the
paper.

Figure 7:  Target risk on the single model and the mixture model.

In the following, we provide additional empirical results for the theoretical
analysis. It notes that in the following experiments, we resize all databases
as $32\times 32\times 3$ and the pixel values of all images are normalized as
$[0,1]$.

We evaluate the target risk (square loss) of a single model on four datasets
(MNIST, SVHN, Fashion, IFashion) under MSFI (See Theorem-2). We plot the
results in Fig. 8-a where ”All” represents the accumulated target risks for
all tasks (left hand side of Equation-7 in the paper). In the next, we train a
single model under MSFI and SFMI setting, respectively. We plot the target
risk on MNIST in Fig. 8-b where ”Early” and ”Recent” denote that the MNIST is
used as the first and third task, respectively. It observes that the model
tends to forget early tasks than recent tasks, as demonstrated in Equation-14
of the paper and discussed in Theorem 3. We also evaluate the accumulated
target risk estimated by DEGM under MSFI lifelong learning, which is shown in
Fig. 8-c. It observes that there have no accumulated errors for each task
during lifelong learning since the number of components match the number of
task, as discussed in Theorem 3.

Figure 8:  ”a” shows the target risk (LHS of Eq.(7) in Theorem-2 of the
paper)) across four tasks under MSFI lifelong learning. ”b” shows the target
risk (LHS of Eq.(7) in Theorem-2 of the paper) on MNIST under MSFI lifelong
learning. ”c” shows the target risk (LHS of Eq.(7) in Theorem-2 of the paper)
estimated by the proposed mixture model under (MSFI) lifelong learning. It
notes that all results are the accumulated target error and we do not
calculate the average result.

In the following, we investigate the performance of DEGM and a single model
when changing the order of tasks. We randomly generate three different orders
: Fashion, SVHN, MNIST, Cifar10, IMNIST, IFashion (FSMCII); Cifar10, IMNIST,
Fashion, MNIST, SVHN, IFashion (CIFMSI); IFashion, IMNIST, Cifar10, Fashion,
SVHN, MNIST (IICFSM). We evaluate the accumulated target risk for all tasks
and the results are presented in Fig. 9 where ”order1”, ”order2” and ”order3”
denote FSMCII, CIFMSI and IICFSM, respectively. It observes that a single
model is sensitive to the choice of orders of tasks since the final
accumulated target risk for all tasks are different when training the model
under different orders of tasks. Although CN-DPM can avoid the forgetting
during the training, the performance is still changed when changing the order
of tasks. This is mainly because newly created components only reuse the
transferable information from the first task and would lead to negative
transfer when learning an entire different task. However, the proposed DEGM is
robust to the change of orders of tasks.

Figure 9: The accumulated target risks of DEGM, a single model and CN-DPM*
with different orders of tasks. “a”, ”b” and ”c” represent the results
achieved by DEGM, a single model and CN-DPM*, respectively.

###  L.3 Evaluation by using other criterion

In addition to SL and NLL, we introduce to use the structural similarity index
measure (SSIM) Hore and Ziou (2010) and the Peak-Signal-to-Noise Ratio (PSNR)
Hore and Ziou (2010) as criteria. We report the results evaluated by the above
criterion in Table 6 and Table 7.

Criteria | SL | SSMI | PSNR
---|---|---|---
BE | LGM | DEGM | DEGM-2 | CN-DPM* | BE | LGM | DEGM | DEGM-2 | CN-DPM* | BE | LGM | DEGM | DEGM-2 | CN-DPM*
MNIST | 26.3 | 685.3 | 22.3 | 22.3 | 21.9 | 0.88 | 0.19 | 0.90 | 0.90 | 0.90 | 21.0 | 7.0 | 21.8 | 21.8 | 21.8
SVHN | 47.0 | 941.7 | 30.1 | 29.0 | 39.3 | 0.58 | 0.20 | 0.66 | 0.67 | 0.61 | 13.7 | 5.0 | 15.5 | 15.7 | 14.3
Fashion | 43.8 | 663.4 | 37.7 | 27.4 | 36.6 | 0.68 | 0.15 | 0.72 | 0.79 | 0.73 | 18.4 | 3.7 | 19.0 | 20.6 | 19.2
IFashion | 45.9 | 1148.4 | 35.6 | 27.4 | 38.4 | 0.72 | 0.11 | 0.76 | 0.81 | 0.76 | 18.2 | 5.0 | 19.4 | 20.6 | 19.1
RMNIST | 27.9 | 704.2 | 20.2 | 22.1 | 25.3 | 0.87 | 0.20 | 0.91 | 0.90 | 0.89 | 16.5 | 7.0 | 22.2 | 21.8 | 21.2
Cifar10 | 994.4 | 1241.1 | 615.3 | 608.1 | 892.1 | 0.29 | 0.23 | 0.49 | 0.50 | 0.34 | 16.5 | 15.4 | 18.9 | 18.9 | 17.0
Average | 197.5 | 897.4 | 126.9 | 122.7 | 175.6 | 0.67 | 0.18 | 0.74 | 0.76 | 0.70 | 18.1 | 7.2 | 19.5 | 19.9 | 18.8
Table 6: The performance of various models under the MSFIRC learning setting. Criteria | SL | SSMI | PSNR
---|---|---|---
BE | LGM | DEGM | DEGM-2 | CN-DPM* | BE | LGM | DEGM | DEGM-2 | CN-DPM* | BE | LGM | DEGM | DEGM-2 | CN-DPM*
CelebA | 213.9 | 535.6 | 229.2 | 217.0 | 215.4 | 0.69 | 0.48 | 0.66 | 0.69 | 0.69 | 23.5 | 19.3 | 23.2 | 23.4 | 23.5
CACD | 414.9 | 814.3 | 368.3 | 281.95 | 347.3 | 0.57 | 0.47 | 0.62 | 0.68 | 0.63 | 20.6 | 17.33 | 21.2 | 22.4 | 21.4
3D-Chair | 649.1 | 2705.9 | 324.0 | 291.46 | 513.8 | 0.73 | 0.42 | 0.84 | 0.86 | 0.79 | 19.0 | 13.54 | 22.4 | 23.1 | 20.5
Omniglot | 875.1 | 5958.9 | 225.6 | 195.7 | 343.2 | 0.73 | 0.22 | 0.92 | 0.93 | 0.89 | 17.9 | 9.2 | 24.0 | 24.6 | 22.1
Sub-ImageNet | 758.4 | 683.1 | 689.6 | 652.8 | 769.1 | 0.37 | 0.42 | 0.41 | 0.43 | 0.37 | 18.5 | 18.9 | 19.0 | 19.2 | 18.5
Car | 745.1 | 583.7 | 588.8 | 565.9 | 709.8 | 0.39 | 0.48 | 0.47 | 0.49 | 0.42 | 18.0 | 19.0 | 19.0 | 19.2 | 18.2
Zappos | 451.1 | 431.2 | 263.4 | 275.8 | 280.7 | 0.68 | 0.60 | 0.75 | 0.74 | 0.73 | 20.0 | 20.2 | 22.4 | 22.3 | 22.1
CUB | 492.0 | 330.2 | 461.3 | 569.6 | 638.6 | 0.35 | 0.48 | 0.45 | 0.43 | 0.35 | 19.0 | 20.9 | 19.3 | 18.6 | 18.0
Average | 575.0 | 1505.4 | 393.8 | 381.3 | 477.2 | 0.60 | 0.45 | 0.64 | 0.66 | 0.61 | 19.6 | 17.3 | 21.3 | 21.6 | 20.5
Table 7: The performance of various models under the CCCOSCZC learning
setting.

###  L.4 Ablation study

First, we evaluate the effectiveness of the proposed dynamic expansion
mechanism used in our model. We train all models under MSFIRC lifelong
learning and present the results in Fig 10a where we compare the average
score, the training times and the memory use. It observes that even if the
DEGM uses few training epochs but does not degenerate the performance. In
addition, DEGM outperforms CN-DPM* that only transfers features from a single
shared model. Furthermore, the proposed DEGM can achieve a similar performance
as the baseline that trains individual VAEs for each task.

Figure 10:  The results for various baselines under MSFIRC. In the left chart,
“A” represent the average square loss and PSNR for all tasks. “B” represent
the overall training times (seconds). “C” represents the number of parameters
($10^{8}$) for various baselines. In the right chart, “A” represent the
average square loss for all tasks. “B” represent number of parameters
($10^{8}$). “C” represents the number of basic components. X-axis represents
different threshold values $thr$

The effects of thresholds in ORVAE

In the following, we investigate the performance of DEGM with different
thresholds $\tau$ under MSFIRC lifelong learning and report results in Fig.
10b. As reduce $\tau$, DEGM tends to increase the number of basic components
and gradually improve performance. It observes that the choice of $\tau=400$
can trade-off between the performance and the model’s complexity since it does
not significantly improve the performance when decreasing $thr$ from 400. In
the next, in order to evaluate the effects of the proposed expansion mechanism
and the adaptive weight, we introduce several new baselines in the following.

DEGM-4:

This baseline generates information flows from all learned components to a new
component. For instance, if a new component receives the information flow from
members of $\mathcal{S}$, we will sum up the latent codes and intermediate
representations from the sub-inference and sub-decoder of these components.
DEGM-4 does not use the adaptive weight.

DEGM-5:

We implement this baseline by considering to create the edges without using
the adaptive weight. The training process for this baseline is described as
follows: Once the $t$-th task learning was finished, As similar done to DEGM,
we have a set of $K$ measures, denoted by $KS=\\{k{s_{1}},\dots,k{s_{K}}\\}$,
which can be used to build the edges from a new node to members of
$\mathcal{G}$. We set a threshold $\tau$ which is used to update $\bf{V}$ such
that if each $ks_{i}<\tau$, then ${\bf{V}}(t+1,{\mathcal{GI}(i)})=1$,
otherwise ${\bf{V}}(t+1,{\mathcal{GI}(i)})=0$. This means that if
$ks_{i}>\tau$, then the construction of a new component does not reuse the
information and parameters from the $i$-th component in DEGM.

DEGM-6:

For this baseline, we consider the adaptive weight for each edge is equal.
This means that the importance of each basic component is treated as the same
for a new task.

DEGM-7:

We implement this baseline by considering to create only a single edge for a
new component to a certain basic component that has the maximum sample log-
likelihood for data of the new task.

CN-DPM*-1;

This baseline builds a new components and creates connections with previously
learned components, as similar to DEGM-4.

CN-DPM*-2:

We implement this baseline by using the large model which contains $1.3\times
10^{9}$ number of parameters.

We report the results in Table 8. It observes that the adaptive weights in the
expansion mechanism can further improve the performance. We also find that
although CN-DPM*-2 uses the more parameters, our DEGM still outperforms CN-
DPM* by a large margin.

Criteria | Dataset | DEGM | DEGM-4 | CN-DPM*-1 | CN-DPM*-2 | DEGM-5 | DEGM-6 | DEGM-7
---|---|---|---|---|---|---|---|---
SL | MNIST | 22.30 | 22.18 | 22.12 | 22.67 | 22.88 | 21.48 | 22.35
SVHN | 30.18 | 30.73 | 40.53 | 38.74 | 30.56 | 29.44 | 29.20
Fashion | 37.73 | 41.22 | 45.03 | 38.51 | 37.65 | 37.26 | 41.29
IFashion | 35.62 | 49.26 | 36.19 | 36.90 | 41.27 | 37.15 | 43.95
RMNIST | 20.23 | 60.72 | 24.79 | 24.39 | 27.86 | 25.73 | 25.78
Cifar10 | 615.34 | 610.38 | 929.55 | 877.35 | 617.34 | 617.36 | 614.48
Average |  126.90 | 135.58 | 183.03 | 173.09 | 129.59 | 128.07 | 129.51
Table 8: The results of various models under MSFIRC lifelong learning.

###  L.5 The number of parameters used in various methods

We list the number of parameters used in various methods in Table 9 and Table
10, respectively. We can observe that the proposed DEGM architecture requires
less parameters than other parameters.

Model | LGM | BE | DEGM | DEGM-2 | CN-DPM*
---|---|---|---|---|---
N | ${1.5\times 10^{8}}$ | ${9.4\times 10^{8}}$ | ${1.6\times 10^{8}}$ | ${8.7\times 10^{8}}$ | ${4.2\times 10^{8}}$
Table 9: The number of parameters of various models under MSFIR learning setting. Model | LGM | BE | DEGM | DEGM-2 | CN-DPM* | LIMix
---|---|---|---|---|---|---
N | ${1.9\times 10^{9}}$ | ${3.9\times 10^{9}}$ | ${3.2\times 10^{8}}$ | ${1.3\times 10^{9}}$ | ${9.4\times 10^{9}}$ | ${9.4\times 10^{9}}$
Table 10: The number of parameters of various models under CCCOSCZC learning
setting.

###  L.6 Visual results

We show the reconstructions from DEGM under MSFIRC and CCCOSCZC lifelong
learning in Fig. 11 and Fig. 12, respectively.

(a) MNIST.

(b) SVHN.

(c) Fashion.

(d) IFasion.

(e) RMNIST.

(f) Cifar10.

(g) Task 1.

(h) Task 2.

(i) Task 3.

(j) Task 4.

(k) Task 5.

(l) Task 6.

Figure 11:  Image reconstructions when using DEGM under the MSFIRC lifelong
learning. The first row represents testing data samples and the second row are
their reconstructions using DEGM.

(a) Task1.

(b) Task2.

(c) Task3.

(d) Task4.

(e) Task5.

(f) Task6.

(g) Task7.

(h) Reconstruction of Task1.

(i) Reconstruction of Task2.

(j) Reconstruction of Task3.

(k) Reconstruction of Task4.

(l) Reconstruction of Task5.

(m) Reconstruction of Task6.

(n) Reconstruction of Task7.

(o) Task8.

(p) Reconstruction of Task8.

Figure 12:  Image reconstructions when using DEGM under the CCCOSCZC lifelong
learning. The first row represents testing data samples and the second row are
their reconstructions using DEGM.

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# Spatial and temporal characteristics of spontaneous parametric down-
conversion with varying focal planes of interacting beams

Richard Bernecker<EMAIL_ADDRESS>Theoretisch-Physikalisches
Institut, Friedrich-Schiller-Universität Jena, D-07743 Jena, Germany.
Baghdasar Baghdasaryan  Theoretisch-Physikalisches Institut, Friedrich-
Schiller-Universität Jena, D-07743 Jena, Germany.  Helmholtz-Institut Jena,
D-07743 Jena, Germany.  Fraunhofer Institute for Applied Optics and Precision
Engineering IOF, Albert-Einstein-Strasse 7, 07745 Jena, Germany  Stephan
Fritzsche  Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität
Jena, D-07743 Jena, Germany.  Helmholtz-Institut Jena, D-07743 Jena, Germany.
Abbe Center of Photonics, Friedrich-Schiller-University Jena, Albert-Einstein-
Str. 6, 07745 Jena, Germany

###### Abstract

Spontaneous parametric down-conversion (SPDC) is a widely used process to
prepare entangled photon pairs. In SPDC, a second-order nonlinear crystal is
pumped by a coherent laser beam to generate photon pairs. The photon pairs are
usually detected by single-mode fibers (SMF), where only photons in a Gaussian
mode can be collected. The collection modes possess typical Gaussian
parameters, namely a beam waist and a focal plane position. The collection
efficiency of photons highly depends on the choice of both parameters. The
exact focal plane position of the pump beam relative to those of the detection
modes is difficult to determine in a real experiment. Usually, theoretical and
experimental studies assume that the focal plane positions of the pump and the
generated beams are positioned in the center of the crystal. The displacement
of beam focal planes can lead to deviations from expected results and the
coupling efficiency into SMF can decrease. In this work, we theoretically
consider variable positions of focal planes and investigate how they influence
the spatial and temporal properties or the purity of photon pairs. We present
SPDC arrangements, in which the knowledge of the exact position of the focal
planes is essential, as well as scenarios, where focal plane displacements do
not contribute significantly to experimental outcomes. These findings are of
particular interest for achieving higher efficiency in SPDC experiments.

###### pacs:

Valid PACS appear here

Quantum-based technologies are increasingly explored and integrated into
today’s applications. In this context, quantum entanglement is an inseparable
part of quantum communication protocols [1, 2]. The process of spontaneous
parametric down-conversion (SPDC) is the most reputable source of photonic
entanglement [3, 4] and provides an experimental platform for fundamental
quantum science [5].

In SPDC, a nonlinear crystal is illuminated with a strong, high-energetic
laser field called pump beam. Photon pairs with lower energies, also known as
signal and idler photons, are subsequently down-converted. The generated
signal and idler photons fulfill the energy $\omega_{p}=\omega_{s}+\omega_{i}$
and the momentum $\bm{k}_{p}=\bm{k}_{s}+\bm{k}_{i}$ conservations. The
momentum conservation, also known as the phase-matching (PM) condition,
ensures constructive interference between the three interacting beams and
inherently determines the spectral and spatial properties of signal and idler
photons. Spectrally and spatially engineered photons are important ingredients
in current research on quantum information [6, 7, 8], quantum computing [9],
and quantum communication [10, 11]. Additionally to the PM, the pump beam
properties also have a large impact on the spectral and spatial properties of
signal and idler photons [12, 13, 14]. The pump characteristics include the
beam width, the focal plane position relative to the crystal, and its spatial
and temporal degrees of freedom (DOFs) [15, 16].

Figure 1: Schematic paths of pump and signal beam in a non-linear crystal. For
the sake of simplicity, only the signal beam is shown, the idler beam can be
similarly imagined. The beams are described as Gaussian beams with beam widths
$w_{p}$ and $w_{s}$. Most calculations assume that the focal planes of the
pump, signal, and idler lie in the center of the crystal $z=0$ as shown in the
left picture. On the contrary, we allow in the right picture that the pump,
signal, and idler focal planes are not fixed. The parameters for focal plane
shifts along the propagation axis are $z_{p}$ for the pump beam and
$z_{s},z_{i}$ for the generated beams.

Besides the generation of photon pairs, optimal experimental verification is
an essential part of quantum fundamental research too. Usually, the spatial
shape of signal and idler photons are detected by multi-plane light conversion
(MPLC) [17, 18], where an arbitrary spatial mode is projected to a Gaussian
mode [19], in order to couple it into a single-mode fiber (SMF) [20, 21]. The
coupling efficiency into SMF depends on the beforehand chosen focal planes and
beams widths of the pump, signal, and idler modes. Optimizing the coupling
efficiency of collecting a photon pair in fundamental Gaussian modes (FGMs) is
a particularly interesting aspect from an experimental perspective [22, 23,
24, 25]. We distinguish between the single-mode coupling efficiency for a
certain frequency (within a narrowband filter bandwidth) [26], and the
spectral brightness, which pertains to the maximal collection probability on a
broadband of frequencies [27, 28].

In the past, there were several theoretical and experimental approaches to
modify the pump or the detection scheme of the generated modes with linear
optical systems [29, 30, 31, 32, 33], in order to improve the pair collection
efficiency. The variation of the pump intensity or the pump beam width has
also been explored to optimize the photon pair collection efficiency or
transverse spatial correlations [34, 35, 36]. In this regard, the manipulation
of the purity between the signal and idler photons via pump focusing was shown
[37, 38]. Moreover, it has been proposed in Refs. [39, 40] how to consider the
change of the angular spectrum of the pump at different positions beyond the
crystal center. In analogy to the change of the pump focal plane position, the
position of the nonlinear crystal has been varied to control the time delay
between signal and idler photons and the coincidence rate [41].

All these considerations were primarily concerned with optimal pump focusing
or the right choice of optical elements in beam paths, in order to achieve
enhanced photon collection efficiency in SPDC. In this work, we theoretically
consider variable focal planes for pump, signal, and idler explicitly and
describe their impact on experimentally measurable quantities such as the
coupling efficiency into SMF, spatial and temporal correlations of generated
photons and the spectral purity between down-converted photons. In this
regard, we will compare scenarios of focal planes fixed at different positions
and investigate if the measurement probability of signal and idler photons
remain unaffected. We discuss setup conditions with noteworthy influence on
the spectral brightness and the coupling efficiency. The alignment of focal
plane positions in these scenarios will require more careful effort in order
to increase the coupling efficiency. We also contemplate scenarios where the
precise position of focal planes becomes insignificant.

##  I Theory

We start our investigation with the theoretical description of the SPDC
process. Our group published a paper on the characterization of spatio-
temporal DOFs in SPDC, where a general expression for the SPDC-state, also
known as biphoton state, has been derived [42] and later verified
experimentally [19]. First, we briefly recap the derivation of the expression
from [42], where the focal planes of the pump, signal, and idler beams are
assumed to be at $z=0$, i.e in the crystal center. This assumption, as shown
in Fig. 1 on the left, is common in theoretical as well as experimental
studies. Next, we extend the expression to consider variable focal planes for
the pump, signal, and idler beams, which is illustrated on the right side of
Fig. 1.

###  I.1 Biphoton state of SPDC decomposed in Laguerre-Gaussian Basis

The common expression of the general biphoton state in the wave vector
representation of the interacting beams is [40]

$\displaystyle\ket{\Psi}_{\text{SPDC}}=\iint$ $\displaystyle
d\bm{q}_{s}\>d\bm{q}_{i}\>d\omega_{s}\>d\omega_{i}\>\Phi(\bm{q}_{s},\bm{q}_{i},\omega_{s},\omega_{i})$
$\displaystyle\times\hat{a}_{s}^{\dagger}(\bm{q}_{s},\omega_{s})\>\hat{a}_{i}^{\dagger}(\bm{q}_{i},\omega_{i})\ket{\text{vac}},$
(1)

where we consider the paraxial approximation by the separation into
longitudinal and transversal components of the wave vector
$\bm{k}=\bm{q}+k_{z}(\omega)\>\bm{z}$, where $z$ is the propagation direction
of the pump beam. The paraxial approximation is valid in most experimental
SPDC setups since typical optical apparatuses support only paraxial rays about
the central axis. The biphoton state (1) is an integral over all possible
transverse wave vectors $\bm{q}_{s,i}$ and frequencies $\omega_{s,i}$ of a
signal and idler pair that is created from the vacuum state $\ket{\text{vac}}$
by creation operators $\hat{a}_{s,i}^{\dagger}(\bm{q}_{s,i},\omega_{s,i})$ of
signal and idler photons, respectively.

The so-called biphoton mode function
$\Phi(\bm{q}_{s},\bm{q}_{i},\omega_{s},\omega_{i})$ encodes the coupling
between the wave vectors of the pump, signal, and idler beams [39]:

$\displaystyle\Phi(\bm{q}_{s},\bm{q}_{i},\omega_{s},\omega_{i})$
$\displaystyle=N\>V_{p}(\bm{q}_{s}+\bm{q}_{i})\>S_{p}(\omega_{s}+\omega_{i})$
$\displaystyle\times\int_{-L/2}^{L/2}dz\>\operatorname{\mathrm{e}}^{i(k_{p,z}-k_{s,z}-k_{i,z})z},$
(2)

where $N$ is the normalization constant, $V_{p}$ is the spatial distribution
of the pump beam, whereas $S_{p}$ characterizes the spectral DOF of the pump.
The integral over the propagation direction $z$ describes the phase mismatch
${\Delta k_{z}=k_{p,z}-k_{s,z}-k_{i,z}}$ in the $z$ direction. The exact
expression of $\Delta k_{z}$ depends on the features of the crystal and the
geometry between the interacting beams and the crystal.

Since discrete modes are easier to detect and manipulate in experiments [43,
44] than continuous modes, the continuous transverse spatial variables in (1)
are often discretized by a set of optical modes. Furthermore, an appropriate
choice of the set can reduce the dimensionality of a state. In Ref. [42],
Laguerre-Gaussian (LG) modes have been used as a basis for the description of
the spatial distribution of the down-converted photons. This choice is
reasonable since LG modes carry well-defined projection of orbital angular
momentum (OAM) [45], which is conserved in collinear SPDC [46, 47, 48]. The
biphoton state decomposed in the LG basis $\ket{p,\ell,\omega}=\int
d\bm{q}\,\mathrm{LG}_{p}^{\ell}(\bm{q})\,\hat{a}^{\dagger}(\bm{q},\omega)\ket{\text{vac}}$
reads then

$\displaystyle\ket{\Psi}_{\text{SPDC}}=\iint d\omega_{s}\>d\omega_{i}$
$\displaystyle\sum_{p_{s},\ell_{s}}\sum_{p_{i},\ell_{i}}C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})$
$\displaystyle\times\ket{p_{s},\ell_{s},\omega_{s}}\ket{p_{i},\ell_{i},\omega_{i}},$
(3)

where the overlap amplitudes $C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}$ of the LG
decomposition are frequency-dependent. The probability to find signal and
idler photons in spatial modes $(p_{s}|\ell_{s})$ and $(p_{i}|\ell_{i})$ at
frequencies $\omega_{s}$ and $\omega_{i}$ is given by
$P_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})=|C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})|^{2}$.
We can also call this quantity the single-mode coupling efficiency. On the
other hand, the maximal value of
$P_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})$ over all possible
energies $\omega_{s}$ and $\omega_{i}$ is called the spectral brightness.

The following assumptions and approximations have been applied in Ref. [42],
in order to derive a comprehensive expression for
$C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})$:

  * •

Pump, signal, and idler beams are focused in the middle of the crystal.

  * •

A small deviation $\Omega$ of generated frequencies from the central frequency
$\omega_{0}$ has been assumed, i.e. $\Omega\ll\omega_{0}$, so that we can
write $\omega=\omega_{0}+\Omega$. The central frequencies fulfill the energy
conservation $\omega_{p,0}=\omega_{s,0}+\omega_{i,0}$.

  * •

In the paraxial regime, where $|\bm{q}|\ll k$, the so-called Fresnel
approximation can be applied on
$k_{z}=\sqrt{k^{2}-\mathinner{\\!\left\lvert\bm{q}\right\rvert}^{2}}$:

$k_{z}=k(\Omega)\sqrt{1-\frac{|\bm{q}|^{2}}{k(\Omega)^{2}}}\approx
k+\frac{\Omega}{u}+\frac{G\Omega^{2}}{2}-\frac{|\bm{q}|^{2}}{2k},$ (4)

with the group velocity $u=1/(\partial k/\partial\Omega)$ and the group
velocity dispersion $G=\partial/\partial\Omega\,(1/u)$ at the corresponding
central frequency.

  * •

Momentum conservation for the central frequencies $\Delta
k=k_{p}-k_{s}-k_{i}=0$ is assumed. When a periodically poled crystal with
poling period $\Lambda$ along the crystal axis is used, this is achieved by
$\Delta k=k_{p}-k_{s}-k_{i}-\frac{2\pi}{\Lambda}=0$.

We shall expand now the formalism from Ref. [42], to support variable
positions of the focal planes for the pump, signal, and idler beams.

###  I.2 Shift of focal plane positions

We briefly recap the angular spectrum propagation of beams. Mathematically,
electromagnetic field distributions can be described by a propagator factor
obtained via the angular spectrum representation in momentum space. This is a
well-investigated formalism, with the following main key ideas. We can choose
the $z$-direction as the propagation axis and write the Fourier transformation
of an arbitrary field $\bm{E}$ in the transverse $x$-$y$-plane of a fixed
point $z=\text{const.}$ as

$\bm{\tilde{E}}(\bm{q};z)\propto\iint_{-\infty}^{\infty}dx\>dy\>\bm{E}(x,y,z)\operatorname{\mathrm{e}}^{-ik_{x}x}\>\operatorname{\mathrm{e}}^{-ik_{y}y}.$
(5)

Here are $\bm{q}=(k_{x},k_{y})$ the transverse wave vector components. The
amplitude in momentum-space $\bm{\tilde{E}}(\bm{q};z)$ can also be used for
the inverse Fourier transform for the field in real space

$\bm{E}(x,y,z)\propto\iint_{-\infty}^{\infty}dk_{x}\>dk_{y}\>\bm{\tilde{E}}(\bm{q};z)\operatorname{\mathrm{e}}^{ik_{x}x}\>\operatorname{\mathrm{e}}^{ik_{y}y}.$
(6)

When we split the field in a spatial and time-dependent part
$\bm{E}=\bm{E}(x,y,z)\operatorname{\mathrm{e}}^{-i\omega t}+\text{c.c}$ and
also write $|\bm{k}|=k=\frac{\omega^{2}n^{2}}{c^{2}}$, the Helmholtz equation
reads as

$\left(\bm{\nabla}^{2}+k^{2}\right)\bm{E}(x,y,z)=0.$ (7)

We insert expression (6) into the Helmholtz equation and arrive at a
differential equation for the spatial evolution:

$(\partial_{z}^{2}+k^{2}-k_{x}^{2}-k_{y}^{2})\>\bm{\tilde{E}}(\bm{q};z)=0.$
(8)

When setting $k_{z}=\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}$, the solution of the
angular spectrum of an electric field evolving along the z-axis can be written
as

$\bm{\tilde{E}}(\bm{q};z)=\bm{\tilde{E}}(\bm{q};0)\>e^{\pm ik_{z}z}$ (9)

(see also Refs. [40, 39]). The positive signs in the exponential indicate a
wave propagation into $z>0$, while the negative sign describes a wave
propagating into the region $z<0$.

We apply Eq. (9) to the pump, signal, and idler beams. The PM function renewed
with the pump, signal, and idler focused at positions $z_{p}$, $z_{s}$, and
$z_{i}$ respectively (see Fig. 1), is now

$\displaystyle\Phi($
$\displaystyle\bm{q}_{s},\bm{q}_{i},\omega_{s},\omega_{i})=V_{p}(\bm{q}_{s}+\bm{q}_{i})\>S_{p}(\omega_{s}+\omega_{i})$
$\displaystyle\times\int_{-L/2}^{L/2}dz\>\operatorname{\mathrm{e}}^{i\left[k_{p,z}(z+z_{p})-k_{s,z}(z+z_{s})-k_{i,z}(z+z_{i})\right]}.$
(10)

Note that we continue to consider paraxial beams for pump, signal, and idler
and use the Fresnel approximation from Eq. (4) for $k_{z}$.

To specify our setup, we assume a Gaussian envelope of pulse duration $T_{0}$
for the spectral part of the pump. Due to
$\omega_{p}-\omega_{p,0}=\Omega_{p}=\Omega_{s}+\Omega_{i}$ we can write

$S_{p}(\Omega_{p})=\frac{T_{0}}{\sqrt{\pi}}\exp{\biggl{(}-\frac{T_{0}^{2}}{4}\>(\Omega_{s}+\Omega_{i})^{2}\biggl{)}}.$
(11)

The spatial distribution of the pump beam is also described as a Gaussian with
beam width $w_{p}$,

$\mathrm{V}(\bm{q}_{\mathrm{s}}+\bm{q}_{\mathrm{i}})\>=\>\frac{w_{p}}{\sqrt{2\pi}}\>\exp{\biggl{(}-\frac{w_{p}^{2}}{4}|\bm{q}_{\mathrm{s}}+\bm{q}_{\mathrm{i}}|^{2}\biggr{)}.}$
(12)

This reduces and simplifies the expression from Ref.[42] enormously. The final
formula reads then

$\displaystyle
C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}\propto\delta_{\ell_{s},-\ell_{i}}\>\exp{\biggl{(}-\frac{T_{0}^{2}}{4}\>(\Omega_{s}+\Omega_{i})^{2}\biggl{)}}\sum_{s=0}^{p_{s}}\sum_{i=0}^{p_{i}}\>(T_{s}^{p_{s},\ell_{s}})^{*}\>(T_{i}^{p_{i},\ell_{i}})^{*}\Gamma[h]\>\Gamma[b]$
$\displaystyle\times\int_{-L/2}^{L/2}dz\>\exp{\biggl{[}iz\biggl{(}\frac{\Omega_{s}+\Omega_{i}}{u_{p}}-\frac{\Omega_{s}}{u_{s}}-\frac{\Omega_{i}}{u_{i}}+\frac{G_{p}\>(\Omega_{s}+\Omega_{i})^{2}}{2}-\frac{G_{s}\>\Omega_{s}^{2}}{2}-\frac{G_{i}\>\Omega_{i}^{2}}{2}\biggl{)}\biggl{]}}\>\frac{D^{\ell_{i}}}{H^{h}\>B^{b}}\>{{}_{2}}{\tilde{F}}_{1}\biggl{[}h,b,1+\ell_{i},\frac{D^{2}}{H\,B}\biggl{]}$
(13)

with the abbreviations

$\displaystyle h$ $\displaystyle=$ $\displaystyle
1+s+\frac{1}{2}\>(\ell_{i}+\mathinner{\\!\left\lvert\ell_{s}\right\rvert}),$
$\displaystyle b$ $\displaystyle=$ $\displaystyle
1+i+\frac{1}{2}\>(\ell_{i}+\mathinner{\\!\left\lvert\ell_{i}\right\rvert}),$
$\displaystyle D$ $\displaystyle=$
$\displaystyle-\frac{w_{p}^{2}}{4}-\frac{i}{2k_{p}}(z+z_{p}),$ $\displaystyle
H$ $\displaystyle=$
$\displaystyle\frac{w_{p}^{2}}{4}+\frac{w_{s}^{2}}{4}-\frac{i}{2}\Bigl{[}\frac{(z+z_{p})}{k_{p}}-\frac{(z+z_{s})}{k_{s}}\Bigl{]},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{w_{p}^{2}}{4}+\frac{w_{i}^{2}}{4}-\frac{i}{2}\Bigl{[}\frac{(z+z_{p})}{k_{p}}-\frac{(z+z_{i})}{k_{i}}\Bigl{]},$
$\displaystyle T^{p,\ell}_{k}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{p+k}(i)^{\ell}}{(p-k)!\>(|\ell|+k)!k!}\>\sqrt{\frac{p!\,(p+|\ell|)!}{\pi}}\,\biggr{(}\frac{w}{\sqrt{2}}\biggl{)}^{2k+|\ell|+1},$

and the regularized hypergeometric function ${{}_{2}}{\tilde{F}}_{1}$ [49].
The collecting widths of the generated signal and idler modes are denoted by
$w_{s}$ and $w_{i}$. The expression (13) gives full insight into the spatial
distribution of the biphoton state decomposed in LG modes and also into the
spatio-temporal coupling in SPDC [50, 51]. Note that the overlap amplitudes
$C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}$ from Eq. (13) depend only on $|\ell|$,
where $\ell=\ell_{s}=-\ell_{i}$ (see Ref. [19]), but we keep the notation of
Eq. (13) for a proper illustration of our results.

##  II Results and discussion

###  II.1 Justifying the choice of the same focal plane positions for signal
and idler, $z_{s}=z_{i}$

In this section, we study the impact of $z_{p}$, $z_{s}$, and $z_{i}$ on the
probability to detect the signal and idler photons in FGMs, or in other words,
the efficiency of direct coupling of generated photons into SMF. In the
following sections, unless otherwise stated, we assume spectrally a
continuous-wave pump such that
$S_{p}(\omega_{s}+\omega_{i})=\delta(\Omega_{s}+\Omega_{i})$ which leads to
the same amount of deviation from the center frequency for signal and idler
photons, $\Omega_{s}=-\Omega_{i}$.

Figure 2: Normalized amplitude of the single-mode coupling for
$\lambda_{s}=$810\text{\,}\mathrm{nm}$$ depending on the pump focal position
$z_{p}$ for a crystal with $L=$30\text{\,}\mathrm{mm}$$ centered at
$z=$0\text{\,}\mathrm{mm}$$. The focal plane positions of signal and idler are
set at $z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$. Two scenarios are examined: (a)
$\gamma=\frac{$10\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}{$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$ and (b)
$\gamma=\frac{$40\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}{$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$. The full width at half
maximum is independent of the crystal length but determined by the beam width
ratio $\gamma$. As $\gamma$ increases, the influence of the pump focal plane
shifts on the normalized amplitude decrease.

In general, we identify that the coupling efficiency into SMF decreases, when
the pump focal plane is displaced from the crystal center (see Fig. 2). The
amplitude becomes more robust to the change of $z_{p}$ if the beam width ratio
$\gamma=\frac{w_{p}}{w_{s}}$ increases ($w_{s}=w_{i}$ is assumed).

Figure 3: The single-mode coupling efficiency for FGMs in dependence of signal
and idler focal plane shits $z_{s},z_{i}$ for different pump focal plane
shifts $z_{p}$. The setup parameters are
$L=$1\text{\,}\mathrm{mm}$,w_{p}=w_{s}=$10\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,w_{i}=$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$\lambda_{s}=$810\text{\,}\mathrm{nm}$$. The black star refers to the pair
($z_{s}|z_{i}$) that maximizes the coupling efficiency. (a) The pump has its
focal plane in the center of the crystal, so optimal coupling would imply that
the focal planes of the signal and idler move in opposite directions. (b) and
(c) If $z_{p}\neq$0\text{\,}\mathrm{mm}$$, the optimum amplitudes are reached
for signal and idler focal planes shifted from the center. These findings are
in line with the advanced-wave picture.

In order to increase the amplitude for a given pump focal plane shift $z_{p}$,
we should adjust the signal and idler focal plane positions. The optimal
choice of a pair $z_{s}$ and $z_{i}$ strongly depends on the beam width of
collection modes, $w_{s}$ and $w_{i}$. We distinguish two scenarios of equal
$w_{s}=w_{i}$ or unequal beam widths.

Fig. 3 represents the coupling efficiency for signal photons filtered at
$\lambda_{s}=$810\text{\,}\mathrm{nm}$$ for unequal beam widths $w_{s}\neq
w_{i}$ as a function of $z_{s}$ and $z_{i}$. The calculations have been
carried out for three different pump focal plane positions $z_{p}$. The star
corresponds to the ($z_{s}|z_{i}$) combination that optimizes the single-mode
coupling efficiency. If $z_{p}$ and for instance, $z_{s}$ are fixed in the
experimental setup, the efficiency strongly depends on the focal plane
position of the corresponding partner $z_{i}$. An edge case is visualized in
Fig. 3 (a), where the pump is fixed in the crystal center. The maximum is
attained when the signal focus plane and the idler focus plane displacements
are equal in magnitude but point in opposite directions, i.e. $z_{i}=-z_{s}$.
The initial dependence of the amplitude on $z_{s}$ and $z_{i}$ from (a) is
more distorted and moves away from the plot center the larger $z_{p}$ is.
Moreover, the maximum amplitudes lay not on the diagonal $z_{s}=z_{i}$ and
move further away for increasing $z_{p}$.

However, in experiments $w_{s}=w_{i}$ is more common and we observe more
symmetric dependencies of the single mode coupling efficiency on $z_{s}$ and
$z_{i}$. Fig. 4 shows the same as Fig. 3 but for a constant focal plane
position $z_{p}=$10\text{\,}\mathrm{mm}$$ and the same beam width for signal
and idler $w_{s}=w_{i}$. We distinguish between two different crystal lengths
$L$ and beam width ratios $\gamma$. Since we choose
$z_{p}\neq$0\text{\,}\mathrm{mm}$$, the area of high efficiency lies not
around $z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$. This area resembles a circle
laying on the diagonal $z_{s}=z_{i}$. The higher the length or the beam width
ratio, the bigger the red circle which means a larger range of $z_{s},z_{i}$
where the amplitude is constant. This enables fixing the focal planes of
signal and idler at the same location.

Figure 4: Like in Fig. 3 The single-mode coupling efficiency for FGMs is shown
in dependence of signal and idler focal plane shits for a fixed pump focal
plane position $z_{p}=$0\text{\,}\mathrm{mm}$$. We considered
$w_{p}=$20\text{\,}\mathrm{\SIUnitSymbolMicro
m}$/$40\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,w_{s}=w_{i}=$20\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,\lambda_{s}=$810\text{\,}\mathrm{nm}$$. The black star refers to the pair
($z_{s}|z_{i}$) that maximizes the coupling efficiency. The high amplitude
areas widen and the exact focal plane position of signal and idler is less
relevant for thicker crystals or higher beam width ratios. Figure 5: Mode
distribution of the biphoton state in LG basis filtered at
$\lambda_{s}=$810\text{\,}\mathrm{mm}$$ for four different arrangements of
focal plane shifts $z_{p}$ and $z_{s}=z_{i}$: (i) pump, signal, and idler
focal planes are assumed to be at $z=$0\text{\,}\mathrm{mm}$$, (ii) only the
pump beam is shifted (by $5\text{\,}\mathrm{mm}$), but signal and idler remain
in the center of the crystal, (iii) pump, signal, and idler are shifted by the
same amount of $5\text{\,}\mathrm{mm}$, (iv) value $z_{s}=z_{i}$ that
maximizes the coincidence amplitude for the given pump shift $z_{p}$. The
upper show a crystal with length $L=$10\text{\,}\mathrm{mm}$$, the lower row
represents $L=$25\text{\,}\mathrm{mm}$$. The mode numbers $(p_{s}|\ell_{s})$
for signal and $(p_{i}|\ell_{i})$ for idler run over $p=0,1,2$ and
$\ell=-2,\dots,2$. The thick bars mark p=0 and the two following thin bars
p=1,2. Each row is normalized by its maximum (corresponding to crystal
length). If the pump is not focused in the center, the signal and idler beams
should be shifted as well, to maximize the coupling efficiency.

Our results show that for a fixed frequency when considering the single-mode
coupling efficiency, $z_{p}=$0\text{\,}\mathrm{mm}$$ not always implies
$z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ for the highest amplitude. This is shown
in more detail in chapter II.2.2. If the total frequency spectrum on contrary
is considered, $z_{p}=$0\text{\,}\mathrm{mm}$$ always implies
$z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ for maximal ”spectral brightness”. The
focal planes of all beams should lie in the center of the crystal.

Our findings that focal plane shifts of the pump, signal, or idler beam from
the crystal center affect the efficiency and have to compensate each other is
consistent with the advanced-wave picture (AWP) [29, 40, 52, 53, 54], which
provides a classical analog to understand biphoton coincidence experiments.

To summarize, the choice of pump, signal, and idler focal plane position can
greatly influence the coupling efficiency. When choosing equal collecting
widths for signal and idler $w_{s}=w_{i}$, it is sufficient to assume signal
and idler focal plane positions at the same spot $z_{s}=z_{i}$ for maximum
efficiency.

###  II.2 Spatial and temporal characteristics for $z_{s}=z_{i}$

Figure 6: Same as in Fig. 5, but for a crystal with length
$L=$1\text{\,}\mathrm{mm}$$ and different beam width ratios,
$\gamma=\frac{w_{p}}{w_{s,i}}=1$ (upper row) and $\gamma=\sqrt{2}$ (lower
row). When all three beam focal planes are positioned in the middle of the
crystal, the coincidence amplitude for $(p_{s}|\ell_{s})=(0|0)$ and
$(p_{i}|\ell_{i})=(0|0)$ is the highest. The amplitude decreases if the focal
plane of the pump beam is shifted. For this particular pump shift exists a
certain shift for signal and idler $z_{s}=z_{i}\neq$0\text{\,}\mathrm{mm}$$
(d), that maximizes the amplitude and improves the efficiency. The focal plane
positions of signal and idler in c) are distant from the maximizing focal
plane shift. The probability of measuring photon pairs in these modes is very
low.

The subsequent sections address how the pump, signal, and idler focal plane
positions influence the spatio-temporal biphoton state. As we discussed in the
previous section, we can set $z_{s}=z_{i}$ for equal signal and idler beam
widths. This assumption provides for our results 99.99 % agreement compared to
the actual maximizing focal plane shits $z_{s}$ and $z_{i}$. We distinguish
four different combinations of $z_{p}$ relative to $z_{s}=z_{i}$:

  1. (i)

The focal plane of the pump, signal, and idler are laying all in the middle of
the crystal, i.e. $z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$.

  2. (ii)

The focal plane of the pump is shifted by a certain amount $z_{p}$
(experimentally perhaps unintentionally and therefore not noticed). However,
signal and idler beams are still positioned at the crystal center,
$z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$.

  3. (iii)

The pump, signal, and idler focal planes are shifted by the same amount as the
pump in (ii) so that they are focused on the same spot, i.e.
$z_{p}=z_{s}=z_{i}\neq$0\text{\,}\mathrm{mm}$$.

  4. (iv)

The pump beam is positioned on the same spot as in (ii) and (iii), but the
focal plane positions of signal and idler are chosen in such a configuration,
that maximizes the amplitude for the Fundamental Gaussian Mode
$|C_{0,0}^{0,0}|^{2}$ for the given $z_{p}$.

In the following, we will probe a ppKTP crystal pumped with a coherent laser
operating at $\lambda_{p}=$405\text{\,}\mathrm{nm}$$. The procedure (i)-(iv)
will be accomplished for different crystal lengths $L$ and beam width ratios
$\gamma$.

####  II.2.1 Influence of $z_{p},z_{s}$, and $z_{i}$ on spatial biphotons
state

Figure 7: Influence of focal plane shifts on the wavelength spectrum for
signal photons. We show crystal configurations with
$L=$1\text{\,}\mathrm{mm}$,$20\text{\,}\mathrm{mm}$$ and
$\gamma=\frac{2}{3},1$. The different focal plane positions of pump, signal,
and idler are illustrated in each plot with different colors. Again, a pump
shift of $z_{p}=$5\text{\,}\mathrm{mm}$$ was chosen. The red curves represent
$z_{s}=z_{i}=z_{s}^{\text{max}}$. The highest spectral brightness is achieved
when the focal planes of pump, signal, and idler are laying in the center of
the crystal. Note that for thinner crystals the spectrum is much wider. It is
also easy to notice that for longer crystals, focal plane shifts have a more
vivid influence on the brightness. The larger the focal plane shift, the
further the maximum from the initial position is shifted.

Fistly we analyze if the spatial DOF of generated photons is affected by the
change of the focal plane positions. We apply the narrowband regime and assume
signal photons filtered at $\lambda_{s}=$810\text{\,}\mathrm{nm}$$. The
normalized coupling efficiency of finding a pair of photons in the Laguerre-
Gaussian modes with radial number $p$ and OAM number $\ell$ are shown in FIGs.
5 and 6. We truncate the infinite space of mode numbers $p$ and $\ell$ to the
subspace of $p_{s,i}\in\\{0,1,2\\}$ and $\ell_{s,i}\in\\{-2,-1,0,1,2\\}$,
where the highest contributions of the overlap amplitudes
$C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}$ occur. The OAM conservation is easy to
see for both figures since only the modes fulfilling the condition
$\ell_{p}=\ell_{s}+\ell_{i}=0$ are non-zero. The scenarios (i)-(iv) are
depicted in four columns, shown from left to right.

Fig. 5 corresponds to the mode distribution for the crystal lengths
$L=$10\text{\,}\mathrm{mm}$$ and $L=$25\text{\,}\mathrm{mm}$$, where the beam
width ratio is fixed to the value $\gamma=\sqrt{2}$. The pump focal plane
shift according to (ii) is $z_{p}=$5\text{\,}\mathrm{mm}$$. The FGMs
$(p_{s}|\ell_{s})=(p_{i}|\ell_{i})=(0|0)$ have always the largest amplitude in
each frame. Moreover, this amplitude is the highest for the crystal with
$L=$10\text{\,}\mathrm{mm}$$ when all beams are center-focused, see case (i).
The probability decreases when $z_{p}\neq$0\text{\,}\mathrm{mm}$$ (ii).
However, this can be compensated by focal plane shifts of signal and idler. If
the signal and idler modes are shifted to the same position as the pump
according to scenario (iii), the amplitude decreases further. The focal plane
position $z_{s}=z_{i}=$2.17\text{\,}\mathrm{mm}$$ in (iv) maximizes the FGMs
for $z_{p}=$5\text{\,}\mathrm{mm}$$. The ratios of the amplitude in comparison
to (i) are in (ii) 0.89, in (iii) 0.76, and in (iv) 0.94. The more distant the
focal plane positions are from $z_{s}^{\text{max}}$, the lower the efficiency.

The situation is different for the crystal length $L=$20\text{\,}\mathrm{mm}$$
in the second row. The scenario (i)
$z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ is no longer the best choice for
achieving the highest efficiency. The ratios relative to scenario (i) now are
the following: (ii) 0.89, (iii) 1.03, and (iv) 1.04. The coupling efficiency
is not maximized at $z_{p}=z_{i}=$0\text{\,}\mathrm{mm}$$, since we filtered
the signal at $810\text{\,}\mathrm{nm}$. We will show in the next section that
the coupling into SMF strongly depends on the considered wavelength.
Particularly, we find the optimal value at
$z_{s}=z_{i}=$5.67\text{\,}\mathrm{mm}$$ that maximize the coupling efficiency
for given $z_{p}=$5\text{\,}\mathrm{mm}$$.

Fig. 6 analyzes the same as Fig. 5, but for for different beam width ratio
values $\gamma=1,\sqrt{2}$. The crystal length remains constant at
$L=$1\text{\,}\mathrm{mm}$$. Each column indicates the scenarios (i)-(iv) of
focal plane positions exactly like those described before. We make similar
observations from left to right as in the upper row of Fig. 5: when all focal
planes are in the center, the highest amplitude is achieved. These
probabilities decrease again for a shifted pump. The optimal shift $z_{s,i}$
for a displaced pump focal plane ($z_{p}=$5\text{\,}\mathrm{mm}$$) is close to
the middle of the crystal with $z_{s}=z_{i}=$0.17\text{\,}\mathrm{mm}$$. The
efficiency is significantly reduced for the shifts
$z_{p}=z_{s}=z_{i}=$5\text{\,}\mathrm{mm}$$, so these focal plane positions
are not recommended. Higher beam width ratios $\gamma$ even allow more mode
combinations, which corresponds to an increasing spiral bandwidth [47].

We can conclude, that the single-mode coupling efficiency of the spatial
spectrum is affected by focal plane position. There is always a certain
combination of $z_{p},z_{s}$, and $z_{i}$ that maximizes the efficiency for a
given setup. However, the positioning of all beam focal planes in the center
of the crystal may not be the most effective choice to achieve the peak
amplitude for all frequencies. The optimal choice of $z_{p},z_{s}$, and
$z_{i}$ depends strongly on the filtered frequency.

####  II.2.2 Influence of $z_{p},z_{s}$, and $z_{i}$ on spectral biphotons
state

Figure 8: (a) The maximizing signal/idler focal shifts $z_{s}^{\text{max}}$
for pump focal shifts $z_{p}$ from $-10\text{\,}\mathrm{mm}$ to
$10\text{\,}\mathrm{mm}$. We compare six different beam width ratios $\gamma$
from $\frac{1}{2}$ to $2$. The larger $\gamma$, the more horizontal the
curves. A horizontal curve indicates that signal and idler focal plane should
be positioned in the crystal center for all shifts of the pump focal plane.
(b) For every point $z_{s}^{\text{max}}(z_{p})$ from above in (a) the
corresponding amplitude is shown. The values are normalized to the maximum
amplitude. The amplitudes are comparatively small for larger beam width ratios
$\gamma$, but a shift of the pump focal plane has only a small influence on
the amplitude. (c) Like in a) the dependence $z_{s}^{\text{max}}(z_{p})$ is
shown. Four different crystal lengths from $1\text{\,}\mathrm{mm}$ to
$30\text{\,}\mathrm{mm}$ are compared. The smaller $L$, the more horizontal
the curves. (d) The corresponding amplitude for every point
$z_{s}^{\text{max}}(z_{p})$ from above in (c) is shown. The amplitudes for
larger $L$ are higher, which is only possible when the demanding PM conditions
in longer crystals are properly fulfilled.

Apart from the spatial DOF, we should also analyze the influence of the focal
plane shifts on the spectral DOF of the biphoton state. We consider here the
spectral response of the Fundamental Gaussian Mode
$|C_{0,0}^{0,0}(\Omega)|^{2}$. It is enough to look only at the spectral
response of the signal mode since the spectrum of signal and idler modes are
symmetric with respect to the central frequency due to the continuous-wave
pump, $\Omega_{s}=-\Omega_{i}$.

Fig. 7 shows $|C_{0,0}^{0,0}(\Omega)|^{2}$ for different focal plane positions
and for different combinations of parameters
$L=$1\text{\,}\mathrm{mm}$,$20\text{\,}\mathrm{mm}$$ and
$\gamma=\frac{2}{3},1$. The four colors in each plot represent four different
setups of combinations of $z_{p}=$5\text{\,}\mathrm{mm}$$ and $z_{s}=z_{i}$
according to scenarios (i)-(iv). The spectrum of signal photons is much
broader for the thin crystal regime on the left compared to a thick crystal
shown in the right column of Fig. 7. In terms of focal plane shifts, the blue
curves corresponding to $z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ show
always the highest brightness. When the pump focal plane is shifted (green,
red, yellow curves), the magnitude of the corresponding amplitudes changes.

Furthermore, in (d), we readily discernible that shifts of signal and idler
directly shape the frequency spectrum and position of the maximum. The larger
the signal and idler shifts are, the more the maximum is moved away from the
value of the blue curve. We observe in Figs. 7 (b) and (d), that for small
wavelengths the blue curve lies under the green and red curves. This implies
that the focusing of all modes in the middle of the crystal is not preferable
anymore at this frequency. However, when considering the possible highest
brightness, the condition $z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ always
provides the highest brightness.

###  II.3 Generalizing the influence of $z_{p},z_{s}$, and $z_{i}$ on the
spectral brighness

We observed from FIGs. 5-7 that for a given shift
$z_{p}=$5\text{\,}\mathrm{mm}$$, a $z_{s}^{\text{max}}$ exists, which
maximizes the coupling efficiency. In this section, we generalize our results
and consider the dependence $z_{s}^{\text{max}}(z_{p})$ for different beam
width ratios $\gamma$ and crystal lengths $L$. The upper plots of Fig. 8 show
the shift $z_{s}=z_{i}$ for a given $z_{p}$ , that maximizes the spectral
brightness. The normalized amplitude is shown at the bottom of the figures.
This means points that overlap vertically belong to the same $z_{p}$ value,
see the example in Fig. 8 (a) and (b). Note that the maximum spectral
brightness is not always achieved at the same frequency for different focal
plane considerations.

In Fig. 8 (a) we plot the maximizing signal and idler focal plane shifts
$z_{s}^{\text{max}}$ for given pump focal shifts in the range from
$-10\text{\,}\mathrm{mm}$ to $10\text{\,}\mathrm{mm}$ for a crystal of length
${L=$20\text{\,}\mathrm{mm}$}$. Six different values for the beam width ratio
are displayed. The maximizing shift $z_{s}^{\text{max}}$ changes linearly with
$z_{p}$, whereby the slope of the line depends on the beam width ratio. The
higher $\gamma$, the smaller the slope of the lines. Hence signal and idler
focal plane can be positioned in the middle at
$z_{s}=$0\text{\,}\mathrm{mm}$$, regardless of $z_{p}$. The shift of the pump
focal plane has no significant impact.

The spectral brightness is highest for $z_{p}=z_{s}=$0\text{\,}\mathrm{mm}$$
for all $\gamma$ in Fig. 8 (b). Whenever the pump focal plane is shifted away
from the crystal center, the amplitude drops. At high values for $\gamma$, the
$z_{s}^{\text{max}}(z_{p})$ dependence becomes almost constant. The
explanation is that a high beam width ratio results in a less divergent pump
with big width and a more constant beam radius over the length of the crystal.
This means no major noticeable change in the system and therefore less impact
on the yield. The optimum beam width ratio for maximum spectral brightness in
Fig. 8 is $\gamma=\frac{3}{2}$.

It is important to mention, that the photons for every
$z_{s}^{\text{max}}(z_{p})$ value are spectrally filtered at the corresponding
maximum. The frequency that maximizes the spectral brightness lies in a very
small range between $809.88\text{\,}\mathrm{nm}$ and
$809.90\text{\,}\mathrm{nm}$.

Similarly, Fig. 8 (c) and (d) show the maximizing beam shifts for given pump
shifts between $-10\text{\,}\mathrm{mm}$ to $10\text{\,}\mathrm{mm}$ at a
constant beam width ratio
$\gamma=\frac{3}{2}=\frac{$30\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}{$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$ displaying four different
crystal lengths. Again, linear lines can be seen in the top chart. The thinner
$L$, the more horizontal these lines are. This corresponds to the expectations
of a thin crystal since the pump beam radius does not change significantly
over the length of the crystal [55, 56]. As a consequence, pump shifts have
almost no effect on the spectral brightness in thin crystals. Pump focal plane
shifts should be taken into account by a proper signal and idler focal plane
positions $z_{s}^{\text{max}}$ in thicker crystals.

###  II.4 Spatio-temporal correlations between signal and idler photons

Figure 9: Spectral purity $\text{Tr}(\rho^{2}_{s\text{,SMF}})$ as a function
of the focal plane position of the pump $z_{p}$ and signal beam $z_{s}$ with
the assumption $z_{s}=z_{i}$ and a setup with crystal length
$L=$30\text{\,}\mathrm{mm}$$, beam width ratio $\gamma=\frac{1}{\sqrt{2}}$ and
pulse duration $T_{0}=$0.5\text{\,}\mathrm{ps}$$. The purity reaches its
maximum when all beams are focused in the crystal center,
$z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$.

In general, we can distinguish two kinds of correlations in the SPDC process:
the correlation between signal and idler photons (see FIGs. 5 and 6) or the
correlation between spatial and spectral DOF of generated photons. The spatio-
spectral correlation implies that the spatial characteristics of signal
(idler) photons can not be considered independently of the spectral DOF, they
are coupled. Mathematically, it means that the biphoton mode function can not
be written as the product state of spatial and spectral DOFs
$\Phi(\bm{q}_{s},\bm{q}_{i},\Omega_{s},\Omega_{i})\neq\Phi_{\bm{q}}(\bm{q}_{s},\bm{q}_{i})\Phi_{\Omega}(\Omega_{s},\Omega_{i})$.
Correspondingly, the correlation between signal and idler photons implies that
the biphoton mode function can not be written as
$\Phi(\bm{q}_{s},\bm{q}_{i},\Omega_{s},\Omega_{i})\neq\Phi_{s}(\bm{q}_{s},\Omega_{s})\Phi_{i}(\bm{q}_{i},\Omega_{i})$.

Here, we quantify explicitly how the focal plane shifts affect both types of
correlations. We consider for this section the Gaussian envelope Eq. (11) for
the spectral DOF of the pump. We look at the purity of the spatial (spectral)
biphoton state [50]

$\displaystyle\text{Tr}(\rho^{2}_{\bm{q}})$ $\displaystyle=\int
d\bm{q}_{s}\>d\Omega_{s}\>d\bm{q}_{i}\>d\Omega_{i}\>d\bm{q}_{s}^{\prime}\>d\Omega_{s}^{\prime}\>d\bm{q}_{i}^{\prime}\>d\Omega_{s}^{\prime}$
$\displaystyle\times\Phi(\bm{q}_{s},\bm{q}_{i},\Omega_{s},\Omega_{i})\>\Phi^{*}(\bm{q}_{s}^{\prime},\bm{q}_{i}^{\prime},\Omega_{s},\Omega_{i})$
$\displaystyle\times\Phi(\bm{q}_{s}^{\prime},\bm{q}_{i}^{\prime},\Omega_{s}^{\prime},\Omega_{i}^{\prime})\>\Phi^{*}(\bm{q}_{s},\bm{q}_{i},\Omega_{s}^{\prime},\Omega_{i}^{\prime})$
(14)

as a measure for the correlations between space and frequency DOF. It turns
our that all dependencies of the spatial purity $\text{Tr}(\rho^{2}_{\bm{q}})$
on $z_{p},z_{s}$, and $z_{i}$ cancel out. Since $z_{s}$ and $z_{i}$ are
parameters associated with detection mechanisms, this seems logical.
Therefore, the spatio-spectral correlation can not be manipulated by the beam
shifts $z_{p},z_{s}$, and $z_{i}$.

The situation is different for the purity of the signal (idler) photon [50]

$\displaystyle\text{Tr}(\rho^{2}_{\text{s}})$ $\displaystyle=\int
d\bm{q}_{s}\>d\Omega_{s}\>d\bm{q}_{i}\>d\Omega_{i}\>d\bm{q}_{s}^{\prime}\>d\Omega_{s}^{\prime}\>d\bm{q}_{i}^{\prime}\>d\Omega_{s}^{\prime}$
$\displaystyle\times\Phi(\bm{q}_{s},\Omega_{s},\bm{q}_{i},\Omega_{i})\>\Phi^{*}(\bm{q}_{s}^{\prime},\Omega_{s}^{\prime},\bm{q}_{i},\Omega_{i})$
$\displaystyle\times\Phi(\bm{q}_{s}^{\prime},\Omega_{s}^{\prime},\bm{q}_{i}^{\prime},\Omega_{i}^{\prime})\>\Phi^{*}(\bm{q}_{s},\Omega_{s},\bm{q}_{i}^{\prime},\Omega_{i}^{\prime}),$
(15)

where its dependence on $z_{p},z_{s}$, and $z_{i}$ does not drop off. The
purity (15) gives the strength of the correlation between signal and idler
photons, i.e., how entangled the two photons are. In the last years, one of
the most important goals of photonic quantum technologies has been the
reduction of the correlation between signal and idler photons. The heralded
pure single photons from SPDC are believed to be a good candidate for an
indistinguishable single-photon source [57, 58, 59], which is required for a
successful photonic boson sampling [60]. Usually, the spatial DOF of the
biphoton state is doped off by just collecting the photons into SMF, which
accepts only the Gaussian mode. We can then talk about the spectral purity of
the biphoton state which can be estimated by

$\displaystyle\text{Tr}(\rho^{2}_{\text{s,SMF}})$
$\displaystyle=\int\>d\Omega_{s}\>d\Omega_{i}\>d\Omega_{s}^{\prime}\>d\Omega_{i}^{\prime}\>$
$\displaystyle\times
C^{0,0}_{0,0}(\Omega_{s},\Omega_{i})\>C^{0,0}_{0,0}(\Omega_{s}^{\prime},\Omega_{i}^{\prime})\>$
$\displaystyle\times[C^{0,0}_{0,0}(\Omega_{s}^{\prime},\Omega_{i})]^{*}\>[C^{0,0}_{0,0}(\Omega_{s},\Omega_{i}^{\prime})]^{*}$
(16)

Fig. 9 shows the dependence of the spectral purity
$\text{Tr}(\rho^{2}_{s\text{,SMF}})$ on $z_{p}$ and $z_{s}$ for a crystal
length $L=$30\text{\,}\mathrm{mm}$$ and pulse duration of
$T_{0}=$0.5\text{\,}\mathrm{ps}$$. The combination
$z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$, which corresponds to the crystal
center, yields the maximum purity. Additionally, high levels of purity can
also be observed along the diagonal $z_{p}=z_{s}=z_{i}$.

##  III Conclusions

In this work, we have assumed paraxial pump and collecting signal and idler
beams with defined beam widths and focal plane positions. We theoretically
investigated the dependence of spatial and temporal DOFs of the biphoton state
on these focal plane positions. In addition, the single-mode coupling
efficiency and the spectral brightness of Fundamental Gaussian Modes were
studied. Generally, the spectral brightness reaches the maximum if all
involved beams are positioned in the center of the crystal. The single-mode
coupling efficiency strongly depends on the frequency: for certain narrow-band
filtered frequencies, positioning all focal planes in the crystal center would
not attain the highest efficiency.

Depending on the setup, small deviations of the focal plane positions from the
crystal center can have a large impact on the coupling efficiency. In sense of
the advanced-wave picture, pump, signal, and idler focal plane shifts must
compensate each other for higher efficiency. However, equal positioning of
signal and idler focal planes is sufficient in most setups. Thus we advise
choosing a suitable signal and idler focal plane position
$z_{s}=z_{i}=z_{s}^{\text{max}}$ for higher efficiency if the pump beam is not
center-focused. Regardless of $z_{p}$, $z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$
can be assumed for high beam width ratios $\gamma$ or short lengths $L$, see
results in section II.3.

We also find that correlations between space and frequency degrees of freedom
are not affected by focal plane shifts. In contrast, the entanglement between
signal and idler photons depends on the focal plane positions of involved
beams. We recommend placing all focal planes of SPDC beams in the center of
the crystal to achieve the highest purity.

##  IV Acknowledgment

The authors thank Carlos Sevilla-Gutiérrez for very helpful discussions and
suggestions.

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