Structural Color and Mueller Matrix Analysis in a Ferrocell

Abstract

This study investigates the magneto-optical properties of a ferrofluid using an accessible Ferrocell device. Our findings demonstrate that the ferrofluid’s behavior is critically dependent on its concentration. At high concentrations, the medium is optically dense, with inter-particle scattering and absorption dominating, which prevents the formation of clear light patterns. However, with intermediate dilution, the system enters a “pattern formation zone” where the magnetic field effectively aligns the nanoparticles, creating complex, visible light patterns like horocycles. The appearance of these patterns provides evidence of field-induced ordering and structural coloration. The colors observed are not due to pigments, but result from the interaction of light with the periodic structures formed by the aligned nanoparticles. Our analysis, supported by the Mueller matrix framework, confirms that the ferrofluid acts as a retarder. The birefringence induced by the magnetic field varies across the film, leading to a chromatic dispersion that selectively suppresses certain wavelengths. This process explains how a specific color, such as blue, can be blocked at one location while others pass through, creating structural colors observed in the patterns.

1. Introduction

Ferrofluids continue to be a captivating area for fundamental research and hold significant promise for diverse technological advancements, primarily owing to their extraordinary response to magnetic fields [1,2,3,4,5,6]. While the broader field of magneto-optics of ferrofluids has seen considerable progress, current research endeavors are actively transitioning from merely observing these effects to meticulously engineering them for sophisticated applications [7,8,9,10,11,12,13,14]. A key focus within this evolving landscape is the in-depth understanding and precise harnessing of the underlying physics that govern structural colors and tunable optical properties in these unique materials.

The formation of intricate luminous patterns within thin ferrofluid films, particularly in devices like the Ferrocell [15,16,17], shown in Figure 1, presents a visually striking phenomenon that garners widespread public interest. Despite the existence of excellent studies characterizing the intrinsic physical properties of ferrofluids, there remains a notable gap in comprehensive research specifically dedicated to explaining the formation of these easily observable luminous patterns. Unraveling the mechanisms behind these macroscopic visual effects, which are directly perceivable to the naked eye, is crucial for advancing both fundamental understanding and practical applications.

Optically, the behavior of ferrofluids is intimately linked to their structure, which in turn is precisely controlled by their composition, particle concentration, and the strength of any applied magnetic field. In the absence of an external magnetic field, a colloidal solution of magnetic nanoparticles is optically isotropic and exhibits no net magnetization. This isotropic state is a direct consequence of the random orientation of the easy axes of the constituent nanoparticles. However, upon exposure to a uniform external magnetic field, the magnetic moment of each nanoparticle experiences a torque, tending to align with the applied field [18,19,20]. This alignment occurs primarily through two distinct mechanisms: Néel relaxation, where the magnetic moment rotates within the nanoparticle’s crystal lattice relative to its easy axis, and Brownian relaxation, involving the physical rotation of the entire nanoparticle to align its easy axis with the external field.

Therefore, our current investigation focuses specifically on devices like the Ferrocell, with particular emphasis on the role of structural colors in shaping various light patterns.This research aims to move beyond simplistic explanations based solely on light scattering and absorption. Instead, it seeks to explore how the magnetic-field-induced self-assembly of nanoparticles into chains or clusters can function as sophisticated photonic crystalsor diffraction gratings, thereby giving rise to the complex and often vivid chromatic effects observed.

2. Materials and Methods

In this work, permanent magnets served as the source of the external magnetic fields applied to the Ferrocell. The ferrofluid within the Ferrocell defines a specific plane. We have used Ferrotec EFH1 (Ferrotec USA Corporation, Livermore, CA, USA), which is a ferrofluid with a light hydrocarbon carrier liquid and a composition of magnetite nanoparticles, with saturation magnetization of 44 mT (440 G), with density at 25 °C of 1.12 × 10³ kg/m³, and initial magnetic susceptibility of 2.64. In the present work, we tested the formation of patterns for different solutions for the composition of the ferrofluid, as shown in the graph in the figure. The ferrofluid and light hydrocarbon were measured independently by volume. They were then mixed and left to rest for one week to ensure a homogeneous particle distribution and stabilization of the solution. Following this period, the solution was placed in the Ferrocell and used continuously over a span of three years. Throughout this time, we observed no changes in the luminous patterns or optical properties of the fluid, which confirms the excellent stability and durability of our solution for the purposes of this study.

To summarize the main results of this test with different solutions, we have the images inserted in the graph of Figure 2, where we have three different scenarios: one without dilution, another with 4% dilution and another with 2% dilution.

In the first scenario, using undiluted ferrofluid, no luminous patterns were observed. The Ferrocell had a uniform reddish-brown appearance. This result suggests that the high concentration of magnetic material (7.7% by volume) causes intense scattering and strong absorption of the incident white light. The saturation of particle aggregation in a magnetic field is likely too high, resulting in a dense, homogeneous medium that prevents visible interference or diffraction effects from appearing. With the ferrofluid diluted at a 1:1 ratio, the concentration of magnetic material drops to 3.85%. In this case, a luminous pattern with clear branches was noted. These branches appeared in the regions where the magnetic field was closest to the magnets. The entire plate took on a beige hue, indicating a greater interaction of light with the now more dispersed particles, allowing some light to pass through and form patterns. In the last case, the concentration of magnetic material is reduced to 2.0%. The Ferrocell became highly transparent, displaying a very clear and white luminous pattern. Most notably, in areas of maximum magnetic field intensity, colored spots with a rust-like hue were observed within the luminous pattern. The appearance of specific colors in regions of high magnetic intensity reinforces the idea that the ferrofluid’s dynamically formed structure, influenced by dilution and the magnetic field, acts similarly to natural structural colors, selectively interacting with white light to produce distinct chromatic effects.

In order to apply the algebra of Mueller matrices in this work, we conducted polarization measurements using the apparatus shown in Figure 3. The quantitative effects of the magnetic field on light’s polarization, which are central to our analysis, can be seen in Figure 4 [21]. For more information and algebraic details on our methodology, please refer to our previous work.

To understand the optics behind the existence of structural colors, our investigation will explore two complementary aspects of photonic crystals. First, we will analyze reflectance versus wavelength in a simplified 1D photonic crystal, specifically examining the TE mode. Second, we will construct photonic band diagrams for a 2D ferrofluid photonic crystal, focusing on the TM mode using a simplified model. This combination offers a comprehensive approach, bridging theoretical concepts of photonic crystal behavior with observable optical phenomena, directly connecting them to the unique properties of ferrofluids.

In Figure 5, we present the spectrum of a Ferrocell illuminated by a lamp, whose spectrum is shown in the inset. Based on literature on ferrofluid optical properties, such as transmittance [22,23], birefringence [24,25] and dichroism [26], we have found that the optical characteristics of magnetic fluids can be precisely controlled by manipulating both nanoparticle concentration and the application of external magnetic fields.

3. Evidence of Structural Colors

In this section, we explore aspects of color formation beyond the illumination of the Ferrocell with white light. We use white light, which is a combination of red, green, and blue wavelengths, as well as monochromatic sources of red (650 nm), green (532 nm), and blue light (405 nm).

3.1. Horocycles and Laser Diffraction at Colored Points—Macroscopic Evidence of StructuralColor

To systematically investigate the magneto-optical effects in the Ferrocell, we utilized the distinctive optical pattern known as the Luminous Horocycle [15,16], as it is shown in Figure 6.

This horocycle pattern is observed within the thin ferrofluid film when illuminated by a point light source emitting a spherical wavefront positioned behind the Ferrocell plate, relative to the observer, and a single magnetic pole is present. Figure 7 further illustrates the phenomena observed with multiple superimposed horocycles generated using white light, obtained at an optimal dilution ratio best suited to luminous pattern formation. In Figure 7a, with one of the magnet’s poles positioned directly against the glass plate and illuminated by a white light source, several overlapping horocycles converge to form two prominent white branches. These branches distinctly converge towards a small region of maximal magnetic field intensity, as further detailed in the inset of Figure 7a, with a yellowish color.

In Figure 7b, we have another magnetic field configuration, which forms three distinct regions of intense magnetic field. These regions appear as colored points, each magnified in the insets just below. The bar graphs indicate the intensity of each color in the composition of these luminous points. We can see that in the yellowish point in Figure 7a, blue is suppressed. In the luminous points of Figure 7b, however, either red or blue is suppressed relative to the green color.

To establish a comparison parameter for understanding the attenuation of each color, we measured the intensity of white light reflected by the Ferrocell. As shown in Figure 8, in the region of the least intense magnetic field from Figure 7a, all three colors are easily obtained without a significant decrease in intensity. This serves as a baseline for our analysis.

3.2. Optical Horocycles

In order to observe the magneto-optical effects across different wavelengths, we have obtained three Luminous Horocycles with monochromatic colors, as depicted in Figure 9 for red (Figure 9a), green (Figure 9b), and blue light(Figure 9c).

We can see that in the region of the most intense magnetic field, at the lower part of the horocycle, the intensities of red and green light remain unchanged, as shown in Figure 9a,b, respectively. However, a significant change occurs in the blue horocycle in Figure 9c, where the intensity of blue light decreases by 18%, giving the impression of a cyan color.

3.3. Laser Diffraction at Colored Points

In combination, the macroscopic visual patterns (horocycles with colors) tell us what is happening (structural color, field-induced patterns), we also explored the microscopic laser diffraction experiments, as it is shown in Figure 9d–f.

The experiments involving laser diffraction at colored points provide direct, quantitative evidence of how the organized internal structure of the ferrofluid is directly interacting with light through interference and diffraction, which is the mechanism behind the observed structural colors. When a monochromatic laser (a single color of light) is directed at these specific colored points within the horocycle, the resulting diffraction pattern offers more detailed information about the underlying microstructure for different aspects, such as verification of nanoparticle alignment and spacing, correlation with laser wavelength, and insights into aggregate morphology and orientation.

Based on the observed patterns, we can say that the particles are randomly spaced in this region and are aligned perpendicularly to the thin film’s plane, which is why we have a halo around the central luminous spot of direct laser incidence. We observed that the halo is more prominent for green light (Figure 9e) compared to red light (Figure 9d) and blue light (Figure 9f).

The intensity of light scattering is highly dependent on a particle’s size relative to the wavelength of the incident light, as well as the spacing between particles. Our observations indicate that the particles in the film are specifically optimized to scatter green light more efficiently than red or blue light.

Red light has a longer wavelength, which may result in less efficient scattering if the particle size is too small in relation to it. Blue light, with its shorter wavelength, is scattered, but its scattering efficiency is likely lower than that of green light. This is because, according to the principles of Mie scattering, the transition point where green light scattering becomes more efficient than blue light scattering typically occurs when the particle diameter is approximately 400−600 nm.

This finding suggests that the longer wavelength of the red laser diminishes its ability to effectively resolve the average particle spacing of 4.3 μm. The loss of efficient blue light scattering begins when the particle diameter becomes larger than the wavelength of blue light and approaches or exceeds the wavelength of green light, which agrees with some results found in the literature [27].

Our results show that we have gone beyond the basic principles of Rayleigh scattering, where particle size is much smaller than the wavelength of light, and entered the Mie scattering regime. The fact that the green halo is more prominent strongly suggests that this system has reached a critical transition point.

4. Discussion and Analysis

Now we explore some aspects of light polarization in this experiment using models and light polarization experiments.

4.1. Simplified Models

Using some photonic crystal (PC) models, first we have studied a model of the Reflectance vs. Wavelength (1D Photonic Crystal, transverse electric (TE) mode), because this directly demonstrates the optical filtering capabilities of a photonic crystal, as it is shown in Figure 10a–c. These plots present the modeled reflectance spectra for a photonic crystal as a function of wavelength. The crystal structure comprises 10 alternating layers of two materials, each with a thickness of 100 nm and refractive indices of n_a = 1 and n_b = 2. The data is calculated for various angles of incidence, θ_in, and for the transverse electric polarization mode.

When light hits the crystal, certain wavelengths are strongly reflected (high reflectance), indicating the presence of a photonic bandgap (PBG) for those wavelengths. This is an easily measurable and intuitive way to visualize a PC’s function.

In addition to the 1D case, we have studied the Photonic Band Diagrams (2D Ferrofluid Photonic Crystal, transverse magnetic (TM) mode, using a simplified model), in Figure 10d with no magnetic field, and in Figure 10e with a magnetic field. These diagrams are the theoretical backbone of photonic crystals. They plot the allowed frequencies (or energies) for light propagation within the crystal against its wave vector (direction of travel). The “gaps” in these bands directly correspond to the photonic bandgaps (PBGs), the frequencies of light that cannot propagate through the crystal. This model helped us in the fundamental understanding, because it provides insight into why the bandgaps exist, stemming from the periodic arrangement of refractive indices. By focusing on a ferrofluid photonic crystal, it directly links the general concept to our research material, showcasing how the magnetic field-induced structures influence light [28,29,30,31,32,33,34,35].

Together, these plots allow us to visually demonstrate the filtering effect (reflectance), and to explain the underlying physics of why that filtering occurs (band diagrams). The 2D plot show how these principles apply to ferrofluids, a material whose optical properties can be actively controlled by external magnetic fields, by using simplified models. The key conclusion here is that the behavior of the ferrofluid depends on concentration. At lower concentrations, it acts as a simple reflector. However, with stronger magnetic fields, the ferrofluid exhibits structural coloration due to its altered reflection properties for different wavelengths of light.

4.2. Mueller Matrix and Ferrofluid

When a magnet’s pole is placed against the Ferrocell, the resulting light pattern, as shown in Figure 11, resembles a Maltese cross. This effect is a result of light polarization. Notably, in the regions of highest magnetic intensity, the blue color is almost completely suppressed, giving those areas a yellowish hue.

To find the Mueller matrix that represents the light coming out of the ferrofluid, we need to consider the initial state of the light and how it is transformed by the optical components. The light’s polarization state is represented by a Stokes vector (S), and the transformation by a material is described by a Mueller matrix (M). The light coming out of the ferrofluid is the result of the initial light passing through the linear polarizer and then the ferrofluid.

The algebraic expression for the Mueller matrix of a ferrofluid film under a radial magnetic field is a matrix whose elements are a function of position, typically described in polar coordinates (r,ϕ). This is because both the orientation and the magnitude of the birefringence change across the film [19,27,30].

Let usdefine the parameters based on the pattern obtained in Figure 11. First we have the orientation angle (θ). The magnetic field lines are radial, and the nanoparticles align with them. Therefore, the fast axis of the birefringence at any point on the film is also radial. In polar coordinates, the angle of a radial vector is simply the polar angle ϕ. So, we have θ(r,ϕ) = ϕ. Another parameter is the retardance (δ), which depends on the magnetic field strength, which decreases as you move away from the center (r), as were show in Figure 4. Thus, the retardance is a function of the radial distance, δ(r). A simple model for the field decrease might be an inverse power law, and thus δ(r)∝1/r^p for some power p.

By substituting these spatially dependent parameters into the general Mueller matrix for a linear retarder [29,30], we get the following matrix for the ferrofluid film:

$$M\left(r,\varphi \right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos 2\left(2\varphi \right)+\sin 2\left(2\varphi \right)\cos \delta \left(r\right)& \cos \left(2\varphi \right)\sin \left(2\varphi \right)\left[1-\cos \delta \left(r\right)\right]& -\sin \left(2\varphi \right)\sin \delta \left(r\right)\\ 0& \cos \left(2\varphi \right)\sin \left(2\varphi \right)\left[1-\cos \delta \left(r\right)\right]& \sin 2\left(2\varphi \right)+\cos 2\left(2\varphi \right)\cos \delta \left(r\right)& \cos \left(2\varphi \right)\sin \mathsf{\delta }\left(r\right)\\ 0& \sin \left(2\varphi \right)\sin \delta \left(r\right)& \cos \left(2\varphi \right)\sin \mathsf{\delta }\left(r\right)& \cos \delta \left(r\right)\end{array}\right)$$

(1)

This matrix shows how the retardance (δ) changes with radial distance from the magnet and how the orientation (ϕ) changes with the polar angle. The combination of these position-dependent parameters is what produces the Maltese cross pattern when viewed between crossed polarizers.

The Stokes vector S_out(r,ϕ) of the light coming from the ferrofluid film at any point (r,ϕ) is the product:

S_out(r,ϕ)=M(r,ϕ)⋅S_LP

(2)

By performing this matrix multiplication, we get the Stokes vector for the light exiting the ferrofluid film:

$$S_{\mathrm{out}}\left(r,\varphi \right)=\left(\begin{array}{c}{S}_0\\ S_1\\ \begin{array}{c}{S}_2\\ S_3\end{array}\end{array}\right)={\displaystyle \frac{{\mathrm{I}}_0}{2}}\left(\begin{array}{c}\begin{array}{c}{\sin }^2\left(2\varphi \right){\sin }^2\left(\delta /2\right)\\ {\sin }^2\left(2\varphi \right){\sin }^2\left(\delta /2\right)\end{array}\\ \begin{array}{c}\begin{array}{c}{\displaystyle \frac{1}{2}}\sin \left(4\varphi \right){\sin }^2\left(\delta /2\right)\\ 0\end{array}\end{array}\end{array}\right)$$

(3)

This vector describes the polarization state of the light at every point (r,ϕ) on the final image, such that each element of this vector represents a physical quantity, which we will see now. The first element S₀ represents the total intensity of the light at that point. The formula I_final = S₀ = I₀/2 $\left[{\sin }^2\left(2\varphi \right){\sin }^2\left({\displaystyle \frac{\delta }{2}}\right)\right]$ shows how the intensity varies across the Maltese cross pattern.This equation is a product of two terms sin²(2ϕ) and sin²(δ/2). The term sin²(2ϕ) is responsible for the Maltese cross pattern. It is zero when the birefringence axis is aligned with the polarizers (ϕ = 0° or 90°), creating the dark arms of the cross. It is maximum when the axis is at 45°, creating the bright lobes. The term sin²(δ/2) determines the brightness of the lobes. Since the retardance δ depends on the magnetic field, the intensity of the lobes will change with radial distance from the magnet. The element S₁ is identical to S₀, indicating that the light emerging from the analyzer is still linearly polarized. The element S₂shows the amount of linear polarization at ±45°, and the sin(4ϕ) term means this component is zero where the bright lobes and dark arms of the cross are, and is maximum in the regions between them. S₃ is related to circular polarization, and itis zero, as the final linear polarizer blocks any circular polarization.

In Figure 12, we present the patterns obtained for two different experimental configurations (Figure 12a,c with no magnetic field).In the first case with radial magnetic field, shown in Figure 12b, a linearly polarized light beam passes through the ferrofluid and is analyzed with a right-handed circular polarizer. In Figure 12c, the same linearly polarized light is analyzed with a left-handed circular polarizer.

For the second case, both the polarizer and the analyzer are circular and have opposite handedness. As shown in Figure 12d, light with a left circular polarization state enters the Ferrocell, and its final polarization is analyzed with a right circular polarizer in Figure 12e. The reverse configuration, with a right circular polarizer at the input and a left circular polarizer as the analyzer, is shown in Figure 12f.

Applying Mueller matrix algebra to these cases, we get very similar positions for the maxima and minima in all four scenarios, if we account for the phase difference between them. This is demonstrated by the Stokes vector below, which represents the case of linearly polarized light passing through the ferrofluid and being analyzed with a right circular polarizer.

$$S_{\mathrm{out}}\left(r,\varphi \right)={\displaystyle \frac{{\mathrm{I}}_0}{2}}\left(\begin{array}{c}\begin{array}{c}1-\sin \left(2\varphi \right)\sin \left(\delta \left(r\right)\right)\\ 0\end{array}\\ \begin{array}{c}0\\ 1-\sin \left(2\varphi \right)\sin \left(\delta \left(r\right)\right)\end{array}\end{array}\right)$$

(4)

The plot for the light intensities for these Stokes vectors is shown in Figure 13.

Because the ferrofluid’s orientation angle ϕ enters the equations through the term sin(2ϕ), the sign flip of the circular polarizer effectively shifts the entire pattern. The angles that produce a bright lobe in one case (where sin(2ϕ) is positive) now produce a dark lobe in the other case. The phase difference between the condition for maximum intensity in the two cases is (3π/2) − (π/2) = π in the 2ϕ domain, which corresponds to a 90-degree spatial shift in the ϕ (polar coordinate) domain. This is why the bright lobes move from the 45° diagonal to the 135° diagonal.

4.3. Suppression of Colors

With the images of the previous experiments, we decided to understand the suppression of colors, like the blue light in more detail, as it is shown in Figure 14. The green color in the dark lobes cannot be explained by a simple monochromatic model. It arises from the fact that the retardance, δ, is also dependent on the wavelength of light (λ), as given by the equation:

$$\delta \left(\lambda , r\right)={\displaystyle \frac{2\pi \mathsf{\Delta }nL}{\lambda }}$$

(5)

Here Δn is the birefringence and L is the thickness of ferrofluid film [19].

In the regions where the dark lobes appear, the overall intensity is low because the light of that wavelength is almost completely blocked by the analyzer. This means that for a given point (r,ϕ), the retardance δ is in a specific range that results in a low value of the term [1 − sin(2ϕ)sinδ(λ,r)]. The fact that other colors are not suppressed is due to chromatic dispersion. The birefringence (Δn) and thus the retardance (δ) are slightly different for each color (wavelength), preventing the condition for complete suppression from being met for all colors simultaneously. In the graph shown in Figure 14, as the magnetic field weakens, we can see the suppression of blue light and a competition between red and green light. Without the magnetic field, the most intense light is blue, as can be seen in Figure 12d.

5. Conclusions

This work demonstrates a powerful and accessible method for studying magneto-optics using only a Ferrocell device. The experiments revealed that the ferrofluid’s magneto-optical behavior is critically dependent on its concentration, exhibiting a clear transition from an optically non-reactive medium to one with complex light-modulating properties.

At high concentrations, the system is dominated by strong inter-particle scattering and absorption, which prevents the formation of clear patterns and causes the ferrofluid to behave as a dense, homogeneous medium. However, with intermediate dilution, a “pattern formation zone” is reached. In this crucial regime, inter-particle forces are balanced, allowing the magnetic field to effectively align the particles and create the initial, visible patterns. Further dilution minimizes scattering, making the system highly transparent and allowing for more direct interaction between light and the dynamically formed structures.

The observation of horocycles with distinct colored points provides several pieces of information, such as the evidence of field-induced ordering, in which the mere formation of these complex, often geometric, light patterns indicates that the applied magnetic field is indeed organizing the otherwise disordered nanoparticles within the ferrofluid. These patterns wouldn’t appear if the nanoparticles remained randomly dispersed. Another aspect is the demonstration of Structural Coloration, in which the appearance of specific colors (blues, greens, or reds) in different parts of the horocycle, especially when illuminated with white light, is compelling evidence of structural coloration. This means the colors are not due to pigments absorbing light, but rather to the light interacting with the physical, quasi-periodic structures formed by the aligned nanoparticles.

Based on our analysis, we conclude that the ferrofluid film acts as an optical retarder. The magnetic dipole field forces the nanoparticles to align, creating a birefringence that is dependent on position. The most compelling finding is the emergence of color in the dark lobes, which highlights the system’s chromatic dispersion. The ferrofluid’s birefringence and, consequently, its retardance, are dependent on the wavelength of light. This means the condition for light suppression is met for blue wavelength at a specific location, while other colors pass through, creating a pattern of structural color.

The suppression of certain colors in a ferrofluid photonic crystal is directly related to the formation and shifting of photonic bandgaps (PBGs). The fundamental connection is that the color we perceive is the light that is not suppressed by a photonic bandgap. The position and width of these gaps are a direct consequence of the quasi-periodic arrangement of the ferrofluid, which is in turn manipulated by the applied magnetic field.

The Mueller matrix framework proves to be a powerful and quantitative tool for characterizing these properties. By modeling the entire optical system with Mueller matrices, we were able to predict and explain the various patterns observed with different combinations of linear and circular polarizers. This approach not only provides a mathematical description of the observed phenomena, but also confirms that birefringence is the core physical property that makes all these effects possible. The birefringence itself is not uniform across the film, with this spatial variation causing the distinct light and dark regions in the patterns.