# Retinex-Diffusion: On Controlling Illumination Conditions in Diffusion
Models via Retinex Theory

Xiaoyan Xing1, Vincent Tao Hu311footnotemark: 1, Jan Hendrik Metzen2
Konrad Groh2, Sezer Karaoglu1, Theo Gevers1

1\. UvA-Bosch Delta Lab 2. Bosch Center for Artificial Intelligence 3. LMU
Munich

###### Abstract

This paper introduces a novel approach to illumination manipulation in
diffusion models, addressing the gap in conditional image generation with a
focus on lighting conditions. We conceptualize the diffusion model as a black-
box image render and strategically decompose its energy function in alignment
with the image formation model. Our method effectively separates and controls
illumination-related properties during the generative process. It generates
images with realistic illumination effects, including cast shadow, soft
shadow, and inter-reflections. Remarkably, it achieves this without the
necessity for learning intrinsic decomposition, finding directions in latent
space, or undergoing additional training with new datasets.

Figure 1: Motivated by the recent state-of-the-art image manipulation method
[9] can not handle the low-level manipulation, such as relighting the scene,
or creating illumination effect (top row). We present physics-guided and
training-free diffusion for controlling illumination conditions in images. In
image synthesis, our method generates photo-realistic illumination conditions
under the proper illumination property guidance. Our model is also able to
perform illumination editing of the original images, such as adding new
illumination to images or face relighting (bottom row). Our approach is
training-free and easily integrated with most pixel-based diffusion models,
enhancing their illumination control capabilities efficiently.

††footnotetext: ∗Work was done at University of Amsterdam.

##  1 Introduction

Generative models have shown their ability to create images that closely
resemble real ones. The advent of conditional diffusion models has further
enhanced the generation of specific semantic content [35, 16, 24, 27, 11],
contents [20, 34, 9, 38], layout [50, 19], etc. However, a notable limitation
is their inability to control illuminations precisely in the generated images.
On the other hand, physically-based rendering pipelines like Blender [10] can
achieve high-fidelity illumination but are time-consuming and lack diversity.
In this paper, we combine the generative capabilities of diffusion models with
the guidance of physically-based models to create a novel approach for
controlling illumination conditions in both generated and real images.

Altering illumination conditions is essential for various computer vision and
computer graphics tasks. Typically, this involves decomposing and recomposing
the intrinsic components of a scene [31, 52, 30, 36, 14, 33]. Recently, the
advancement of the generative models provide a way to solve such tasks end-to-
end, e.g., finding the corresponding direction in the Style-GAN’s latent space
[6, 7]. However, these methods often require extensive datasets containing
various intrinsic labels such as surface normals, depth, light sources, or the
direction searching. In contrast, we propose a more straightforward approach
to illumination condition control by harnessing the power of diffusion models
guided by physical principles (Table 1).

In this paper, we propose harnessing the generative process of the diffusion
model as a self-contained rendering pipeline, enabling the manipulation of
illumination conditions through physics-driven guidance. This approach allows
us to circumvent the intricate tasks of decomposing and recomposing scene
intrinsic components. Consequently, it can be seamlessly applied to diverse
datasets and lighting scenarios without the need for additional training,
tuning, or extra data labels.

In summary, the contributions are as follows:

  * •

We regard the diffusion model as a black-box image render and decompose its
energy function according to the image formation model.

  * •

We introduce illumination-controllable image synthesis and geometry-preserving
real image relighting, both guided by physics-based principles.

  * •

Our work provides photorealistic control of illumination in both generated and
real images, delivering results that are on par with, and in some aspects
surpass, models tailored to specific data domains.

  * •

Our approach is entirely devoid of training requirements and does not rely on
any CGI techniques to attain controllable illumination conditions.

##  2 Related work

Conditional Diffusion Models. Diffusion models perform well in various tasks
due to their controllability [50]. This includes image content [34], image
layout [38], audio content [32], and human motion generation [45]. However,
current diffusion models often struggle with fine-grained control over
illumination as illustrated in Figure 1. Our work leverages the knowledge of
illumination to formulate flexible energy functions. Classifier-guidance [13]
demonstrates that diffusion models can be guided by pretraining a noisy-data
based classifier. Classifier-free guidance [17] further eliminates the need
for extra pretraining by randomly dropping out the guidance signal during
training. Self-guided diffusion models [19] even remove the annotation process
altogether. More recently, ControlNet [50] and its variants has been used to
address the illumination condition in diffusion models [25, 48], yet those
methods usually require training Controlnet on extra dataset, resulting in a
complex and costly implementation.

Our method is complementary to all these approaches. To fully leverage the
potential of pretrained diffusion models, we propose an additional energy
function based on illumination theory to guide the generation of the diffusion
models.

Diffusion Based Image manipulation. A substantial body of literature is
dedicated to image manipulation and editing using diffusion models [24, 9, 16,
34, 35]. While these methods harness the remarkable generative capabilities of
diffusion models to modify high to mid-level image attributes such as
semantics [18, 27] and layouts [38, 50], the control of lower-level features,
especially illumination, has been unexplored. To this end, we introduce a
novel illumination manipulation method tailored for conditional image
generation based on illumination. Our approach involves decomposing the
illumination-related properties during the generative process and applying
targeted guidance based on these properties.

Notably, our training-free method can be easily integrated into major
diffusion models, enhancing their ability to adjust illumination conditions
on-the-fly. This contrasts with [36], which is limited to human faces and
relies on trained and pre-trained parameters.

Intrinsic Image Decomposition. Intrinsic image decomposition methods [31, 52,
33, 2, 21, 29, 51, 46, 12, 4, 5, 3] aim to decompose an image into
illumination-relevant and illumination-irrelevant parts. Recent diffusion-
based methods [49, 26] have shown impressive decomposition results, indicating
that diffusion models have an awareness of intrinsic.

Our method builds on one of the earliest vision techniques, retinex theory
[28], for solving intrinsic decomposition. It extracts the illumination
property of the image and approximates it as the illumination-irrelevant
component of the image at a very low cost. We employ the simple decomposition
results as hints to guide the pre-trained diffusion models to generate images
with designated illumination.

| Neural | Directional | DiFaReli | Ours
---|---|---|---|---
| rendering[52, 31] | search [7, 6] | [36] |
Neural backbone | CNN, Transformer | Style-GAN | Diffusion | Diffusion
label-free | light mask, depth | ✓ | light source, cam paras | ✓
End to End | $\times$ | ✓ | $\times$ | ✓
Different types of data | $\times$ | $\times$ | $\times$ | ✓
Diverse ouputs | $\times$ | Some of them | $\times$ | ✓

Table 1: Comparison with previous illumination control methods.

##  3 Preliminaries

To illustrate the challenges in modifying illumination attributes within
diffusion models, we begin by comparing the image synthesis process of
diffusion models with the image formation of real-world images.

###  3.1 Real-world Image Formation

The image formation can be modelled by [40]:

$\displaystyle
I=m(\mathbf{n},\mathbf{l})\int_{\omega}{f_{c}(\lambda)e(\lambda)\rho(\lambda)d\lambda}$
(1)

where $\mathbf{n}$ represents the surface normal vector, and $\mathbf{l}$
denotes the direction of the light source. The term $m$ represents the
geometric relationship in the interaction. Additionally, $\lambda$ indicates
the wavelength, $f_{c}(\lambda)$ the camera spectral sensitivity, $e(\lambda)$
the spectral power distribution of the illuminant, $\rho$ the surface
reflectance.

Assuming the sensor response is linear and the wavelength of visible spectrum
is narrow ($\lambda_{I}$), Equation (1) is simplified as:

$I=m(\mathbf{n},\mathbf{l}){e(\lambda_{I})\rho(\lambda_{I})}.$ (2)

Then, at point $\mathbf{x}$ on the surface, the decomposition of
$I(\mathbf{x})$ can be approximated by:

$I(\mathbf{x})=R(\mathbf{x})\odot{S(\mathbf{x})},$ (3)

where $\odot$ represents the Hadamard product, $R(\mathbf{x})$ is the
reflectance, which is illumination-invariant, and $S(\mathbf{x})$ is the
shading, which is illumination-variant. By manipulating shading, one can
control the illumination properties in a physics-based manner.

###  3.2 Diffusion Image Generation

Diffusion models [41, 18, 44] gradually perturb data using a forward diffusion
process and then reverse the process to reconstruct the original data, as
explained in previous studies [43, 18, 13, 1]. Let $q({\bm{x}}_{0})$ denote
the unknown data distribution in ${\mathbb{R}}^{D}$. The forward diffusion
process, indexed by time $t$ as $\\{{\bm{x}}_{t}\\}_{t\in[0,T]}$, can be
succinctly represented by the following forward Stochastic Differential
Equation (SDE):

$\displaystyle d{\bm{x}}={\bm{f}}({\bm{x}},t)dt+g(t)d{\bm{w}},$ (4)

where ${\bm{w}}\in\mathbb{R}^{D}$ is a standard Wiener process,
${\bm{f}}(\cdot,t):\mathbb{R}^{D}\to\mathbb{R}^{D}$ is the drift coefficient
and $g(t)\in\mathbb{R}$ is the diffusion coefficient. The $f({\bm{x}},t)$ and
$g(t)$ are related to the noise size and determine the perturbation kernel
$q_{t|0}({\bm{x}}_{t}|{\bm{x}}_{0})$ from time $0$ to $t$.

Let $q_{t}({\bm{x}})$ be the marginal distribution of the SDE at time $t$ in
Equation (4). Its time reversal can be described by another SDE [44]:

$\displaystyle\mathrm{d}{\bm{x}}=[{\bm{f}}({\bm{x}},t)-g(t)^{2}{\bm{s}}({\bm{x}},t)]\mathrm{d}t+g(t)\mathrm{d}\overline{{\bm{w}}},$
(5)

where $\overline{{\bm{w}}}$ is a reverse-time standard Wiener process with
$\mathrm{d}t$ as an infinitesimal negative timestep, and
${\bm{s}}({\bm{x}},t)=\nabla_{\bm{x}}\log q_{t}({\bm{x}})$ represents the
score. The score, similar to energy, allows us to introduce an additional
energy function ${\mathcal{E}}(\cdot,\cdot,\cdot)$ into the reverse SDE
process for the specific guidance. We will further discuss the design of the
energy function for illumination-based guidance in later sections.

##  4 Method

As illustrated in Figure 2, our method performs two tasks: 1) it can control
the illumination conditions of the generated image, and 2) it can apply new
lighting conditions on real images. To accomplish these task, we first
reformulate the energy function in the diffusion process. Then, we introduce
the illumination guidance in image synthesis. Finally, we propose geometry
preserved relighting for real images. Notably, this pipeline requires no
further training, nor extra data labels or CGI techniques.

Figure 2: Overall diagram of the proposed illumination control diffusion
method. Top, the illumination guidance image generation; Bottom, image
relighting. $x_{t}$ represents the image at time step $t$, $\epsilon_{\theta}$
is the pre-trained U-Net [39]. $y_{s}$ and $y_{c}$ are prompts for
illumination guidance and the geometry persevering guidance.

###  4.1 Illumination Energy Decomposition

We reconsider the reversed SDE processing of diffusion image generation based
on the image formation model. As mentioned in the preliminary, an image can be
decomposed into illumination-variant $I$ and illumination-invariant $R$ parts.
Therefore, we decompose the energy function
${\mathcal{E}}({\bm{y}},{\bm{x}},t)$ by the sum of two log potential functions
[8]:

$\displaystyle{\mathcal{E}}({\bm{y}},{\bm{x}},t)$
$\displaystyle=\lambda_{I}{\mathcal{E}}_{I}({\bm{y}},{\bm{x}},t)+\lambda_{R}{\mathcal{E}}_{R}({\bm{y}},{\bm{x}},t)$
(6)
$\displaystyle=\lambda_{I}\mathbb{E}_{q_{t|0}({\bm{x}}_{t}|{\bm{x}})}{\mathcal{S}}_{I}({\bm{y}},{\bm{x}}_{t},t)$
$\displaystyle+\lambda_{R}\mathbb{E}_{q_{t|0}({\bm{x}}_{t}|{\bm{x}})}{\mathcal{S}}_{R}({\bm{y}},{\bm{x}}_{t},t),$

where ${\bm{y}}$ is our target guidance for energy,
${\mathcal{E}}_{I}(\cdot,\cdot,\cdot):\mathbb{R}^{D}\times\mathbb{R}^{D}\times\mathbb{R}\rightarrow\mathbb{R}$
and
${\mathcal{E}}_{R}(\cdot,\cdot,\cdot):\mathbb{R}^{D}\times\mathbb{R}^{D}\times\mathbb{R}\rightarrow\mathbb{R}$
are the log potential functions, ${\bm{x}}_{t}$ is the perturbed source image
in the forward SDE, $q_{t|0}(\cdot|\cdot)$ is the perturbation kernel from
time $0$ to time $t$ in the forward SDE,
${\mathcal{S}}_{R}(\cdot,\cdot,\cdot):\mathbb{R}^{D}\times\mathbb{R}^{D}\times\mathbb{R}\rightarrow\mathbb{R}$
and
${\mathcal{S}}_{I}(\cdot,\cdot,\cdot):\mathbb{R}^{D}\times\mathbb{R}^{D}\times\mathbb{R}\rightarrow\mathbb{R}$
are functions measuring similarity between the target guidance and perturbed
source image. $\lambda_{R}\in\mathbb{R}{>0}$ and
$\lambda_{S}\in\mathbb{R}{>0}$ are weighting hyper-parameters for
illumination-variant and illumination-invariant components, respectively.

In the reverse process, given by Equation (5), and adopting a step size of
$h$, the iteration rule from $s$ to $t=s-h$ is as follows when applying our
new illumination-based energy function:

$\displaystyle{\bm{x}}_{t}$
$\displaystyle={\bm{x}}_{s}-[{\bm{f}}({\bm{x}},s)-g(s)^{2}({\bm{s}}({\bm{x}}_{s},s)-\nabla_{\bm{x}}{\mathcal{E}}({\bm{y}},{\bm{x}},s))]h$
(7) $\displaystyle+g(s)\sqrt{h}{\bm{z}},$

where ${\bm{z}}\sim{\mathcal{N}}({\bm{0}},{\bm{I}})$.The expectation in
${\mathcal{E}}({\bm{y}},{\bm{x}},s)$ is estimated by the Monte Carlo method of
a single sample for efficiency. In the following section, we will introduce
the construction of the energy function based on related illumination
theories.

###  4.2 Illumination-Energy Guided Generation

#### Prompt for Illumination Energy

Illumination is related to the brightness of a pixel. [36] shows that by
matching the mean value difference between the reversed sampling and generated
sampling, the consistency of global brightness of human portrait can be
realized. This can be explained by Equation (3), where the reflectance $R$ of
the human portrait can be approximated as a constant. Therefore, the intensity
of pixel is directly reflecting illumination $S$. However, this simplification
falls short when dealing with more complex surface reflectance. Therefore, an
additional step of illumination property extraction is required.

We modify the multi-scale retinex model [37] to extract illumination
properties. The illumination is computed from the denosied image $x_{t}$ at
the candidate step $t$. First $x_{t}\in\mathbb{R}^{H\times{W}\times 3}$ is
filtered by 2D Gaussians at multiple scales, to capture the image’s varying
level of geometry and illumination. Subsequently, these blurred images are
combined using either uniform or user-defined weights. The processing is
formed as:

$f_{s}(m_{t},n_{t})=\sum_{r,g,b}\sum_{k=1}^{N}w_{k}\cdot\left(\frac{1}{2\pi\sigma_{k}^{2}}e^{-\frac{i^{2}+j^{2}}{2\sigma_{k}^{2}}}*x_{t}(m,n)\right),$
(8)

where $f_{s}(m_{t},n_{t})$ is the estimated lighting at pixel $(m,n)$ at given
timestep $t$, $\\{i,j\\}$ represents the position of the Gaussian center,
$\sigma_{k}$ is the standard deviation for the $k^{th}$ scale, and $*$ donates
the convolution.

#### Illumination Energy Guidance

We model the target illumination condition by parameterizing the illumination
map as a composition of $N$ 2D Gaussian functions $G(\cdot)$ as:

$y_{s}=\sum_{i=1}^{N}\alpha_{i}G(\mu_{i},\Sigma_{i}),$ (9)

where its mean $\mu_{i}$ represents the position of the light source (for the
visible light source) or the location of the brightest parts of the image (for
the invisible light source), and its covariance matrix ($\Sigma_{i}$)
describes the spread and directionality of the light. $\alpha_{i}$ is the
weight of corresponding Gaussian, which is subject to $\Sigma{\alpha_{i}}=1$.

After the illumination property $f_{s}(.)$ and the illumination prompt $y_{s}$
are computed, their difference can be quantified using any differentiable
similarity metric. In practice, the pixel-wise mean square error is used to
calculate this difference, which is defined as:

${\mathcal{S}}_{I}(y_{s},{\bm{x}},t)=\sum_{x\in{\bm{x}}}||f_{s}(x_{t})-y_{s}||_{2}.$
(10)

###  4.3 Illumination-Energy on Real-world Image

The illumination-based guidance generation can be extended to modify or add
lighting conditions in real-world images, as shown in Figure 2. This involves
two key steps: firstly, an inverse processing stage to convert the input image
into a latent code, and secondly, a reflectance-based correction to maintain
the original image geometry while altering the lighting. This approach enables
flexible lighting manipulation while preserving the original image’s core
structure and details.

#### Real-image inversion

Given an input image $x_{0}\in\mathbb{R}^{H\times{W}\times 3}$, we utilize the
inverse process of DDIM [42] converts it to a latent
$x_{z}\in\mathbb{R}^{H\times{W}\times 3}$ at the final forward DDIM step $z$.
Hence, given $x_{z}$ as input, the illumination-guided generative process is
able to change the light condition of the real image. Yet, the edited image
may not have the exact same geometry, since the guidance on shading may also
disturb the geometry.

#### Geometry persevering guidance

Another key contribution of our method is the reflectance-based correction for
preserving the geometry of the edited image. Similar to the condition on the
illumination property, the geometry property can be regarded as part of
illumination-invariant component ${R}$. And it can be represented as the
surface reflectance difference. We apply a gradient-based method to extract
such difference by calculating the cross color ratios (CCR) [15] of the
surface.

Given an image $x_{0}\in\mathbb{R}^{H\times{W}\times 3}$ to be edited and two
neighbouring pixels $p_{1}$, $p_{2}$. The CCR’s are calculated as:

$M_{RG}=\frac{R_{p_{1}}G_{p_{2}}}{R_{p_{2}}G_{p_{1}}},M_{RB}=\frac{R_{p_{1}}B_{p_{2}}}{R_{p_{2}}B_{p_{1}}},M_{GB}=\frac{G_{p_{1}}B_{p_{2}}}{G_{p_{2}}B_{p_{1}}},$
(11)

where, $M_{RG},M_{RB}$ and $M_{GB}$ are the CCR for the color channel pairs
$(R,G)$, $(R,B)$ and $(G,B)$, respectively. To simplify the writing, we use
$M_{RG}$ to represent the CCR for all channel pairs, in the following
statements. Given the image formation in Equation (2), and the illumination
condition are close in a small neighborhood [12], we obtain:

$\displaystyle\log{M_{RG}}=\log(\rho^{R_{p_{1}}})+\log(\rho^{G_{p_{2}}})$ (12)
$\displaystyle-\log(\rho^{R_{p_{2}}})-\log(\rho^{G_{p_{1}}}).$

Therefore, the CCR are illumination invariant and only depends on reflectance
transitions. For more mathematical details, please refer to the supplementary
material section. The final normalized target CCR matrix
$\textbf{C}_{x_{0}}\in\mathbb{R}^{H\times{W}\times 3}$ is
[$M_{RG},M_{RB},M_{GB}$]. The operation to compute the CCR matrix for a image
is expressed as: $f_{c}(\cdot)$.

Similarly, the CCR matrix of the generated image at time step $t$ can be also
calculated in the same manner, denoted by $\textbf{C}_{x_{t}}=f_{c}(x_{t})$.

In a manner similar to the illumination prompting, we define the geometry-
related prompt as $y_{c}$. Hence, the discrepancy between the target CCR
$y_{c}$ and the computed CCR $f_{c}(x_{t})$ at sampling step $t$ is quantified
as follows:

${\mathcal{S}}_{R}(y_{c},{\bm{x}}_{t},t)=\sum_{x\in{\bm{x}}}||f_{c}(x_{t})-y_{c})||_{2}.$
(13)

Then, the anti-gradient of the reflectance difference
$-\nabla_{\textbf{C}_{t}}\mathcal{L}$ is sent to the diffusion process
together with the anti-gradient of the illumination difference
$-\nabla_{S_{t}}\mathcal{L}$, to achieve the geometry preserved relighting in
the real image editing.

##  5 Experiments

#### Pre-trained diffusion models.

Our approach leverages the generative capabilities of pre-trained diffusion
models. A summary of the models used in our study, along with the datasets on
which they are pre-trained, is as follows:

  * •

ADM [13]: pretrained on LSUN-Bedroom[47] for generating and editing the
diverse illumination conditions of bedrooms.

  * •

EDM [23]: pretrained on FFHQ dataset for generating diverse illumination
conditions of human faces.

  * •

DDIM [42]: pretrained on CelebA-HQ [22] for editing illumination conditions of
human faces.

Figure 3: Illumination Property-Guided Image Generation: Each pair of columns
displays the generated image alongside its corresponding illumination feature.
The initial two columns show the original image without illumination guidance
and its illumination feature. Subsequent columns illustrate images generated
under various specific lighting conditions, with the illumination direction
indicated by a sphere. Figure 4: Illumination effect control of an invisible
light source. Given the same illumination direction prompt, our method is able
to generate multiple variants of illumination effect with respect to the
strength of the light source.

###  5.1 Illumination Guidance in Image Synthesis

First, we demonstrate the effectiveness of our method in generating photo-
realistic images with different illumination variants, including different
light direction, shadow, and strength of illumination. Then, we evaluate the
illumination guidance by comparing it with the human-instructed image to image
generative model [9]. In the case of the human-instructed model, we employ
text prompts that specify fixed illumination conditions to generate images. In
contrast, our method directly integrates illumination prompts to guide the
generation process.

Figure 3 and Figure 5 illustrate the images generated under varying
illumination condition guidance. The associated illumination features are
displayed alongside the images. The results showcase the photo-realistic
generation of illumination conditions, such as turning on a lamp or increasing
outdoor illumination, and so on.

Figure 4 highlights our method’s application in controlling the illumination
effect of a new light source. By using consistent illumination direction
prompts, it illustrates the ability to produce a range of illumination
effects. These variations correspond to the intensity of the light source,
illustrating the flexibility of our method in adjusting light intensity within
a scene.

Figure 6 visually demonstrates the results of increasing the illumination
coming from the window. These results can be compared with those obtained
using human-guided image-to-image generative methods, Instruct-Pix2Pix [9],
which struggles on generating the image with the prompted illumination, our
method generates vivid illumination with our illumination guided generation.
The hyper-parameters of [9] are carefully selected to obtain the stable
results.

Table 2 presents the quantitative evaluation of our method for generating
images under specified illumination conditions. We use MSE as the metric to
measure the discrepancy between the illumination extracted from the generated
images and the given illumination prompt. Across three distinct illumination
conditions, our method consistently achieves a better MSE score compared to
the baseline method without guidance and [9] with our carefully designed
prompt, underscoring the effectiveness of ours.

| Illumination | Illumination | Illumination
---|---|---|---
| condition #1 | condition #2 | condition #3
Instruct-Pix2Pix [9] | 0.0896 | 0.0835 | 0.0346
ADM [13] | 0.0887 | 0.0823 | 0.0344
Ours (ADM+illum. guided) | 0.0757 | 0.0706 | 0.0311

Table 2: Quantitative results of the illumination map generated by Instruct-
Pix2Pix[9](w. text prompt guidance), ADM (w/o. illumination guidance) and ours
(w. illumination guidance).  Figure 5: Illumination Property-Guided Image
Generation (Using EDM [23]): Each pair of columns displays the generated image
alongside its corresponding illumination feature. Illumination direction
indicated by a sphere.  Figure 6: Visual comparison with state-of-the-art
image to image diffusion model [9] in the context of generating new lighting
conditions.

###  5.2 Illumination Editing in Real-world Image

We further evaluate our methods on illumination editing in synthetic and real-
world images. Figure 7 demonstrates our method’s capability in illumination
editing on various images. The sequence comprises the original input image,
its inversion, and the final relighted image, each accompanied by their
corresponding illumination features. Remarkably, our method skillfully adjusts
the illumination direction while maintaining the original geometry, showcasing
its effectiveness even on colorful surfaces.

Figure 8 shows real-world image relighting using our method, which operates
without learned intrinsics [31] or directional searches in latent space [6].
The loss of high-frequency detail is attributed to the limitations of DDIM
inversion, as our guidance does not significantly alter the geometry.

Figure 9 showcases our real-face relighting results. Our method surpasses the
performance of the state-of-the-art image-to-image manipulation method by [9]
(we carefully designed their text prompt.). Additionally, our results are
comparable to those obtained by the specialized face relighting method of
[36].

Figure 7: Geometry-Preserved Image Relighting: The sequence from left to right
displays the original input image, the inverted image, and the image
relighting results under three different lighting conditions (denoted as
Relight 1, Relight 2, and Relight 3). This progression demonstrates the
effectiveness of our method in maintaining geometric consistency while
altering illumination conditions. Illumination direction indicated by a
sphere. Figure 8: Real indoor image relighting comparison: The left column
displays real images captured with invisible light turned on and off,
illustrating the ground truth. The middle-left column showcases images
relighted by the method proposed by Li et al. [31], which utilizes multiple
sub-networks for intrinsic decomposition and generates new images based on
these modified intrinsics. The middle-right column shows StylitGAN’s [6]
relighting, applying directional search in the latent space. The right column
presents our results using illumination-guided diffusion, notably achieving
relighting without learned intrinsic decomposition, directional search, or CGI
techniques. Figure 9: Real face image relighting comparison: The left column
presents the real face image for relighting. Middle-left column, the lateset
diffusion based face relighting method [36], which is conditioned on multiple
intrinsic and extrinsic components, and needs multiple pretrained methods to
decompose those components. Middle-right column, human instructed image to
image generative method [9], fails badly on relighting. Right column, our
method is able to relight the face without learned intrinsic decomposition,
directional search, or CGI techniques.  Figure 10: Ablation comparison of the
illumination property extraction. Top row: illumination guided generation
without extracting the illumination property. Bottom row: illumination guided
generation with the extracted illumination property.

###  5.3 Ablation Study & Discussion

Illumination property extraction. As introduced in Section 4.2 , by extracting
the illumination property, one can distinguish the illumination condition from
the composited image generation. Here, we evaluate the illumination guidance
without extracting the illumination property. As shown in Figure 10, we guide
the image synthesis process based on the same illumination prompts, while we
gradually increase the lighting range of the light prompts, the content of the
image guided by the non-illumination specific property shifts rapidly. Also,
images are much blurry compared to the ones with extracting illumination
properties. Such results clearly showcase the effectiveness of the
illumination property extraction.

Sampler | Batch | Sampling | Batch Time (s) | Batch Time (s)
---|---|---|---|---
| Size | steps | w/o. guidance | w. guidance
ADM [13] | 16 | 100 | 102.63 | 110.85
DDIM [42] | 8 | 40 | 11.59 | 15.63
EDM [23] | 64 | 15 | 4.17 | 4.35

Table 3: Comparison of generation times (per batch) with and without
illumination guidance using ADM [13], DDIM [42], and EDM [23] on a Pytorch
framework. Tests were conducted on an Nvidia RTX4090 GPU.

Computational efficiency. The computational cost analysis of our method (Table
3), reveals minimal additional burden on the guided diffusion model.
Specifically, the inference time for image generation/editing, using standard
resolutions, batch sizes, and sampling steps, shows only a marginal increase.
An increment of 8.2 seconds is shown for ADM [13], 0.17 seconds for EDM [23],
and 4.04 seconds for DDIM [42], affirming the efficiency of ours.

Other intrinsic decomposition methods. Theoretically, the retinex model can be
replaced by any other decomposition models, including learning-based models.
We employ the multiscale retinex model for following reason: 1) it is
naturally differentiable, and can be directly used to calculate the score; 2)
it is computational cheap, since it only contains few convolution operations,
it will not significantly increase the sampling time.

##  6 Conclusion

We proposed a novel and physics-based, training-free method for manipulating
illumination in both diffusion-generated and real images. Our method realizes
accurate illumination-conditioned image generation by reforming the energy
function of diffusion models based on the image formation model. Our approach
does not require any type of extra training or direction research. It is easy
to be embedded into current diffusion models. By prompting the illumination-
related feature, the diffusion model is able to generate/adjust illumination-
related semantics for proper illumination conditions, such as turning on the
lamp and opening the window.

Limitation. Such illumination control is not always accurate, and it is
aligned to the learned data distribution of pretrained diffusion models. A
deeper investigation on extracting intrinsics of the diffusion models is
needed in the future.

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Supplementary Material

##  Appendix A Cross Color Ratios

Section 4.3 introduced one can use cross color ratios to represent part of the
illumination-invariant components. Here we provide the detail to prove the CCR
is illumination-invariant. Taking the logarithm on both side of the Equation
(11), we get:

$\log{M_{RG}}=\log{R_{p_{1}}}+\log{G_{p_{2}}}-\log{R_{p_{2}}}-\log{G_{p_{1}}}.$
(14)

Given the image formation in Equation (1) and [4], at certain point, we
obtain:

$C_{p_{1}}=m_{b}(\mathbf{n},\mathbf{s}){e^{C_{p_{1}}}(\lambda)\rho_{b}^{C_{p_{1}}}(\lambda)}+\\\
m_{s}(\mathbf{n},\mathbf{s},\mathbf{v}){e^{C_{p_{1}}}(\lambda)\rho_{s}^{C_{p_{1}}}(\lambda)},$
(15)

where $C_{p_{1}}$ represents the color channel $C$ at pixel $p_{1}$, for an
input image. $\rho_{b}$ and $\rho_{s}$ represent the body diffuse refletance
and specular refletance respectively. Since the two neighbouring pixels
$p_{1}$ and $p_{2}$, the same illumination condition can be assumed.
Therefore,

$e^{C_{p_{1}}}=e^{C_{p_{2}}}.$ (16)

Then, Equation (12) can be derived. It is important to note that the Cross-
Color Ratio (CCR) is solely influenced by the surface reflectance.
Consequently, the CCR remains invariant to changes in illumination.


# Electrically tunable spin-orbit coupled photonic lattice in a liquid crystal
microcavity

Marcin Muszyński  Institute of Experimental Physics, Faculty of Physics,
University of Warsaw, Poland  Przemysław Oliwa  Institute of Experimental
Physics, Faculty of Physics, University of Warsaw, Poland  Pavel Kokhanchik
Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal,
F-63000 Clermont-Ferrand, France  Piotr Kapuściński  Institute of Experimental
Physics, Faculty of Physics, University of Warsaw, Poland  Eva Oton  Institute
of Applied Physics, Military University of Technology, Warsaw, Poland  Rafał
Mazur  Przemysław Morawiak  Wiktor Piecek  Institute of Applied Physics,
Military University of Technology, Warsaw, Poland  Przemysław Kula  Institute
of Chemistry, Military University of Technology, Warsaw, Poland  Witold
Bardyszewski  Institute of Theoretical Physics, Faculty of Physics, University
of Warsaw, Poland  Barbara Piętka  Institute of Experimental Physics, Faculty
of Physics, University of Warsaw, Poland  Daniil Bobylev  Université Clermont
Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal, F-63000 Clermont-
Ferrand, France  Dmitry Solnyshkov  Université Clermont Auvergne, Clermont
Auvergne INP, CNRS, Institut Pascal, F-63000 Clermont-Ferrand, France
Institut Universitaire de France (IUF), 75231 Paris, France  Guillaume
Malpuech<EMAIL_ADDRESS>Université Clermont Auvergne, Clermont
Auvergne INP, CNRS, Institut Pascal, F-63000 Clermont-Ferrand, France  Jacek
Szczytko<EMAIL_ADDRESS>Institute of Experimental Physics, Faculty
of Physics, University of Warsaw, Poland

###### Abstract

We create a one-dimensional photonic crystal with strong polarization
dependence and tunable by an applied electric field. We accomplish this in a
planar microcavity by embedding a cholesteric liquid crystal (LC), which
spontaneously forms a uniform lying helix (ULH). The applied voltage controls
the orientation of the LC molecules and, consequently, the strength of a
polarization-dependent periodic potential. It leads to opening or closing of
photonic band gaps in the dispersion of the massive photons in the
microcavity. In addition, when the ULH structure possesses a molecular tilt,
it induces a spin-orbit coupling between the lattice bands of different
parity. This interband spin-orbit coupling (ISOC) is analogous to optical
activity and can be treated as a synthetic non-Abelian gauge potential.
Finally, we show that doping the LC with dyes allows us to achieve lasing that
inherits all the above-mentioned tunable properties of LC microcavity,
including dual and circularly-polarized lasing.

##  I Introduction

The physics of microcavities turned out to be extremely rich within the last
decades and directly related to the diversity of materials that can be
embedded inside. These embedded materials can deeply modify the cavity mode
properties, and conversely, the material response can be affected by the
cavity, even beyond its optical properties. A large variety of semiconductor
materials have been utilized, ranging from inorganic semiconductors [1, 2] to
organics [3], 2D materials [4], moiré heterostructures [5] or DNA molecules
[6]. One example is when semiconductor excitons strongly couple to cavity
modes, giving rise to hybrid exciton-polariton modes [7, 8] with mixed light-
matter properties. One can also cite quantum Hall materials, which can have
their unique topological insulator character broken by the coupling to the
cavity modes [9], and also polariton chemistry, where chemical reactions are
modified by the strong coupling of molecules with the confined light modes
[10]. The essential degrees of freedom that have to be controlled in photonic
systems, and, in particular, in microcavities, are the polarization and
orbital angular momentum of light. In that perspective, the spin-orbit
coupling (SOC) of light appears as a crucial tool [11, 12, 13, 14, 15]. In
microcavities, an intrinsic light SOC is due to the TE-TM spliting of photonic
modes which is at the origin of the optical spin Hall effect [12, 16] and of
the topologically non-trivial nature of photonic modes in microcavities [17].
Another key ingredient for photonic mode engineering in these systems is the
in-plane potential implementing a periodic lattice. The most efficient
approach to date is to etch the cavities to realize lattices of micropillars
[18, 19, 20]. Combining topologically non-trivial photonic modes induced by
SOC and appropriate periodic potentials led to the observation of photonic
topological insulators [21, 22], topological lasers [23, 24], and lasers
carrying non-zero orbital angular momentum [25].

The materials known for their unique capabilities to control light
polarization on demand, and therefore perspective for topological photonics,
are liquid crystals (LCs). Their photonic properties have been studied for
more than one century, but mostly in the context of light reflection and
transmission, which gave rise to various applications such as displays,
optical filters, adaptive optical elements, lasers, etc [26, 27]. Recently, a
nematic liquid crystal was embedded in a microcavity, which allowed to
synthesize a tunable photonic SOC [28] and to observe a variety of original
effects such as photonic persistent spin helix [29, 30] or Hermitian and non-
Hermitian topological transitions [31]. Another LC phase which not explored so
far in that context is the cholesteric LC (CLC), which is induced by doping a
nematic LC with a chiral dopant. As a result, the LC molecules self-assemble
with a twist, producing a helical structure called uniform lying helix (ULH).
The helical axis can be oriented parallel to the substrate in a structure by
imposing a topological frustration. Naturally, a lying helix induces a
periodic variation of refractive index. Moreover, CLCs are known to possess a
quick response to an applied electric field, through the flexoelectric effect
[32].

In this work, we report the fabrication of structures, where a CLC is embedded
in a microcavity. We demonstrate that the resulting ULH induces an in-plane
periodic potential acting on the photonic modes. It results in the formation
of a spin-polarized lattice band structure, reversibly tunable by an external
voltage controlling the CLC molecular orientation. We show that when two
cavity modes are brought in resonance, photonic SOC efficiently couples the
periodic potential bands of different parity and linear polarization. This
realizes an analogue of a 1D chain with two components per site, being tunable
and showing topological properties similar to the Su-Schrieffer-Heeger chain.
Finally, by introducing a dye within the microcavity, we demonstrate the
capability of our platform to achieve spin-polarized and dual lasing.

##  II Results

Figure 1: ULH texture inside optical microcavity. (a) Scheme of the dye-doped
optical microcavity with an embedded stabilized ULH structure. The rows of
molecules whose directors are oriented vertically are colored in red to
indicate the period of the lattice. (b) Polarizing optical microscopy image of
light transmitted through a ULH texture embedded inside LC microcavity.

The fabrication of our samples depicted in Fig. 1a is described in the Methods
section. The samples are based on an optical microcavity composed of two
SiO2/TiO2 distributed Bragg reflectors (DBRs), covered with transparent
electrodes of indium tin oxide (ITO). The cavity is filled with a nematic LC
and a chiral dopant, which induces a helical twist in the LC matrix, producing
a supramolecular helical structure with a helical pitch (double period, $p$)
of the order of several micrometers [33, 34]. By treating the orienting layer
with homeotropic anchoring, the surfaces impose a topological frustration that
induces a ULH, with the helical axes parallel to the substrate. Cross-
polarized transmission microscopy shows a variable periodic refractive index
(Fig. 1b) confirming the ULH configuration [35]. The variety of possible
applications of the proposed system increases with the integration of light
emitters, so we doped the system with the organic laser dye pyrromethene 580
(P580), which, when externally excited, can transit the system to a nonlinear
regime with emitted polarized laser light.

The proposed optical platform can be fabricated using a range of different
liquid crystal mixtures. To demonstrate the versatility of our approach and
various ULH-induced effects, we prepared two samples. Sample A, with a higher
birefringence of the CLC mixture, was used in the first part of the work to
demonstrate band structure tunability by an external electric field. Sample B
has a CLC mixture with lower birefringence but exhibits a deeper photonic
potential thanks to the higher homogeneity along the cavity axis $z$. It was
used in the second part for generating a 1D lattice in the presence of SOC and
for demonstrating lasing.

Figure 2: The tunable band structure. Illustrative representation of the LC
director arrangement inside the microcavity and the corresponding schematic
dispersion relations of the horizontal ($H$, pink) and vertical ($V$, green)
modes for (a) zero and (b) non-zero voltage. Effective refractive index
ellipses for $H$ and $V$ polarizations showing periodical dependence along the
ULH axis $y$ over half pitch $p$ and obtained by averaging director
distributions shown in (a) and (b) along microcavity axis $z$ for (c) zero and
(d) non-zero voltage, respectively. Reflectivity spectra of (e-g) horizontally
and (h-j) vertically polarized band structures measured for different voltages
(indicated in top right corners) applied to the sample A.

First, we study sample A. An optical cavity transforms photons into 2D massive
quasiparticles [36]. These quasiparticles experience a potential defined by
the effective refractive index obtained by averaging along the microcavity
axis $z$. This averaging can be easily performed for a known LC director
distribution inside the microcavity. The director distribution in the absence
of an external voltage naturally consists of a perfect ULH aligned with the
$y$ axis and superimposed director fluctuations from the perfect ULH-defined
orientation (Fig. 2a) [37]. The effective refractive index distribution
obtained by averaging along $z$ for both polarizations is shown in Fig. 2c in
the form of effective refractive index ellipses. Remarkably, only $H$
polarization (pink, along $x$ axis) has a periodic variation of refractive
index (potential), while $V$ polarization has a constant refractive index
(green, along $y$ axis). This should result in gapped $H$-polarized dispersion
and continuous $V$-polarized dispersion, as shown in the right part of Fig.
2a. Under a non-zero applied voltage, the molecules inside the microcavity are
reoriented according to the so-called dielectric effect [38]. It induces a
gradual increase of the refractive index contrast for both polarizations with
growing voltage, thus deepening both potentials, opening a gap for $V$
polarization, and increasing the already existing gap for $H$ polarization
(Fig. 2d). For more details, see Section I.A of [39].

Our experiment based on angular-dependent reflectivity measurements (see
Methods and Section II in [39]) confirms these expectations. Without any
voltage applied to the sample (Fig. 2e,h), the bandgap is visible only in the
$H$ polarization, while the spectrum remains gapless (within the spectrometer
resolution and line broadening) in the $V$ polarization. The so-called
$s$-band of lowest energy is observed in the $H$ polarization (denoted
$H_{s}$, $E\approx 2.1$ eV). Under non-zero external voltage, an energy gap
emerges in the $V$ polarization as well, as shown in Fig. 2i,j, opening from 0
meV at 0.0 V to 20 meV at 3.7 V. The whole band structure in $V$ polarization
is shifted towards higher energies. The process of gap opening is reversible,
as the liquid crystal director fluctuations recover when the applied voltage
is turned off. We also observed the increase of the gap for $H$ polarization,
from 12 meV at 0.0 V to 25 meV at 3.7 V (Fig. 2f,g). Beside the symmetric
$s$-band, another gap opens in $H$ polarization for higher energies (around
2.14 eV), thus forming an additional antisymmetric $p$-band (denoted $H_{p}$),
as it is known in solid state physics and photonic structures [24].

Figure 3: Spin-orbit coupling in ULH LC microcavity. (a) (top) Illustration of
the LC director distribution inside the microcavity and (bottom) effective
refractive index ellipses. $sp$-hybridized bands obtained from the tight-
binding Hamiltonian Eq. (2) with colors corresponding to (b) $S_{1}$ and (c)
$S_{3}$ Stokes parameters ($S_{2}=0$ everywhere); the inset shows a scheme of
hybridization. The $k$-space reflectivity spectra measured for (d) total
intensity ($S_{0}$ Stokes parameter) and orthogonal linear (e) horizontal $H$
and (f) vertical $V$ polarization of light. The degree of polarization is
expressed by Stokes parameters $S_{1}$, $S_{2}$, and $S_{3}$ in (g,h,i)
reciprocal and (j,k,l) real space, respectively. The left panels of the Stokes
parameter maps correspond to the experimental data, while the right panels
represent the semi-analytical model (see Section V in [39]). The ISOC-mixed
eigenstates appear near 2.0 eV as a result of coupling between $p$-band of $H$
polarization and $s$-band of $V$ polarization.

Therefore, we have demonstrated that sample A shows an on-demand
reconfigurability of the band structure. It is important to stress that the
strong optical activity (OA) of a CLC helix [40] is suppressed for sample A
because of the mirrors, restoring the inversion symmetry. Consequently, the
$H$ and $V$ modes are uncoupled. However, this is not always the case as we
show below. If the inversion symmetry is broken, then the linear polarizations
should be coupled by an anti-symmetric SOC term, in the simplest case, linear
in the wavevector. In this general case, an effective Hamiltonian describing
the cavity modes of two polarizations $H$ and $V$ in the polarization-
dependent periodic potential created by the ULH on the ($H$,$V$) basis in real
space reads:

$H_{cont}=\begin{pmatrix}-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
y^{2}}+\frac{A_{H}}{2}[1-\cos{\left(2qy\right)}]-\frac{\delta}{2}&-2i\alpha\frac{\partial}{\partial
y}\\\ -2i\alpha\frac{\partial}{\partial
y}&-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
y^{2}}+\frac{A_{V}}{2}[1-\cos{\left(2qy\right)}]+\frac{\delta}{2}\end{pmatrix},$
(1)

with $\hbar$ the reduced Planck constant, $m$ the mass of cavity photon,
$\delta$ the linear birefringence, $q=2\pi/p$, and $p$ the pitch of the ULH.
$A_{H,V}$ are the potential amplitudes for $H$ and $V$ polarizations. As
discussed for sample A, $A_{V}$ can be switched on and off, and $A_{H}$ can be
modified by applying a voltage. The coefficient $\alpha$ describes the SOC
resulting from the OA, which is zero for sample A, as explained above. This
coupling term is called the Rashba-Dresselhaus SOC (RDSOC), since it appears
in the case of balanced Rashba and Dresselhaus SOC contributions in
electronics. The physics offered by this term has already been studied in the
context of birefringent microcavities [28, 41], although the microscopic
origin of the term was different.

In order to combine the effects of the periodic potential and the SOC, we
prepared sample B with a different CLC material. The origin of SOC for sample
B lies in the particular texture of the director distribution inside the
microcavity at zero voltage (Fig. 3a, top), different from sample A. This
distribution combines the homeotropic (vertical) anchoring of the molecules at
the mirrors, perfect ULH structure, and a tilt of the molecules around the $x$
axis (the role of director fluctuations is not crucial for sample B). Both
straight and tilted director distributions of samples A and B, respectively,
are stable. The particular choice of one of them depends on the used LC,
deposition techniques, anchoring strength, etc [42]. Because of the presence
of the tilt, the inversion symmetry of the system is broken, and the SOC,
suppressed for a straight molecular orientation of sample A, is restored in
sample B (see section I.B of [39] for details). The necessity of the tilt for
SOC appearance is additionally confirmed by the exact solution of Maxwell’s
equations based on the Finite Element Method (see Section III of [39]). In
sample B, both polarizations have a non-zero effective refractive index
contrast ($A_{H,V}\neq 0)$ even at zero voltage (Fig. 3a, bottom), which opens
multiple gaps in their dispersions.

In that regime, it is possible to go further and derive from the continuous
Hamiltonian (1) a tight-binding Hamiltonian describing the SOC between the
orthogonally-polarized bands of the periodic potential, as detailed in Section
IV of [39]. Indeed, in the absence of the SOC, each polarization exhibits a
band structure formed from its own ladder of states: $s$, $p$, $d$, and so on.
The SOC term is an odd function of $k$, and so it couples only the localized
states of opposite parity. For instance, the $s$-band in $V$ polarization
($V_{s}$) and the $p$-band in $H$ polarization $H_{p}$ are coupled by the SOC,
giving rise to what can be called an interband SOC (ISOC). It is worth noting
that this contrasts with earlier works on SOC in birefringent cavities, in
which the opposite parity of states was satisfied by coupling even and odd $H$
and $V$ modes [28, 41]. Here, the $H$ and $V$ modes have the same parity,
therefore the even subband $s$ is coupled to the odd subband $p$. The
corresponding tight-binding Hamiltonian reads:

$H=\begin{pmatrix}E_{H_{p}}-2t_{pp}\cos{(k_{y}p/2)}&-i\alpha\left[t_{sp,0}-2t_{sp,1}\cos{(k_{y}p/2)}\right]\\\
c.c.&E_{V_{s}}-2t_{ss}\cos{(k_{y}p/2)}\end{pmatrix},$ (2)

where $E_{H_{p}}$ and $E_{V_{s}}$ are the onsite energies of $H_{p}$ and
$V_{s}$, respectively, $t_{pp}$ and $t_{ss}$ are the nearest-neighbor
couplings between $p$ states of $H$ and $s$ states of $V$, respectively,
$t_{sp,0}$ and $t_{sp,1}$ are the onsite and nearest-neighbor couplings
between $H_{p}$ and $V_{s}$ (ISOC terms), respectively. All couplings are
calculated from the Hamiltonian (1) (including the ISOC terms) using the
standard perturbation theory on the basis of $s$ and $p$ states of a harmonic
oscillator. Explicit expressions are provided in Section IV of [39]. In the
limit of strongly localized modes ($\alpha t_{sp,1},\ t_{ss},\ t_{pp}\ll\alpha
t_{sp,0}$), the spectrum of the Hamiltonian (2) is shown in Fig. 3b,c, with
color representing the Stokes parameters $S_{1}$ and $S_{3}$.

We performed the experiment for sample B in order to confirm our predictions
(Fig. 3d-l, left panels). Fig. 3d-f present the angle-resolved reflectivity
spectra measured for total intensity (Stokes parameter $S_{0}$), horizontal
$H$, and vertical $V$ polarizations of detected light, respectively. The right
parts of the panels are obtained by a model beyond the tight-binding limit
based on a semi-analytical solution of Maxwell’s equations [43] using the
coefficients of the Fourier decomposition of the dielectric tensor of the
material inside the cavity as fitting parameters (details are given in Section
V of [39].

The presence of a deeper photonic potential, compared to Fig. 2, leads to the
opening of a greater number of photonic bandgaps, as well as a deeper
localization of light: a large effective mass of the trapped photons makes the
lowest bands almost flat. In a system with uncoupled $H$ and $V$
polarizations, each of them should possess a band structure with its own
ladder of states. However, we observe that the states around 2 eV, expected to
be $H_{p}$ and $V_{s}$, show up in both polarizations. Fig. 3g,h,i show the
corresponding Stokes parameters $S_{1}$, $S_{2}$, and $S_{3}$, measured in the
reflectivity in $k$-space. It is evident that each of these two coupled states
contains both $H$ and $V$ contributions. Furthermore, in the circular
polarization degree $S_{3}$ in Fig. 3i, these states form a check pattern
which is extremely well reproduced by the tight-binding model (Fig 3b,c) and
the semi-analytical model (right part of the panel). These features are also
well reproduced by the exact numerical solution of Maxwell’s equations with a
ULH dielectric tensor including molecular tilt (Section III of [39]). In
addition, Fig. S3 of [39] shows the influence of the detuning between the
bands $H_{p}$ and $V_{s}$ on the degree of circular polarization $S_{3}$
measured in $k$-space.

Stokes parameters of spatially-resolved ($r$-space) transmission spectrum are
shown in Fig. 3j-l. The lowest energy states in Fig. 3j portray the spatial
periodicity of the refractive index corresponding to the half-pitch of ULH.
The check pattern similar to that in Fig. 3i is visible in the Stokes
parameter $S_{2}$ (Fig. 3k) for $sp$-hybridized states at energies around 2
eV, which further proves the presence of the ISOC in the system.

As an ultimate demonstration of the viability and usefulness of our design and
the possibilities that it offers, we harnessed the spin-polarized bands to
manipulate the polarization of laser emission. For an optically-pumped dye-
doped nematic LC microcavity, lasing takes place at the minimum energy of the
$H$-polarized mode, which better overlaps spectrally with the dye gain curve
[30]. Due to the presence of the photonic potential and the coupling of the
$H_{p}$ and $V_{s}$ modes, for the ULH microcavity, the energy transfer
between the split states provides lasing in both orthogonal linear
polarizations of light. Fig. 4a shows the Stokes parameter $S_{1}$ of angle-
resolved emission spectrum above the lasing threshold for pumping energy of
0.36 µJ/pulse (see Fig. S4 in Section II of [39] for photoluminescence
intensity and linewidth as a function of pumping power). Our experiments
demonstrate that lasing occurs not only for the ground state $H_{s}$, but also
for the ISOC mixed states, preserving the polarization structure of the bands.
In this regime, the characteristic check pattern in the circular polarization
degree $S_{3}$ is inherited (Fig. 4b).

Figure 4: Polarization conserving lasing from ULH LC microcavity. Stokes
parameters (a) $S_{1}$ and (b) $S_{3}$ of angle-resolved emission spectra
collected above the lasing threshold for 0.36 µJ/pulse pumping energy.

##  III Discussion

The effective $2\times 2$ Hamiltonian (1) describing the interaction of bands
in the continuous limit has a particularly interesting interpretation in terms
of a non-Abelian gauge potential [28] (similar to the Yang-Mills gauge
potential [44, 45]), described by the RDSOC constant $\alpha$. Its non-Abelian
nature allows it to produce effects, such as Zitterbewegung [46], even for a
constant gauge potential. The ULH, which is known to exhibit various
topological defects [47] characterized by the change of the RDSOC magnitude
and orientation, offers even broader opportunities: the variation of RDSOC
(corresponding to a non-zero gauge field) can be expected to lead to spin-
polarized photonic Landau levels [48] exhibiting non-zero orbital angular
momentum, localized at the topological defects of the LC texture. The change
of sign of the RDSOC has already been observed in the present sample (not
shown), and the observation of the Landau levels will be a subject for future
works.

Another important outlook of our work is related to the topological properties
of the tight-binding Hamiltonian (2): similar to the tight-binding Hamiltonian
of the Su-Schrieffer-Heeger model [49], it can exhibit a topological
transition with a change of the winding number [50] when $t_{sp,0}=2t_{sp,1}$,
corresponding to the formation of the topological edge states for a finite
chain. While such transition is not observed for the $s$ and $p$ bands in our
sample, it might potentially be observed for different bands having another
tunneling coefficient ratio. In general, the observation of RDSOC in a 1D
lattice opens many exciting possibilities, such as the control of the
amplitude and phase of the tunneling coefficients [51] and, combined with
polarization-dependent losses and gain, it could allow the observation of
exceptional points [31] and of the non-Hermitian skin effect [52].

Finally, the results obtained under strong pumping (Fig. 4) demonstrate that
our ULH microcavity can serve as a device with simultaneous laser emission of
linearly and circularly polarized non-degenerated lines (dual lasing [53]).

##  IV Conclusions

We have demonstrated a versatile photonic platform based on a cholesteric
liquid crystal optical microcavity showing an embedded self-organizing
photonic crystal potential. The simple and efficient manufacturing process
allows us to obtain, on a macroscopic scale, an electrically-tunable one-
dimensional periodic lattice with spin-orbit coupling. In the absence of SOC,
the resulting bands are linearly polarized and can be dynamically tuned by the
external electric field. The SOC present in the system causes the bands of the
periodic lattice of different polarization and parity to couple, leading to
the hybridization of states. Our results are well described by the solution of
Maxwell’s equations, but also by a comprehensive effective Hamiltonian. We
utilized ISOC-hybridized states to create a spin-polarized (circularly-
polarized) laser operating at room temperature. This work paves the way for
studying the combined effects of SOC, or, more generally, non-Abelian gauge
fields, and periodic lattices, including possible topological transitions,
with practical applications that include dual spin-polarized lasing.

##  V Methods

###  V.1 Sample preparation

Both samples (A and B) contained the DBRs centered at 550 nm with 6 pairs of
SiO2/TiO2 layers, which were deposited onto commercially available ultra-flat
polished glass substrates for STN technology with transparent ITO electrodes.
The two DBR substrates were assembled together in a sandwich-like cell,
separated by calibrated silica spacers ranging between 2 $\upmu$m and 3.5
$\upmu$m and creating a sample wedge. An orienting polymer layer with
homeotropic anchoring was used to align the liquid crystal. The liquid crystal
mixtures were prepared with a nematic base, a chiral dopant and a laser dye
dopant: Sample A (Fig. 2) contained a highly birefringent liquid crystal
mixture W-2091 ($n_{o}=1.565$, $n_{e}=1.907$, $\Delta n=0.342$ at
$\lambda=600\,\mathrm{nm}$) with a chiral dopant at a concentration of 0.65%,
while sample B (Fig. 3-4) contained liquid crystal mixture E7 ($n_{o}=1.522$,
$n_{e}=1.737$, $\Delta n=0.215$ at $\lambda=600\,\mathrm{nm}$) and a chiral
dopant concentration of 0.70%. The chiral dopant used was R-5011 (Daken
Chemical) and both mixtures were doped with organic dye P580 at a
concentration of 1%. Producing a stable ULH structure requires precise control
of the confinement ratio. The confinement ratio is the relationship between
the cavity thickness $d$ and helical pitch $p$: $C=d/p$, and it was kept in a
range of $C=1.85-2.75$. This factor is crucial for obtaining a periodic
structure with a well-defined alignment and a pitch as small as possible, and
it allowed the formation of uniform ULH structures with a decreased number of
dislocations. By controlling the chiral dopant concentration, we are able to
adjust the ULH pitch.

###  V.2 Experimental details

The optical measurements are performed at room temperature. The scheme of the
experimental setup is shown in Fig. S2 (Section II of [39]). In reflectivity
measurements, the microscope objective with 60x magnification, numerical
aperture NA = 0.7, and coverglass correction ring is used for both the
excitation and the collection of light. The spot has a diameter of around 10
$\upmu$m. In transmission and photoluminescence measurements, an additional
microscope objective with 5x magnification and NA = 0.15 is used for
excitation. The set of a quarter-wave plate, a half-wave plate, and a linear
polarizer is used to collect light in the selected polarization. For emission
measurements, the sample is excited by a $Q$-switched diode-pumped laser with
a shot-on-demand experiment (532-nm center wavelength, 2-ns pulse duration).
To tune the dispersion relation of the cavity modes, the LC microcavity is
addressed by an AC waveform generator with a 100 Hz square signal and varying
amplitude of 0 – 5 V.

## Acknowledgments

This work was supported by the National Science Centre grants
2019/35/B/ST3/04147, 2019/33/B/ST5/02658 and 2022/47/B/ST3/02411,
2023/51/B/ST3/03025, and the Ministry of National Defense Republic of Poland
Program – Research Grant MUT Project 13-995 and MUT University grant (UGB) for
the Laboratory of Crystals Physics and Technology for the year 2021 and the
European Union’s Horizon 2020 program, through a FET Open research and
innovation action under the grant agreements No. 964770 (TopoLight).
Additional support was provided by the ANR Labex GaNext (ANR-11-LABX-0014),
the ANR program "Investissements d’Avenir" through the IDEX-ISITE initiative
16-IDEX-0001 (CAP 20-25), and the ANR project "NEWAVE" (ANR-21-CE24-0019).

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