# Point2Pix: Photo-Realistic Point Cloud Rendering via Neural Radiance Fields

Tao Hu1 Xiaogang Xu1111Corresponding author. Shu Liu2 Jiaya Jia1,2
1 The Chinese University of Hong Kong 2 SmartMore
{taohu, xgxu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Synthesizing photo-realistic images from a point cloud is challenging because
of the sparsity of point cloud representation. Recent Neural Radiance Fields
and extensions are proposed to synthesize realistic images from 2D input. In
this paper, we present Point2Pix as a novel point renderer to link the 3D
sparse point clouds with 2D dense image pixels. Taking advantage of the point
cloud 3D prior and NeRF rendering pipeline, our method can synthesize high-
quality images from colored point clouds, generally for novel indoor scenes.
To improve the efficiency of ray sampling, we propose point-guided sampling,
which focuses on valid samples. Also, we present Point Encoding to build
Multi-scale Radiance Fields that provide discriminative 3D point features.
Finally, we propose Fusion Encoding to efficiently synthesize high-quality
images. Extensive experiments on the ScanNet and ArkitScenes datasets
demonstrate the effectiveness and generalization.

##  1 Introduction

Point cloud rendering aims to synthesize images from point clouds at given
camera parameters, which has been frequently utilized in 3D visualization,
navigation, and augmented reality. There are many advantages to point cloud
representation, such as flexible shape and general 3D prior. However, since
point clouds are generally produced by 3D scanners (RGBD or LiDAR) [14, 3, 9]
or by Multi-View Stereo (MVS) from images [15, 4, 52], the points are usually
sparsely distributed in 3D scenes. Although traditional graphics-based
renderers [42, 58, 38, 50] can render point clouds to images without training
or finetuning, the quality is not satisfying with hole artifacts and missing
details [10].

Recently, Neural Radiance Fields (NeRF) [29] were proposed for 3D
representation and high-fidelity novel view synthesis. It employs an implicit
function to directly map each point’s spatial information (location and
direction) to attributes (color and density). However, most NeRF-based methods
[23, 54, 55, 49] are scene-specific, thus taking much time to train from
scratch for novel scenes in abundant multi-view images, which limits the
practical applications.

In this work, we bridge the gap between point clouds and NeRF, thus proposing
a novel point cloud renderer, called Point2Pix, to synthesize photo-realistic
images from colored point clouds. Compared with most NeRF-based methods [29,
23, 54, 53], ours does not necessarily require multi-view images or fine-
tuning procedures for indoor scenes.

First, point clouds are treated as underlying anchors of NeRF. The training
process of NeRF is to learn 3D point attributes from given locations. Because
there is no mapping ground truth for multi-view images, NeRF-based methods
[29, 23] indirectly train their networks with a pixel reconstruction loss.
Note that point clouds are exactly made up of points with location and
attributes, thus can provide training pairs for the mapping function to
conduct supervised learning and improve performance.

Then, point clouds can also improve the efficiency of ray sampling. NeRF-based
methods [29, 23, 54] learn the 3D shape and structure from multi-view images,
which do not involve geometric prior in novel scenes. Thus, dense uniform
sampling [5, 46] or coarse-to-fine sampling [29, 54] were used to synthesize
high-quality images. These strategies are inefficient because most of the
locations in 3D scenes are empty [23, 31]. Since point clouds represent a
relatively fine shape of 3D scenes, the area around existing points deserves
more attention. Based on this observation, we propose a point-guided sampling
strategy to mainly focus on the local area of points in the point cloud. It
can significantly reduce the number of required samples while maintaining
decent synthesis accuracy.

Further, point-based networks can provide 3D features prior to subsequent
applications, general for novel scenes. Although many methods [36, 37, 18, 7]
have been proposed for various point cloud understanding tasks, they are
usually designed for existing points. In this work, we propose Multi-scale
Radiance Fields, including Point Encoder and MLP networks, to extract multi-
scale features for any location in the scene. These 3D point features are
discriminative and general, ensuring finetuning-free rendering. Also, inspired
by recent NeRF-based image generators [19, 57, 22], we render the 3D point
features as multi-scale feature maps. Our fusion decoder gradually synthesizes
high-resolution images. It can not only fill the possible holes but also
improve the quality of rendered images. Our main contributions are summarized
as follows.

  * •

We propose Point2Pix to link point clouds with image space, which renders
point clouds into photo-realistic images.

  * •

We present an efficient ray sampling strategy and fusion decoder to greatly
decrease the number of samples in each ray and the total number of rays, thus
accelerating the rendering process.

  * •

We propose Multi-scale Radiance Fields, which extract discriminative 3D prior
for arbitrary 3D locations.

  * •

Extensive experiments and ablation studies on indoor datasets demonstrate the
effectiveness and generalization of the proposed method.

##  2 Related Work

In this section, we briefly review the related works, including various point
renderers, point-based networks in extracting 3D point features, and NeRF-
based synthesis.

###  2.1 Point-based Rendering

Traditional Point Rendering [38, 42, 58] is based on computer graphics, which
generates images from point clouds by simulating the physical process of
imaging, considering geometry [44], material [12], BRDF [16], and lighting
[21]. The rendering pipeline is general for arbitrary scenes. But it cannot
fill missing points and thus generate vacant pixels.

In the deep learning era, neural-based point renderers [11, 10, 41] made great
progress in generating images from point clouds. They first extract multi-
scale projected features from point clouds and then use a decoder to generate
images. NPBG [11] augments each point by a neural descriptor to encode local
geometry and appearance. NPCR [10] represents point features by 3D Convolution
Neural Network (CNN), and converts the 3D representation to multi-plane
images. Different from these methods, our Point2Pix combines a more
discriminative point encoder with the renderer pipeline of NeRF, thus
achieving better performance.

###  2.2 Point-based Networks

Point-based networks have been developed for many years [36, 37, 8, 17, 18,
7]. For general point understanding, PointNet [36] utilizes point-wise Multi-
Layer Perception (MLP) and Pooing to extract features for 3D classification
and semantic segmentation, while not capturing local structures. PointNet++
[37] introduces hierarchical feature learning to encode local point features.
Although 3D CNN can also deal with point cloud data after voxelization, the
maximal resolution of 3D volumes is low because of huge memory consumption and
slow computation. Thus, sparse 3D CNNs [17, 45, 20, 7] draw more attention.
SparseConvNet [18] differs from previous sparse convolutional networks since
it uses a rectangular grid, instead of dilating the observation. In this
paper, we adopt a more efficient sparse 3D CNN – MinkowskiEngine [7] – as the
basic point encoder to extract 3D prior from point clouds.

###  2.3 NeRF-based Synthesis

NeRF [29] well balances neural network and physical rendering, thus achieving
state-of-the-art performance in novel view synthesis. To handle dynamic scenes
where objects move, a deformable function is learned to align different frames
[33, 24]. For human reconstruction and synthesis, many methods, such as
Neural-Body[35], Neural-Actor [25], and Anim-NeRF [6], introduce the
parameterized human model SMPL [26] as a strong 3D prior and achieve
impressive performance. To generate high-resolution images, methods of [32,
22, 19, 57] first render low-resolution feature maps instead of images, then
upsample features to the final images via 2D CNN. NeRF’s input is only multi-
view images and camera parameters, when combined with 3D prior, such as depth
and point cloud, the performance can be further improved [51, 1, 31, 13]. Our
model also combines deep learning with physical rendering, while taking more
advantage of point clouds as 3D priors to render decent-quality images in
general indoor scenes.

##  3 Our Approach

For a point cloud
$\textbf{P}=\cup_{k=1}^{K}\\{\textbf{p}_{k},\textbf{c}_{k}\\}$ with $K$
points, $\textbf{p}_{k}=(x_{k},y_{k},z_{k})\in\mathbb{R}^{3}$ and
corresponding colors $\textbf{c}_{k}=(r_{k},g_{k},b_{k})\in\mathbb{R}^{3}$,
our goal is to synthesize a high-fidelity image $\hat{I}$ at the given camera
parameter V via our proposed renderer (Point2Pix) $\mathcal{R}$, formulated as

$I=\mathcal{R}(\textbf{P},V),$ (1)

where $V$ is represented by $H\times W$ rays r. Each ray starts from camera
center o in pixel direction d.

To begin with, we introduce the background knowledge of NeRF in Sec. 3.1.
Then, we propose an efficient point-based ray sampling strategy in Sec. 3.2.
We also build a network to extract multi-scale 3D prior from point clouds in
Sec. 3.3. Finally, we show how to combine the point feature with NeRF to
render target images in Sec. 3.4. The overview of our framework can be seen in
Fig. 1.

Figure 1: Overview of our proposed Point2Pix. (a) Multi-scale Radiance Fields.
For an input point cloud, we first extract multiple 3D feature volumes in four
scales. Next, for any queried point, we first linearly interpolate the coarse
features from these feature volumes and infer the final features through MLP
networks. (b) Fusion and Upsampling. Four 2D feature maps are respectively
rendered through NeRF. They are fused with the previous 2D CNN output and then
are upsampled by 2 times. (c) Fusion Decoding. We finally design a neural
renderer to gradually synthesize target images from the projected feature
maps.

###  3.1 Preliminary

Given multi-view camera-calibrated images of a scene, NeRF [29] synthesizes
high-quality novel view images. It mainly consists of ray sampling, implicit
function, and volume rendering.

Ray Sampling. Starting from the camera center o, ray sampling is the process
that obtains a series of positions $\textbf{x}_{i}$ along ray r with direction
d as

$\textbf{x}_{i}=\textbf{o}+z_{i}\cdot\textbf{d},\quad i=1,2,...,N,$ (2)

where $z_{i}$ is the sampling depth and $N$ is the number of samples on each
ray.

Implicit Function. An implicit function $f_{\theta}$ is trained as a mapping
from each queried location $\textbf{x}_{i}$ and direction d to corresponding
colors $\textbf{c}_{i}=(r_{i},g_{i},b_{i})$ and density $\sigma_{i}$, as

$(\textbf{c}_{i},\sigma_{i})=f_{\theta}(\textbf{x}_{i},\textbf{d}),$ (3)

where $f_{\theta}$ is an MLP network, and $\theta$ is its parameter.

Volume Rendering. Each ray (or pixel) color $\hat{\textbf{c}}$ is calculated
via volume rendering [28] as

$\displaystyle\hat{\textbf{c}}$
$\displaystyle=\sum_{i=1}^{N}T_{i}\alpha_{i}\textbf{c}_{i},$ (4)
$\displaystyle\alpha_{i}$ $\displaystyle=1-\textbf{
exp}(-\sigma_{i}\delta_{i}),$ $\displaystyle T_{i}$
$\displaystyle=\textbf{exp}(-\sum_{j=1}^{i-1}\sigma_{j}\delta_{j}),$

where $\delta_{j}$ is the distance between neighbor samples along the ray r.

###  3.2 Point-guided Sampling

According to Eq. (4), increasing the number of samples $N$ along each ray can
generate more realistic results [29]. However, the required computing
resources and running time will also linearly grow. Our model is based on a
point cloud with a relatively fine shape prior. Thus, we propose point-guided
sampling to achieve more efficient ray sampling through the guidance of the
point cloud.

For any queried point $\textbf{x}_{i}$, we first find the nearest neighbour
point $\textbf{p}_{i}$, then check whether $\textbf{x}_{i}$ is located in
$\textbf{p}_{i}$’s ball area with radius $r$ or not, as

$\parallel\textbf{p}_{i}-\textbf{x}_{i}\parallel_{2}\leq r.$ (5)

If the above condition is satisfied, we treat the queried point
$\textbf{x}_{i}$ as a valid sample and obtain the point feature in Sec. 3.3.
If there is no valid sample along one ray, we adopt uniform sampling from
default near to far depth. As illustrated in Fig. 2. Compared with previous
uniform and coarse-to-fine sampling [29, 5, 46], our sampling strategy reduces
computation and memory costs.

Figure 2: The proposed point-guided sampling. For any queried point
$\textbf{x}_{i}$, we find its nearest point $\textbf{p}_{i}$ in the point
cloud. If $\textbf{x}_{i}$ is located in the ball area (with radius $r$) of
$\textbf{p}_{i}$, it is a valid sample. Invalid samples are omitted to improve
sampling efficiency.

###  3.3 Multi-scale Radiance Fields

We extract discriminative 3D point and ray features via constructing Multi-
scale Radiance Fields, including Point Encoding and NeRF-based Feature
Rendering.

Point Encoding. Point Encoding is to output a discriminative 3D point feature
for each valid sample $\textbf{x}_{i}$. We adopt 3D sparse UNet from the
Minkowski Engine (ME) [7] as the backbone of our point encoder. ME is an auto-
differentiation library to build a 3D CNN for sparse tensors, that are
converted from point clouds. As illustrated in Fig. 1, our point encoder
extracts multiple 3D feature volumes from the raw point cloud in $L$ different
scales. We select $F_{l}$ at scale $l$ to construct multi-scale radiance
fields.

For each valid sample $\textbf{x}_{i}$ at each scale $l$, we query feature in
$F_{l}$ to obtain the interpolated feature $F_{i}^{l}$ and employ an implicit
function $\Phi_{l}$ to infer density $\sigma_{i}^{l}$ and final point feature
$f_{i}^{l}$ for $\textbf{x}_{i}$ as

$(\sigma_{i}^{l},\textbf{f}_{i}^{l})=\Phi_{l}(\textbf{F}_{i}^{l})=\Phi_{l}(\textbf{F}^{l}[\textbf{x}_{i}]).$
(6)

NeRF-based Feature Rendering. Then we render the queried 3D point features to
2D feature maps and generates images at different scales. At each feature
scale $l$, we aggregate density $\sigma_{i}^{l}$ and feature
$\textbf{f}_{i}^{l}$ to generate 2D feature map $\textbf{f}^{l}$ by volume
rendering of NeRF [29] as

$\displaystyle\textbf{f}^{l}$
$\displaystyle=\sum_{i=1}^{N}\textbf{w}_{i}^{l}\textbf{f}_{i}^{l},$ (7)
$\displaystyle\textbf{w}_{i}^{l}$
$\displaystyle=\textbf{exp}(-\sum_{j=1}^{i-1}\sigma_{j}^{l}\delta_{j})(1-\textbf{exp}(-\sigma_{i}^{l}\delta_{i})).$

So far, we obtained $L$ rendering feature maps
$\\{\textbf{f}^{l}\\}\in\mathbb{R}^{C_{l}\times H_{l}\times W_{l}}$, where
$H_{l}$ and $W_{l}$ respectively represent the feature height and width, and
$C_{l}$ is the number of channels.

###  3.4 Fusion Decoding

Although we propose an efficient ray sampling strategy to reduce memory
consumption, it still requires more than 20 GB GPU memory to render target
images with the size of 480 $\times$ 640, as shown in Tab. 4. In addition,
there are still many holes to be filled in the 2D-rendered image space. To
address these issues, we design a Fusion Decoder as a neural renderer that
synthesizes final images from these rendered feature maps
$\textbf{f}^{l},l\in[1,L]$ by conditional convolution and upsampling modules.

Fusion. Our conditional convolution is to fuse the previous layer’s feature
$\mathcal{F}^{l-1}$ with the rendered features $\textbf{f}^{l}$, which treats
the rendered feature at each scale $l$ as the conditional input. This module
is inspired by SPADE [34], while we use Layer Normalization [2]. Specifically,
as shown in Eq. (3.4), for the rendered feature map $\textbf{f}^{l}$, we
calculate the conditional parameters, including scale $\gamma$ and bias
$\beta$, by a Conv2D module. Then for feature $\mathcal{F}^{l-1}$ from the
previous stage, we normalize it by Layer Normalization and scale it by
$\gamma$. Finally, the fused feature $\mathcal{F}^{l}$ is obtained by adding
the bias $\beta$ as

$\displaystyle(\gamma,\beta)$ $\displaystyle=\textbf{Conv2D}(\textbf{f}^{l}),$
(8) $\displaystyle\mathcal{F}^{l-1}$
$\displaystyle=\gamma\cdot\textbf{LayerNorm}(\mathcal{F}^{l-1})+\beta.$

Upsampling. We adopt PixelShuffle [43] as our upsampling modules that upsample
the fused feature $\mathcal{F}^{l-1}$ by $2$ times at each stage, instead of
using bilinear or nearest interpolation. PixelShuffle [43] is frequently
adopted in the super-resolution task, which utilizes a convolution layer to
extend the channel size and reshape them into the spatial size, as

$\displaystyle\mathcal{F}^{l}$
$\displaystyle=\textbf{Pixelshuffle}(\textbf{Conv2D}(\mathcal{F}^{l-1}),2).$
(9)

ToRGB. Finally, we introduce the decoder of the present large-scale generator,
like [39], as a post-process to generate the final rendering image $\hat{I}$
for the whole point cloud renderer.

Dataset ScanNet [9] ARKitScenes [3] Metrics PSNR $\uparrow$ SSIM $\uparrow$
LPIPS $\downarrow$ PSNR$\uparrow$ SSIM$\uparrow$ LPIPS$\downarrow$ Pytorch3D
[38] 13.62 0.528 0.779 15.21 0.581 0.756 Pix2PixHD [47] 15.59 0.601 0.611
15.94 0.636 0.605 NPCR [10] 16.22 0.659 0.574 16.84 0.661 0.518 NPBG++ [11]
16.81 0.671 0.585 17.23 0.692 0.511 ADOP [41] 16.83 0.699 0.577 17.32 0.707
0.495 Point-NeRF [51] 17.53 0.685 0.517 17.61 0.715 0.508 Point2Pix (Ours)
18.47 0.723 0.484 18.84 0.734 0.471

Table 1: Comparing our method with different point renderers on the ScanNet
[9] and ARkitScenes [3] datasets. There is no finetuning process in this
experiment, which demonstrates the generalization in novel scenes.

###  3.5 Loss Function

For the NeRF-based rendering images and neural rendering images, their
optimization goals are the ground-truth images with target camera parameters.
We employ point cloud loss, NeRF rendering loss, neural rendering loss, and
perceptual loss to train the parameters of the proposed point encoder and
fusion decoder as

$\mathcal{L}=\lambda_{pc}\mathcal{L}_{pc}+\lambda_{nr}\mathcal{L}_{nr}+\lambda_{per}\mathcal{L}_{per},$
(10)

where $\lambda_{pc}$, $\lambda_{nr}$, and $\lambda_{per}$ respectively control
weights of these losses.

Point Cloud Loss. All points $\textbf{p}_{k}$ in raw point clouds provide
groud-truth mapping from locations $\textbf{x}_{k}$ to densities
$\hat{\sigma}_{k}$ and colors $\hat{\textbf{c}}_{k}$. Denoting the queried 3D
densities and colors from Point2pix in point $\textbf{p}_{k}$ as
$\textbf{c}_{k}$ and $\sigma_{k}$, the point cloud loss can be represented as

$\mathcal{L}_{pc}=\sum_{k=1}^{K}(\parallel\hat{\textbf{c}}_{k}-\textbf{c}_{k}\parallel^{2}+\frac{1}{D}\max(0,D-\sigma_{k})).$
(11)

We encourage the predicted densities at $\textbf{p}_{k}$ to be greater than a
threshold $D$.

Neural Rendering Loss. $\mathcal{L}_{nr}$ is the MSE between rendered images
$I$ from fusion decoder and ground truths $\hat{I}$ as

$\mathcal{L}_{nr}=\parallel\hat{I}-I\parallel^{2}_{2}.$ (12)

Perceptual Loss. $\mathcal{L}_{per}$ is a frequently used loss in image
synthesis, which improves the realism of generation, as

$\mathcal{L}_{per}=\sum_{l=1}^{L}\parallel\phi(\hat{I}^{l})-\phi(I^{l})\parallel_{1},$
(13)

where $\phi(\cdot)$ means extracting VGG features.

##  4 Experiments

In this section, we conduct experiments to demonstrate the effectiveness of
our proposed method. First, we introduce the indoor datasets and evaluation
metrics. Then, we quantitatively and qualitatively compare the proposed method
with state-of-the-art point cloud renderers to show our advantages. Next,
ablation studies are performed to validate the effect of each proposed module,
including point-guided sampling, point encoder, and fusion decoder. Finally,
we apply our method to point cloud applications.

###  4.1 Experimental Settings

Datasets. We perform experiments on indoor datasets containing point cloud and
multi-view images, including ScanNet [9] and ARKitScenes [3]. ScanNet [9] is
an RGBD scanned dataset, which contains 2.5 million images at different views
in 1,513 scenes. The dataset has been annotated with calibrated cameras and
colored point clouds. We split the first 1,200 scenes as a training set and
the rest as a testing set. ARkitScenes [3] is a 3D indoor-scene understanding
dataset, whose scenes are captured by Apple iPad Pro. There are around 5,000
scenes, and we choose the first 4,500 scenes for training and the 500 scenes
for testing.

Metrics. We adopt three common metrics, including PSNR, SSIM [48], and LPIPS
[56], to evaluate the performance of Point2Pix. They measure the
reconstruction accuracy between the rendered images and ground-truth images.

Evaluation. We evaluate the rendering quality of different methods in two
aspects, including non-finetuning and finetuning. The non-finetuning
evaluation means directly measuring the rendering quality in the testing
datasets. As for the finetuning evaluation, methods can refine their results
on specific scenes to improve performance. Since finetuning evaluation in each
case usually consumes much resources and time, we randomly choose 8 testing
scenes from ScanNet datasets and the same number from the ArkitScene dataset
for finetuning evaluation.

Implementation Details. We adopt MinkUnet14A as our Point Encoder. The radius
$r$ for point-guided sampling is $0.08$ meters. The maximal number $N$ of
samples for each ray is 128. We extract feature volumes with $L=4$ scales.
Thus the scales are $\dfrac{1}{8},\dfrac{1}{4},\dfrac{1}{2}$, and $1$
respectively. The resolution of the final rendered images is $640\times 480$.
During training, the initial learning rate is $0.004$ with AdamW [27]
optimizer. It exponentially decays to $0.0004$ till the end (by 500 epochs).
We set $\lambda_{pc}=0.1$, $\lambda_{nr}=1.0$, and $\lambda_{per}=0.1$
empirically. The density threshold $D=10$. We train our model on 4 NVIDIA
Titan-V GPUs, and the batch size is 1 for each GPU.

###  4.2 Comparison with Point Renderers

We first compare our method with different point rendering methods by non-
finetuning evaluation. After training, all methods are directly tested in
novel scenes. The competitors include graphics-based point cloud renderer
Pytorch3D [38], previous neural-based point renderers NPBG [11] and NPCR [10],
and image generator Pix2PixHD [47]. For Pix2PixHD, we use it as an image-to-
image generator to translate the rendered images from graphics-based point
renderer [38] to the ground-truth images. The evaluation results are shown in
Tab. 1. Ours achieves significantly higher accuracy than other solutions,
which reflects the great advantage in practical applications.

Figure 3: Qualitative comparison between different point renderers on the
ScanNet [9].

###  4.3 Comparison with NeRF-based Synthesis

We also compare the proposed method with NeRF-based synthesis in both non-
finetuning and finetuning evaluations. In this experiment, we adopt the same
coarse-to-fine ray sampling as NeRF-based methods [29, 54] for a fair
comparison. The results are illustrated in Tab. 2.

To achieve general view synthesis in novel scenes, MVSNeRF [5] and IBRNet [46]
combine image prior with NeRF, while ours combines the point cloud prior. For
a fair comparison, we also pre-train these two methods on the same pretraining
ScanNet dataset. Our method achieves the highest performance among all.
Although NeRF [29], NSVF [23], and PlenOctrees [54] can achieve competitive
accuracy, their training time is much longer. When training Instant-NGP [30],
Plenoxels [53], Point-NeRF [51], and our Point2Pix in 20 minutes, ours
achieves better performance. This experiment demonstrates the advantage of
point cloud prior when combined with NeRF.

Method Time PSNR($\uparrow$) SSIM ($\uparrow$) LPIPS($\downarrow$) Point-NeRF
[51] 0 mins 17.53 0.685 0.517 Point2Pix (Ours) 0 mins 18.47 0.723 0.484 NeRF
[29] $\sim$30 hours 21.33 0.788 0.355 NSVF [23] $\sim$40 hours 22.47 0.791
0.337 PlenOctrees [54] $\sim$30 hours 22.02 0.795 0.341 Instant-NGP [30] 20
mins 21.94 0.775 0.363 Plenoxels [53] 20 mins 22.35 0.780 0.346 Point-NeRF
[51] 20 mins 22.55 0.792 0.336 Point2Pix (Ours) 20 mins 23.02 0.815 0.318

Table 2: Comparing our method with NeRF-based methods on the ScanNet dataset
[9]. “Time” means the average finetuning time for all scenes.

###  4.4 Qualitative Comparison

We also qualitatively compare our Point2Pix with other point renderers and
NeRF-based synthesis. The visualization is illustrated in Fig. 3 and Fig. 4.
The Graphics-based point renderer Pytorch3D [38] usually generates images with
holes because of sparse points. Due to missing 3D prior, the generated images
are not realistic. Ours achieves the best visual quality, which shows
Point2Pix’s superiority.

Figure 4: Qualitative comparison between different point renderers and NeRF-
based methods on the ArkitScenes [3] dataset.

###  4.5 Ablation Studies

Effect of Point-guided Sampling. To prove the efficiency of our point-guided
sampling, we replace our sampling with other strategies used in NeRF-based
methods, including uniform [5, 46] and coarse-to-fine sampling [29, 54].
Uniform sampling means uniformly obtaining $N$ points on each ray in near-to-
far depth. Coarse-to-fine sampling is proposed by the original NeRF [29]. The
coarse stage uniformly samples $\frac{N}{2}$ points and the fine stages
samples $\frac{N}{2}$ points according to the probability from the coarse
stage.

Our point-guided sampling uniformly samples $N$ points while only inferring
the valid samples. The results are shown in Tab. 3. Since our sampling
strategy considers the point cloud 3D prior, we significantly decrease the
average number of samples and reduce the rendering time and GPU memory.

Ray Sampling # Val. Sampl. PSNR ($\uparrow$) Render Time (seconds,
$\downarrow$) Train Memory (GB, $\downarrow$) Uniform 128 17.96 3.13 8.54
Coarse-to-Fine 128 18.69 5.78 9.17 Point-Guide (Ours) 15.6 18.47 0.92 6.68

Table 3: Comparison between different ray sampling strategies on ScanNet [9].
In our method, the mean sampling number for each ray is only $15.6$, which is
significantly less than the others.

Effect of Multi-scale Radiance Fields. Previous methods, like GIRAFFE [32],
HeadNeRF [22], StyleNeRF, and CIPS-3D [57], also render feature maps via NeRF,
which can effectively reduce the memory and rendering time. However, different
from ours, they only adopt a single scale of radiance fields. We validate our
Multi-scale Radiance Fields in this experiment and show results in Tab. 4.

We first study the effect of the number of rays. In the condition of a single
scale, if we directly render the image at the final resolution by NeRF, the
memory consumption is heavy, and the running time is long. Decreasing sampled
rays and increasing upsampling scale can promote the synthesis quality and
rendering efficiency. With the increasing number of NeRF scales, accuracy
further improves, which validates the effectiveness of our multi-scale
radiance fields. We finally adopt the combination in the last column to
balance the accuracy and time.

# Scales 1 1 1 2 4 # Rays $640\times 480$ $320\times 240$ $80\times 60$
$80\times 60$ $80\times 60$ Upsampling $\times 1$ $\times 2$ $\times 8$
$\times 8$ $\times 8$ PSNR ($\uparrow$) 17.49 17.86 18.05 18.16 18.47
Rendering Time (seconds, $\downarrow$) 13.12 3.56 0.85 0.92 0.96 Training
Memory (GB, $\downarrow$) 22.93 11.12 6.27 6.68 6.95

Table 4: Effect of different combinations in terms of the number of scales,
number of rays and scale of upsampling. We choose the combination in the last
column, which achieves the best performance and is also efficient.

Selection of Point Encoder. In our Point2Pix, the Point Encoder is the
backbone to provide multi-scale 3D features. In the literature of point cloud
analysis, many point-based networks [36, 37, 18, 7] were proposed. We compare
different backbones in this experiment.

The candidate networks include PointNet++ [37], SparseConvNet [18], and
MinkUnet [7] (MinkUnet14A and MinkUnet34C). We evaluate them by only changing
the point encoder, and the results are shown in Tab. 5. PointNet++ [37]
consumes the largest memory and takes the longest rendering time, while the
final synthesized accuracy is low. MinkUnet achieves the best results and is
faster than SparseConvNet [18]. We select MinkUnet14A as our point encoder
since it is more efficient than MinkUnet34C.

Point Encoder Training Memory (GB, $\downarrow$) Rendering Time (seconds,
$\downarrow$) PSNR (dB, $\uparrow$) PointNet++ [37] 9.14 3.41 16.53
SparseConvNet [18] 8.18 2.18 18.26 MinkUnet14A [7] 6.95 0.96 18.47 MinkUnet34C
[7] 8.26 1.45 18.45

Table 5: Comparison between different point encoders. We adopt MinkUnet14A [7]
to extract the basic 3D prior since it achieves accurate synthesized results
and is also lightweight.

Effect of Fusion Decoder. Previous neural point encoders usually adopt image-
to-image translator [40, 47] to render images from projected feature maps. We
conduct this experiment to analyze the effect of our Fusion Decoder. We
construct different alternatives by combining different decoder and fusion
strategies, as shown in Tab. 6.

The combination between U-Net [40] and concatenation strategy is the most
frequently adopted [11, 10, 41], while its performance is not high. When
replacing the decoder with PixelShuffle [43], accuracy improves. It shows that
the neural renderer does not require large respective fields as U-Net. When
the concatenate strategy is replaced with our proposed fusion model, the
performance is further improved, showing the rationality of our design.

Decoder Fusion Strategy PSNR ($\uparrow$) SSIM ($\uparrow$) LPIPS
($\downarrow$) U-Net [40] Concatenate 18.04 0.708 0.511 PixelShuffle [43]
Concatenate 18.13 0.712 0.499 PixelShuffle [43] SPADE (LayerNorm) 18.47 0.723
0.484

Table 6: Comparison between different neural renderers.

Effect of Point Cloud Loss. We perform this experiment to validate the effect
of point cloud loss. By setting different loss weight $\lambda_{pc}$, we
obtain the results in Tab. 7. We conclude that point cloud loss indeed
promotes the mapping process from point features to 3D attributes, thus
improving the performance. Interestingly, larger $\lambda_{pc}$ does not
necessarily achieve better accuracy, which demonstrates the minor difference
between point cloud attributes with NeRF’s attributes.

$\lambda_{pc}$ 0.0 0.1 1.0 PSNR (dB, $\uparrow$) 18.23 18.47 18.30

Table 7: The Effect of point cloud loss.

###  4.6 Application: Point Cloud Sampling

Since our point encoder can extract multi-scale point features and predict the
density and color attributes, we upsample the raw point cloud by dense
sampling and predict corresponding 3D attributes in the nearby area of
existing points. As illustrated in Fig. 5, although there are no ground-truth
dense points for us to perform supervised learning, our Point2Pix can still
in-paint the missing points and insert many details for input point clouds.

Figure 5: Our Point2Pix application in point cloud in-painting and upsampling.
We upsample and fix the missing part of the raw point cloud by random and
dense sampling around existing points.

##  5 Conclusion

In this paper, we have proposed a general point renderer, which can be
directly utilized to render photo-realistic images in indoor scenes. We
introduce the advantages of point cloud representation to NeRF where existing
points provide ground truth pairs during training, the point area can guide
the ray sampling process and the 3D prior feature can be generalized to novel
scenes. We propose Multi-scale Radiance Fields to extract discriminative 3D
features, point-guided sampling to efficiently reduce the number of valid
samples, and a Fusion Decoder to synthesize realistic images. Experiments and
ablation studies demonstrate that our Point2Pix achieves state-of-the-art
synthesized performance. Our Point2Pix can also be directly employed to
upsample and in-paint the raw point cloud for indoor scenes.

Limitation and Future Work. There are still common limitations in our proposed
Point2Pix. First, the overall rendering time is long compared with recent
caching-based rendering [53]. Second, apart from indoor scenes, it is still
difficult to directly render photo-realistic images for arbitrary
environments. In future work, we will accelerate the rendering speed by
combining it with other 3D scene representations, such as octrees. We will
also extend our present work to more real-world situations, like the human
body.

## Acknowledgments

This work is partially supported by Shenzhen Science and Technology Program
KQTD20210811090149095.

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# Phase noise of a microwave photonic channel: direct-current versus external
electro-optic modulation ††thanks: This work is partially funded by the Agence
Nationale de Recherche (ANR) Programme d’Investissement d’Avenir under the
following grants: ANR-11-EQPX-0033-OSC-IMP (Oscillator IMP project)
ANR-10-LABX-48-01 (FIRST-TF network), and ANR-17-EURE-00002 (EIPHI); and by
grants from the Région Bourgogne Franche Comté intended to support the above.
††thanks:  The authors are with Université de Franche Comté (uFC), Centre
National de la Recherche Scientifique (CNRS), Institut FEMTO-ST, and
Supmicrotech-ENSMM, Besancon, France.  ††thanks: Enrico Rubiola is also with
the Divsion of Quantum Metrology and Nanotechnology, Istituto Nazionale di
Ricerca Metrologica (INRiM), Torino, Italy. ††thanks: _Corresponding author_ :
R. Boudot<EMAIL_ADDRESS>
Damien Teyssieux, Jacques Millo, Enrico Rubiola, and Rodolphe Boudot

###### Abstract

We characterize the phase noise of a microwave photonic channel, where a 10
GHz signal is carried by an intensity-modulated light beam over a short
optical fiber, and detected. Two options are compared, (i) an electro-optic
modulator (EOM), and (ii) the direct modulation of the laser current. The 1.55
$\mu$m laser and the detector are the same. The effect of experimental
parameters is investigated, the main of which are the microwave power and the
laser bias current. The main result is that the upper bound of the phase
flicker is $-117$ dBrad2 in the case of the EOM, limited by the background
noise of the setup. In contrast, with direct modulation of the laser, the
flicker is of $-114$ to $-100$ dBrad2, depending on the laser bias current
(50–90 mA), and the highest noise occurs at the lowest bias. Our results are
of interest in communications, radar systems, instrumentation and metrology.

##  I Introduction

The generation of low-phase-noise microwaves from optics has found great
interest and extensive use in applications such as high-performance Doppler
radar systems [1], communications [2], low-timing jitter analog-digital
conversion [3], and time and frequency metrology [4, 5]. Additionally, low
noise microwaves carried by an optical beam are required in vapor cell atomic
clocks based on coherent population trapping (CPT) [6] because the phase noise
of the microwave field which interrogates the atoms can limit the clock’s
short-term stability [7].

The purest microwaves are nowadays obtained by frequency-division from cavity-
stabilized lasers using optical frequency combs (OFCs) [8]. The phase noise of
such systems is of the order of $-$172 and $-$107 dBrad2/Hz at 100 kHz and 1
Hz Fourier frequency, respectively, compliant with zeptosecond-level time
fluctuations [9]. However, ultra-stable optical cavities and OFCs are rather
large and fragile pieces of equipment, compared to regular oscillators and
synthesizers. The cavity requires extreme mechanical and thermal stability,
while the OFC relies on the generation of octave-wide supercontinuumm light
and on the simultaneous stabilization of repetition rate and carrier-envelope
frequency offset. Therefore, moving such systems outside metrological labs is
a challenge. Simpler and compact systems have been proposed [10], relying on
the use of a free-running monolithic femtosecond laser [11], or on the
transfer oscillator technique [12, 5], possibly coupled with soliton-
microcombs [13, 14].

The optoelectronic oscillator (OEO) is an alternative option for the
generation of low noise microwaves [15, 16, 17, 18]. The OEO is a delay-line
oscillator, where the microwave delay is implemented with a photonic channel
(laser, intensity modulator, optical fiber and photodetector). The low loss of
the fiber (0.2 dB/km at 1.55 $\mu$m wavelength) enables the implementation of
a long delay, which is equivalent to a large-$Q$ resonator in the feedback
loop. For example, a delay of 10 $\mu$s (2 km fiber) is equivalent to
$Q=~{}3.14{\times}10^{5}$ at 10 GHz carrier, out of reach for room-temperature
resonators (Ref. [19, Chap. 3-4]).

Figure 1: Simplified schematic of the experimental setup used for phase noise
measurements at 10 GHz of the microwave-modulated laser system. The 10 GHz
microwave reference is split into two arms. Setting the switch in position ‘1’
(actually, moving a semirigid cable and terminating the unused input), the
microwave modulates the laser power via the bias current, and the MZ EOM is
replaced with a fiber patch. In position ‘2,’ the laser is in CW mode, and the
beam is modulated by the EOM. The fast photodiode (FPD) extracts the
microwave, which is amplified and compared to the reference. With the RF and
LO inputs saturated and in quadrature, the mixer works as a phase-to-voltage
converter. The Fast Fourier Transform (FFT) analyzer measures the power
spectral density averaged over a convenient number of acquisitions, which
smooths the phase-noise plots.

Two main modulation configurations are seen in OEOs and in CPT clocks, namely,
(i) a fibered Mach-Zehnder electro-optic modulator (EOM) at the output of a CW
laser diode, and (ii) the direct modulation (DM) of a laser diode. The former
option is most often used, and is encountered in the demonstration of state-
of-the-art OEOs [18] and CPT-based atomic clocks [6]. That said, there are
good reasons to abandon the EOM in favor of the direct modulation. The EOM is
large ($\approx 10$ cm long) and expensive, it has large optical loss (4–5 dB,
plus 3 dB intrinsic loss due to the idle point at half power), shows large
temperature sensitivity because of the $\mathrm{LiNbO_{3}}$ waveguide, and
requires large microwave power ($\approx 50{-}100$ mW). In turn, this is
detrimental to the thermal stability, and stabilization of the bias point may
be necessary [20]. Conversely, the direct modulation of the laser requires
only an appropriate network to combine DC bias and microwave in the laser
diode. Several OEO structures based on directly modulated distributed-feedback
lasers [21, 23, 22], vertical-cavity surface-emitting lasers [24] and
microsquare [25] lasers have been demonstrated, with phase noise levels
reaching $-129~{}\mathrm{dBrad^{2}/Hz}$ at 10 kHz Fourier frequency for a 10
GHz carrier. With the same motivation, high-performance CPT-based atomic
clocks using directly-modulated lasers have been reported, exhibiting
competitive short-term stability [26].

The studies discussed focus on the characterization of the overall phase noise
of the output microwave signal, with comparatively little attention to the
noise contribution of the photonic channel, i.e., microwave-to-microwave via
modulated light. Yet, the quantitative knowledge of such contribution is
necessary to assess ultimate phase noise limit. The phase noise of optical
links using DM lasers was considered in [27], but reporting a limited number
of experimental cases. Other studies focus on the numerical [28] or electrical
modelling [29, 30] of DM laser systems. A theoretical analysis on the noise of
links using externally-modulated lasers is reported in [31]. Reference [32]
demonstrates a technique to mitigate the microwave phase noise of a DM laser
using feedback on the laser current. Reference [33] reports on a self-
sustained microwave oscillator, based on a EOM and on a CPT cesium cell as the
resonator in the feedback loop, with a phase noise in agreement with the
Leeson model [34, 19].

In this article, we compare the phase noise of a 10 GHz photonic channel in
the two relevant configurations mentioned, CW laser followed by an EOM, and
directly modulated laser. The flicker phase noise is lower than $-117$ dBrad2
in the case of the EOM-based setup. This is an upper bound, limited by the
background of the measurement system. For comparison, the flicker of high-
speed photodetectors similar to ours is of $-120$ dBrad2 or lower [35]. We
found no data about the flicker of the EOM, but our experience suggests that
it is negligible at this scale. In contrast, the direct modulation gives a
flicker of $-114$ dBrad2 at 90 mA bias current, progressively increasing to
$-100$ dBrad2 when the bias current is reduced to 50 mA. The higher noise,
compared to the CW laser plus EOM, is clearly due to the modulation of the
laser bias current.

##  II Experimental setup

Figure 1 shows a schematic of the experimental setup. The reference microwave
source is a Rohde & Schwarz SMB100A. The laser source is a single-mode
pigtailed distributed feedback (DFB) diode laser (Gooch and Housego AA0701)
with internal isolator, emitting at 1550 nm wavelength. The laser is driven by
a low noise current controller [40] and is integrated in a Butterfly package
with embedded thermistor and Peltier cooler. The laser has an efficiency of
$\sim 0.19$ W/A beyond the 10 mA threshold, delivering 16 mW optical power at
90 mA bias current. The laser is followed by a voltage-controlled optical
attenuator (VOA, IDIL COCOM03898, not shown on Fig. 1), which is used to set
the same optical power at the photodetector input for the two configurations.
The EOM (iXblue MXAN-LN-20, $V_{\pi}$ $\simeq$ 5.5 V) has no temperature
control, and no active control of the bias voltage $V_{b}$. It has a
polarization controller at the input. The fast photodiode, a Discovery DSC30,
is followed by four cascaded HMC606 amplifiers with 9 dB attenuation between
$2^{\mathrm{nd}}$ and $3^{\mathrm{rd}}$, providing a total gain of 35 dB and a
noise figure of 4.8 dB. The output power at the amplifier output is measured
with a power meter (Rohde-Schwarz NRVS) tapping the signal with a 10-dB
directional coupler (Macom PN2020). The mixer (Miteq DB0218) is followed by a
1.9 MHz lowpass filter (Mini-Circuits SLP-1.9+) which eliminates the 2nd
harmonics (20 GHz) and unnecessary high-frequency noise, followed by a 40 dB
low-noise amplifier optimized for lowest flicker [39]. The fast Fourier
transform (FFT) analyzer is a HP3562A. However old, it features efficient
logarithmic frequency resolution. A line stretcher (Arra 9426B) in the lower
arm of the system enables fine adjustment of the quadrature relation at the
mixer inputs. The background noise of the setup is measured by replacing the
photonic channel with a variable attenuator (Arra 6803-10A), so that the
microwave power remains the same as in the two configurations under test.

Figure 2: Phase noise at 10 GHz with the EOM-based laser system. (a) Fixed
bias current ($I_{b}$ = 90 mA) and different microwave powers $P_{\mu W}$. (b)
Fixed microwave power ($P_{\mu W}$ = 18 dBm) and various $I_{b}$ values. The
background noise of the setup is also reported ($P_{\mu W}$ = 18 dBm).  Figure
3: Phase noise at 10 GHz with the DM-laser system, for $P_{\mu W}$ = 18 dBm,
and different values of $I_{b}$. The background noise of the setup is also
reported ($I_{b}$ = 90 mA). TABLE I: Experimental conditions for plots shown
in Fig. 2–3. The quantities $P_{\mu W}$, $P_{o}$ , $P_{\mu 0}$ and
$k_{\varphi}$ are the microwave power at the synthesizer output, the optical
power at the photodiode input, the microwave power at the amplifier input, and
the mixer phase-to-voltage gain, respectively.  (A) parameters of Fig. 2(a)
---
$P_{\mu W}$ (dBm) |  $P_{o}$ (mW) |  $P_{\mu 0}$ (dBm) |  $k_{\varphi}$ (mV/rad)
10 | 3.75 |  $-$30.65 | 179
12 | 3.75 |  $-$28.65 | 223
14 | 3.75 |  $-$26.65 | 268
16 | 3.75 |  $-$24.7 | 320
18 | 3.75 |  $-$23 | 371
(B) parameters of Fig. 2(b)
$I_{b}$ (mA) |  $P_{o}$ (mW) |  $P_{\mu 0}$ (dBm) |  $k_{\varphi}$ (mV/rad)
50 | 1.8 |  $-$30.6 | 207
60 | 2.3 |  $-$27.9 | 258
70 | 2.8 |  $-$25.8 | 296
80 | 3.3 |  $-$24.2 | 334
90 | 3.75 |  $-$23 | 371
(C) parameters of Fig. 3
$I_{b}$ (mA) |  $P_{o}$ (mW) |  $P_{\mu 0}$ (dBm) |  $k_{\varphi}$ (mV/rad)
50 | 0.45 |  $-$30.6 | 200
60 | 0.85 |  $-$27.9 | 245
70 | 1.3 |  $-$25.8 | 290
80 | 1.95 |  $-$24.2 | 323
90 | 2.7 |  $-$23 | 363

##  III Experimental results

We express the phase noise in terms of power spectral density $S_{\varphi}(f)$
of the random phase $\varphi(t)$, as a function of the Fourier (modulation)
frequency $f$. Its physical dimension is rad2/Hz. The usual polynomial
approximation of $S_{\varphi}(f)$ is in our case limited to flicker and white
terms, $\mathsf{b}_{-1}/f$ and $\mathsf{b}_{0}$, respectively. Their units are
$\mathrm{rad^{2}}$ for $\mathsf{b}_{-1}$, and $\mathrm{rad^{2}/Hz}$ for
$\mathsf{b}_{0}$. Albeit the quantity $\mathscr{L}(f)$ is more often seen in
the literature, defined as $\tfrac{1}{2}S_{\varphi}(f)$ [37], $S_{\varphi}(f)$
should be preferred because it is expressed in SI units, while
$\mathscr{L}(f)$ is not. We encourage reading Ref. [38] for a tutorial on
phase noise and Ref. [37] for the commonly agreed terminology.

Figure 2(a) shows the phase noise $S_{\varphi}(f)$ of the EOM configuration at
10 GHz carrier frequency, with fixed laser bias current ($I_{b}=90$ mA) and
different values of the microwave power $P_{\mu W}$ from the microwave source.
The experimental conditions are reported in Table I(A). The flicker
coefficient is $-117~{}\mathrm{dBrad}^{2}$, almost independent of the
microwave power. This fact is consistent with the concept of phase modulation
from a near-DC $1/f$ fluctuation that we have described in [36]. The value
reported is only an upper bound because it equals the background noise of the
setup. Nonetheless, a slight degradation shows up at the lowest power, $P_{\mu
W}=10~{}\mathrm{dBm}$. More sophisticated methods [41] can be envisioned for
determining the flicker limitation set by the EOM. However, the level we
measured is already lower than the phase noise of state-of-the-art microwave
sources [9, 42, 43].

Still on Fig. 2(a), we see that the white region of the background noise is
$-141~{}\mathrm{dBrad^{2}/Hz}$ at 90 mA bias and at maximum microwave power.
This value is satisfactory, to the extent that it is a few dB lower than that
of the photonic channel. The background noise is due to the amplifier noise
(thermal noise and noise figure), and to the shot noise. The former is
$S_{\text{amp}}=FkT=1.25{\times}10^{-20}~{}\mathrm{W/Hz}$ where $F=3$ (4.8 dB)
is the noise factor, and $kT=4{\times}10^{-21}\mathrm{~{}J}$ is the thermal
energy at room temperature. The latter is
$S_{\text{sh}}=2qIR=2.5{\times}10^{-20}$ W/Hz on the $R=25~{}\Omega$ load (the
shot current sees two 50 $\Omega$ loads in parallel, one inside the diode and
one inside the amplifier), but only half of this is transferred to the
amplifier. Thus, amplifier and shot noise give nearly equal contributions, and
the overall noise is $N=2.5{\times}10^{-20}~{}\mathrm{W/Hz}$. According to
[36], the phase noise is $S_{\varphi}(f)=N/P_{\mu
0}=5{\times}10^{-15}~{}\mathrm{rad^{2}/Hz}$ ($-143~{}\mathrm{dBrad^{2}/Hz}$)
with $P_{o}=5~{}\mu W$ ($-23~{}\mathrm{dBm}$). There is a discrepancy of 2 dB.
This is due to a technical problem inside the amplifier. A separate test on
the amplifier alone shows that the white noise exceeds $FkT/P_{\mu 0}$ when
the input power at one of the four stages is comparable or higher than the
compression point, resulting in reduced gain.

Then, we measured the phase noise for different values of $I_{b}$, from 50 mA
to 90 mA in 10 mA steps. The results shown in Fig. 2(b) are obtained in the
conditions detailed in Table I(B). The flicker coefficient is
$-117~{}\mathrm{dBrad^{2}}$ for all $I_{b}$, except for a small degradation at
50 mA. A notable degradation is observed, at fixed microwave power when the
EOM is operated off the maximum-sensitivity condition by changing the bias
voltage $V_{b}$. For example, at $V_{b}$ = 0.6 V (1.3 V below the optimal
operation point), the flicker was degraded to $-100~{}\mathrm{dBrad^{2}}$.

Figure 4: Comparison of phase noise spectra obtained for the DM-and EOM-based
laser systems. The setup background noise is also reported for information.
(a) $I_{b}$ = 90 mA and $P_{\mu W}$ = 18 dBm. (b) $I_{b}$ = 50 mA and $P_{\mu
W}$ = 18 dBm.  Figure 5: Phase noise at 10 GHz at $f$ = 1 Hz (a) and $f$ = 10
kHz (b) for the DM-laser, the EOM-system and the measurement setup.

Figure 3 shows results obtained with DM of the laser current, at fixed
microwave power ($P_{\mu W}$ = 18 dBm) and for several values of $I_{b}$ from
50 mA to 90 mA in 10 mA steps. The lowest phase flicker,
$-114~{}\mathrm{dBrad^{2}}$, is obtained at 90 mA. Reducing the bias current
results in a relevant degradation of the flicker, up to
$-100~{}\mathrm{dBrad^{2}}$ at 50 mA bias. This behavior is due to the
increase of the laser low-frequency amplitude and frequency noise occurring at
low bias current, up-converted to the microwave frequency [29, 30, 28].

Figure 4 provides a direct comparison between the two options, EOM and DM. At
the highest bias value (Fig. 4 (a), $I_{b}=90$ mA and $P_{\mu W}=18$ dBm), the
DM suffers from slightly higher flicker, while the white noise is nearly equal
and just a few dB higher than the background noise of the setup. In contrast,
at low bias (Fig.4 (a), $I_{b}=50$ mA and $P_{\mu W}=18$ dBm) the flicker
noise of the DM laser gets 15 dB higher, while the EOM system is not affected.
The white noise of both increases but, surprisingly, the DM features lower
white noise.

Finally, Fig. 5 shows the typical phase flicker (phase noise at $f=1$ Hz) and
white (phase noise at $f=10$ kHz) for the two options, measured with $P_{\mu
W}=18$ dBm. The flicker noise of microwaves obtained with the DM-laser is
higher than with the EOM system for all bias current values. For bias values
lower than 80 mA, the noise floor obtained with the EOM is higher than with
the DM-laser system and found to increase when $I_{b}$ decreases.

##  IV Conclusions

We have reported on the phase noise of a photonic microwave channel at 10 GHz,
comparing the cases where direct modulation (DM) of the laser current or
external electro-optic modulation is used. The laser system using the EOM
features a flicker noise coefficient lower than $-117~{}\mathrm{dBrad}^{2}$ in
a large range of laser bias current, and microwave power. This is an upper
bound because measurement is limited by the background noise of the setup.
Conversely, the white noise floor of the setup is not a limitation, even
accounting for the 2-dB excess noise due to the microwave amplifier. The
flicker noise of the DM scheme is higher than that of the EOM scheme. We
observed $-114$ dBrad2 for a bias current of 90 mA, degraded to $-100$ dBrad2
for a bias current of 50 mA. For both configurations, flicker phase noise
levels obtained under proper conditions are compliant with the transfer of the
most stable microwave signals.

## Funding

This work has been partly funded by Agence Nationale de la Recherche (ANR) in
the frame of the LabeX FIRST-TF (Grant ANR 10-LABX-0048), EquipeX Oscillator
IMP (Grant ANR 11-EQPX-0033), and EIPHI Graduate school (Grant
ANR-17-EURE-0002) projects.

## Acknowledgments

The authors thank Vincent Giordano (Institut FEMTO-ST) for fruitful
discussions and reading of the manuscript, and Ghaya Baili (Thales RT) and
Brian Sinquin (Institut FOTON), for the providing of some bibliography
references.

## Disclosures

The authors declare no conflicts of interest.

## Data availability statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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