🔬 optics

Modulation in degree of cross-polarization at Young's interferometer illuminated by non-uniformly polarized electromagnetic fields

This paper investigates Young's interferometer illuminated by non-uniformly polarized fields to demonstrate that the degree of cross-polarization at the observation plane equals the average degree of polarization at the pinholes, while the electromagnetic degree of coherence can be controlled by the pinholes' degree of polarization, offering potential applications in ghost and lensless imaging.

Original authors: Rajneesh Joshi, Gyaprasad

Published 2026-08-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rajneesh Joshi, Gyaprasad

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine light not just as a bright beam that lets you see your reflection, but as a bustling crowd of invisible messengers, each carrying a tiny, spinning arrow. In the world of optics, these messengers are electromagnetic waves, and their "spinning" is what we call polarization. Sometimes, all the messengers spin in perfect unison (fully polarized), sometimes they spin in total chaos (unpolarized), and often, they do a mix of both. Another key trait is coherence, which is like a group of dancers moving in perfect rhythm; if they are coherent, they step together, but if they are incoherent, they are just a chaotic jumble of steps. Scientists have long known that these two traits—how the messengers spin and how well they dance together—are deeply connected. Understanding this connection is like having a master key to unlock new ways of seeing the world, from creating "ghost" images that appear out of thin air to peering through foggy atmospheres. But what happens when you take a light source that is a chaotic mix of different spinning styles and split it down the middle? That is the mystery this paper sets out to solve.

This study dives into a classic setup known as Young's interferometer, which is essentially a fancy way of splitting light through two tiny pinholes to see how it behaves when it recombines. The researchers, Rajneesh Joshi and Gyaprasad, decided to play a trick on this setup: instead of using a perfectly uniform light source, they imagined a scenario where the light hitting the two pinholes has different degrees of polarization. Think of it like having two buckets of water; one bucket is filled with perfectly aligned, straight streams (highly polarized), while the other is filled with swirling, chaotic splashes (unpolarized). When these two very different "streams" of light meet again at a screen, how does their combined personality look?

The paper finds that the resulting light at the observation screen has a property called the Degree of Cross-Polarization (DoCP). In simple terms, this measures how "mixed up" the polarization is between two different points in space. The authors discovered something beautifully simple: if you look at the light right in the middle of the interference pattern (where the two paths are equal), the DoCP is exactly the average of the polarization levels at the two pinholes. It's like mixing a cup of strong coffee with a cup of water; the result is a cup of medium-strength coffee. If one pinhole has light that is 0% polarized (completely chaotic) and the other is 100% polarized (perfectly ordered), the light at the center will be 50% polarized. This confirms that the DoCP is essentially a "two-point" version of the standard polarization we usually measure at a single spot.

However, the story gets a bit more rhythmic when you move away from the center. The paper shows that as you look at different spots on the screen, the DoCP and another property called the Electromagnetic Degree of Coherence (EM DoC) don't just stay flat; they wiggle or modulate in a smooth, wave-like pattern. This happens because the light waves from the two pinholes are interfering with each other, creating a pattern of bright and dark spots, but this time, the "brightness" is actually a measure of how polarized the light is. The researchers calculated that this wiggling depends on the distance between the pinholes and the angle at which you look at the screen.

Crucially, the paper rules out the idea that this behavior is random or unpredictable. Instead, it proves that the behavior is strictly governed by the polarization levels at the source. For instance, if the light at both pinholes is perfectly uniform (both 100% polarized), the DoCP at the screen simply matches that 100% value, just as you would expect. But if the polarization is uneven, the "average" rule takes over at the center, and the "wiggling" takes over elsewhere. The authors also note that the EM DoC (how well the light waves dance together) can be controlled by changing the polarization at the pinholes. If both pinholes have fully polarized light, the coherence hits its maximum value of 1. If both are completely unpolarized, the coherence drops to a minimum of roughly 0.71 (specifically $\sqrt{1/2}$ ).

In the end, this paper acts as a guidebook for understanding how light behaves when it's a bit messy. It shows that even when light is a chaotic mix of different spinning styles, nature has a simple rule: the combined effect is often just the average of its parts, with a rhythmic dance added in for good measure. These findings aren't just theoretical math; the authors suggest they could be useful for improving "ghost imaging" (taking pictures of objects using light that never actually touched the object) and studying how light correlations work in complex environments. It's a reminder that even in the chaotic dance of light, there is a predictable, average rhythm waiting to be found.

Technical Summary: Modulation in Degree of Cross-Polarization at Young's Interferometer Illuminated by Non-Uniformly Polarized Electromagnetic Fields

Problem Statement
The statistical properties of light fields, specifically the degree of coherence (DoC), degree of polarization (DoP), and degree of cross-polarization (DoCP), are interdependent quantities governed by electric field correlations. While DoP is a one-point correlation property and DoC/DoCP are two-point properties, their behavior under non-uniform polarization conditions remains a subject of investigation. Existing literature has established relationships for uniformly polarized beams and incoherent sources, but a comprehensive theoretical framework describing the statistical properties at observation points for incoherent light beams with non-uniform polarization (where the DoP varies spatially) at the source pinholes is required. The paper addresses the need to characterize how the DoCP and electromagnetic degree of coherence (EM DoC) behave in a Young's two-pinhole interference experiment when the input fields at the pinholes possess different degrees of polarization.

Methodology
The study employs a theoretical approach based on Wolf's unified theory of coherence and polarization. The authors utilize the Cross-Spectral Density (CSD) matrix to describe the correlations of random electromagnetic fields.

Theoretical Formulation: The authors define the CSD matrix elements and relate them to generalized Stokes parameters ( $S_0, S_1, S_2, S_3$ ) and intensity-normalized generalized Stokes parameters ( $\mu_n$ ).
Definitions: The EM DoC ( $\mu$ ) and DoCP ( $P$ ) are expressed in terms of these normalized Stokes parameters. The DoCP is defined as a two-point generalization of the DoP.
Experimental Model: The study models a Young's two-pinhole interference experiment (Figure 1) where the source is incoherent but exhibits non-uniform polarization at the two pinholes ( $\rho_1$ and $\rho_2$ ).
Derivation: Using the paraxial approximation, the electric field components at observation points ( $r_1, r_2$ ) are derived as superpositions of fields from the two pinholes. The CSD matrix at the observation plane is calculated by substituting these field expressions into the correlation definition.
Analysis: The derived expressions for the coherence Stokes parameters at the observation plane are used to calculate the DoCP and EM DoC. The analysis specifically examines the case where the light is incoherent at the pinholes (vanishing two-point correlation functions at the source) and investigates the dependence on the difference in direction cosines ( $\Delta u_x$ ) and the DoPs at the pinholes.

Key Contributions and Results
The paper derives analytical expressions for the DoCP and EM DoC at the observation plane of a Young's interferometer illuminated by incoherent, non-uniformly polarized light.

Sinusoidal Modulation: The results indicate that both the DoCP and EM DoC at the observation points exhibit sinusoidal modulation with respect to the parameter $\Delta u_x$ (related to the observation geometry).
Average Value Relationship: Under the specific condition where $\Delta u_x = 0$ (trivial case), the study derives that the DoCP at the observation plane is exactly the arithmetic average of the DoPs at the two pinholes:
$P(r_1, r_2) = \frac{P(\rho_1) + P(\rho_2)}{2}$
Generalization of DoP: The findings verify that DoCP is a two-point generalization of DoP. In the case of uniform polarization at the pinholes ( $P(\rho_1) = P(\rho_2)$ ), the DoCP at the observation plane equals the DoP of the pinholes.
Control of EM DoC: The EM DoC at the observation points is shown to be controllable by the DoPs at the pinholes. The EM DoC reaches its maximum value of 1 only when both pinholes contain fully polarized light ( $P=1$ ). The minimum value is $\sqrt{1/2}$ , occurring when both pinholes contain completely unpolarized light ( $P=0$ ).
Numerical Illustration: The paper provides graphical representations (Figures 2 and 3) showing the variation of DoCP and EM DoC as functions of the DoPs at the pinholes. For instance, if one pinhole is unpolarized ( $P=0$ ) and the other is fully polarized ( $P=1$ ), the resulting DoCP is 0.5, and the EM DoC is approximately 0.79.

Significance and Claims
The authors state that these findings offer a theoretical basis for understanding the statistical properties of vector beams in interference setups with non-uniform polarization. The paper claims that the derived relationships may be useful for:

Classical Ghost Imaging: Utilizing intensity correlations in non-uniformly polarized fields.
Lensless Imaging: Applications relying on coherence and polarization correlations.
Stokes Correlations: Further study of correlations between Stokes parameters.
Characterization and Manipulation: The results provide insight into the characterization and manipulation of vector vortex beams and the measurement of intensity correlations between polarization components.

The study concludes by emphasizing that the DoCP serves as a generalized form of DoP and that the EM DoC is directly influenced by the polarization states at the source pinholes, providing a method to control coherence properties through polarization engineering at the source.

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