Overview: Study of magnetization temperature in ferromagnetic crystals

EuO and EuS Materials are used in study. As per Weiss mean field approximation (MFA), below certain critical temperature(Tc) ferromagnetic materials have spontaneous magnetization — i.e., a sizable macroscopic magnetic moment even in the absence of an external magnetic field. The effective field acting on a magnet in a ferromagnetic medium is H+gM(T), term gM(T) is called self-consistent molecular field.

Magnetization M is given by,

 

Langevin function describes the average alignment of magnetic moments with an applied magnetic field at a given temperature.

x=uB/kT

For very large T, x becomes small and Langevin function becomes

For H = 0 and T slightly below TC , The equation  becomes

Brillouin function is a quantum mechanical function that describes the magnetization of a system of spins in response to an applied magnetic field at a given temperature.

The study shows that M(T) exhibits anomalous scaling near Tc, with a scaling index β≈1/3, consistent with experimental data for EuO and EuS.

This result differs from the classical MFA prediction of β=1/2

The Weiss-Heisenberg MFA value of the Curie Temperature Tc wh ≈ 86.6K in EuO was observed.(about 20 % larger than its experimental value Tc exp≈ 69.8 K). spin-wave included  MFA equations match with experimental data across all temperature ranges. 

Source: https://arxiv.org/abs/2412.10124

Read More

Summary of Article: The topological aberrations of twisted light

 

• Topology describes the study of properties of spaces that are invariant under any continuous deformation.

• Topological aberration: It contains a high-order optical vortex which experiences not only geometrical shifts, but an additional splitting of its high-order vortex into a constellation of unit-charge vortices.

• Multiple optical vortices indicate  the presence of more than one optical vortex in a light beam. Each optical vortex is a point or region where the intensity of light is zero, and the phase of the light waves spirals around this point, creating a "twisted" or helical wavefront. These beams are characterized by their helical wavefronts.

• Goos-Hänchen (GH) Shift: The GH shift is a lateral displacement of a reflected light beam along the plane of incidence. Instead of reflecting exactly along the predicted path, the beam's central position shifts slightly parallel to the interface. GH shift arises from changes in the reflection coefficient of the interface, which vary with the incidence angle. These changes affect the beam's overall phase, leading to a shift.

• 

• ΔGH​ is the lateral shift (Goos-Hänchen shift), λ is the wavelength of the light. ϕr​ is the phase of the reflection coefficient, θi​ is the angle of incidence.

• Imbert-Fedorov (IF) Shift : It is a transverse displacement of a reflected  beam that occurs perpendicular to the plane of incidence.

• In the context of vortex constellations, the coordinates of the vortices can be represented as complex numbers. The authors use Elementary Symmetric Polynomials(ESP)  to summarize these coordinates and understand how they change under reflection.

• 

• vectors eI and eR contain the ESPs of the input and aberrated constellations, respectively.

• Wirtinger Derivative: It helps how a complex function (or light beam) changes, especially when dealing with distortions or shifts in the beam's structure.

• Above equations applying for this experiment,

• 

• R’ and R” are first and second Wirtinger derivatives of R(χ) at χ = χ* = 0.

• Aberrations usually change the elementary symmetric polynomials (ESP), which describe the positions of the vortices in a group (constellation). From these changes, we can directly figure out the angular Wirtinger derivatives related to the aberration.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10305470/

https://www.nature.com/articles/s41467-024-52529-6

Read More

Summary of Article: Temporal signal processing method for optical metasurfaces

The temporal signal: Temporal signal varies with time. It is used to identify patterns, or changes in audio signals, video frames, or sensor readings. In optics, it is used for shaping, filtering, or differentiating light signals to data transmission or image processing.

The metasurface is very thin material that can manipulate electromagnetic waves in ways that normal materials can't. At the nanoscale, metasurfaces can change how light bends and focuses.

Nonlocal metasurface: It manipulates light in ways that depend on the light's current position and where the light has been before. It  detects changes over time in a light signal. study shows that they can efficiently perform operations like first-order differentiation of signals.

Here metasurfaces material is TiO2-coated glass substrates is used. The Fourier transform of an input signal is calculated then multiplying it by the metasurface transfer function calculated and then applying the inverse Fourier transform.

The transfer function tω dictates how the metasurfaces affect different frequency components of the impinging pulse. 

 

metasurface with a transfer function

signal Sin(t) ( a square pulse) is encoded in the envelope of an electromagnetic wave impinging on a metasurface. The envelope of the transmitted wave is the first-order derivative of the input pulse.

Assuming  electric field pulse created by the pulse shaper is given by the sum of two gaussian pulses,

CCT Cross-Correlation Trace : CCT helps determine how the output signal produced by the metasurface correlates with the input signal. By comparing these signals, researchers can evaluate how effectively the metasurface performs differentiation.

CCT of the input field

CCT of output field

 

Reference: https://www.nature.com/articles/s44310-024-00039-0

Read More

Summary of Article Related to How Taylor Series is used in Non linear time-varying systems

 

The time-varying system is approximated as a polynomial function of time, enabling the derivation of a time-varying control law.

Nonlinear Time-Varying Systems (NLTV): These systems are difficult to control due to their complexity. Traditional methods such as linearization scheduling transform NLTV systems into simpler LTI models by designing control laws at specific operating points.

Limitations of Traditional Methods: While linearization scheduling and gain scheduling(GN) are successful for simple NLTV systems, stability is a problem for operating points. This limitation becomes more critical when the system operates over a wide range of conditions, such as in flight vehicle guidance.

Methodology:

The first-order Taylor expansion is used to approximate the system's dynamics, allowing the design of a modified time-varying control law. This law considers time-dependent changes in the system's behavior, ensuring improved stability and performance.

Example - Inverted Pendulum: Inverted pendulum with an oscillating center of mass is a classical control problem where the pendulum's angle and angular rate must be controlled to maintain stability.

The general formation of the NLTV system can be written as 

where, the function 𝑓(𝑥, 𝑡) is continuous and differentiable to variables. And the singularly perturbed system of system (1) can be written as

where, 𝜖 is the parameter of the singular perturbation, which describes the time-scale separation, such as

the ratio between the bandwidths or between the time constants of the slow and fast dynamics

To improve accuracy, the first-order Taylor expansion of A(t) and B(t)around t0is used:

Using the Taylor series approximation, the modified LTV system becomes

The stability is analyzed using the Singular Perturbation Margin (SPM) method.  

It is useful for systems where time-scale separation exists, such as systems with both fast and slow dynamic components (e.g., spacecraft attitude control, flight dynamics, and robotic systems). The SPM helps in determining the stability of such systems by the interaction between the slow and fast dynamics.

The simulation result of the system controlled by GS method (dashed line) oscillates more violently than by the new time-varying method (solid line),  The new time-varying method can increase the capability of the singular perturbation of the controlled system.

Read More

Use of Taylor series and Fourier Transform

(Digital Image Correlation-DIC is a non contact technique to measure the surface displacement of a specimen by comparing digital images taken at different stages of deformation)

1. Taylor Series vs. Direct Fourier Transform: The paper introduces the Taylor series image reconstruction method, which increases efficiency over the conventional DFT-based method. Using third- and fourth-order Taylor series expansions leads to computational improvements by factors of 57 and 46, respectively.

2. Higher-order expansions improve the accuracy of deformation measurements and provide better noise resistance compared to the DFT method. This is particularly useful when Gaussian noise is introduced in the images.

3. Simulations were conducted using different strains and noise levels. The results showed that the Taylor series method performs comparably to or better than the DFT method in terms of accuracy, with significantly faster computation speeds.

4. Real-world Application: The proposed method was validated through experiments on a silicone rubber specimen under uniaxial tension. The displacement and strain measurements were consistent with theoretical expectations, highlighting the method's practical utility.

As per experiments, the standard deviations of displacement under different strains and noise levels, showing that the Taylor series method handles noise better than the Direct Fourier Transform method.

Conclusion:

The study demonstrates that replacing DFT with Taylor series expansions for image reconstruction in SDIC leads to substantial gains in computational speed while maintaining or improving the accuracy of strain and displacement 

measurements. This method is especially beneficial in handling large deformations and noisy conditions, making it highly applicable for real-time and high-precision measurements in various engineering and scientific fields

Read More