The Poincare Sphere Representation

Introduction to Poincare Sphere

The Poincare sphere is a powerful geometric representation for any polarization state, first introduced by H. Poincare in 1892. Since any elliptically polarized light can be completely determined by two azimuthal angles, which can be represented by longitude and latitude on a sphere, each point on the sphere corresponds to a unique polarization state. These points on the sphere represent various combinations of polarization states, forming a complete mapping that has become essential in fields from lcd monitor design to optical communications.

Figure 1: Basic structure of the Poincare sphere

This elegant mathematical construct simplifies the analysis of polarization transformations, making it invaluable in practical applications such as lcd monitor technology, where precise control of polarization states is crucial for image formation. By representing each polarization state as a point on the sphere's surface, complex polarization phenomena can be visualized and manipulated with remarkable clarity.

1. Description Using Stokes Parameters

The polarization state of any plane monochromatic light wave can be represented by two mutually perpendicular vibrations. Taking xOy as the vibration plane, these two vibrations can be expressed as:

There are four Stokes parameters, defined as:

Based on equations (1.28) and (1.29), we can introduce a three-dimensional space with mutually perpendicular axes S₁, S₂, and S₃. In this space, for any monochromatic light wave of known intensity (fixed S₀), we can construct a sphere: with its center at the origin and radius S₀. Thus, any point M on this sphere represents a specific polarization state of the light wave.

Figure 1.12: Representation method of polarization states for any point on the Poincare sphere

As can be seen in Figure 1.12, the coordinates S₁, S₂, and S₃ of any point M on the sphere have the following relationships with S₀, 2θ, and 2β:

This sphere, defined and used to describe the polarization state of monochromatic light waves of specific intensity, is known as the Poincare sphere. Its mathematical elegance has made it indispensable in modern optics, from lcd monitor design to fiber optic communication systems, where precise polarization control is essential. The ability to map complex polarization states to simple geometric points simplifies both analysis and practical implementation of polarization-based technologies.

2. Characteristics of the Poincare Sphere

Using the above definitions and basic formulas of spherical trigonometry, we can prove that the Poincare sphere has the following characteristics, which form the foundation of its utility in optical systems including lcd monitor technology:

① Equatorial Points and Linear Polarization

Different points on the equator of the Poincare sphere represent linearly polarized light with different vibration directions, as shown in Figure 1.13. At these points, S₃ = 0, which implies either a=0, b=0, or φ=kπ, where k can only be zero or any integer. At the intersection of the OS₁ axis with the sphere, S₂ and S₃ are also zero, with S₀ = S₁, so a = 0. This point A represents a linear polarization state parallel to Oy (OS₃).

Similarly, at the intersection point B of the OS₂ axis with the sphere, equation (1.28) shows that it represents a linear polarization state at 45° to Oy (OS₃). The point A' on the sphere diametrically opposite to point A represents a linear polarization state at 90° to Oy (OS₃), i.e., parallel to the Ox (OS₁) direction. It can be easily proven that any point C on the equator corresponding to 2θ represents a linear polarization state at an angle θ to Oy (OS₃), a principle exploited in lcd monitor technology to control pixel brightness through polarization rotation.

Figure 1.13: Linear polarization states at points on the Poincare sphere equator

③ Elliptical Polarization Representation

Any specific point M (S₁, S₂, S₃) on the Poincare sphere uniquely determines a, b, and φ through the relationships defined earlier. Therefore, a specific point M represents a specific elliptical polarization state, except at the poles and equator. From equations (1.28) and (1.30), we can derive:

tan2θ = S₂/S₁ = (2ab cosφ)/(b² - a²) (1.31)

θ is precisely the angle between the principal axis of the elliptical vibration, formed by the synthesis of two mutually perpendicular vibrations expressed in equation (1.28), and the Oy direction when a, b, and φ are determined. Since all points on the same meridian of the Poincare sphere have the same 2θ, these points represent different elliptical polarization states, but their ellipses have the same principal axis orientation, all at an angle θ to Oy.

From equations (1.28) and (1.30), we can also derive:

tan2β = S₃/√(S₁² + S₂²) = (2ab sinφ)/√[(b² - a²)² + (2ab cosφ)²] (1.32)

The tangent of β represents the axis ratio of the elliptical vibration formed by the synthesis of two mutually perpendicular vibrations expressed in equation (1.28) when a, b, and φ are determined. Therefore, points on the same parallel (with the same 2β) of the Poincare sphere represent different elliptical polarization states, and these ellipses have the same axis ratio, or the same eccentricity, as shown in Figure 1.13. This property is particularly useful in lcd monitor design, where precise control of elliptical polarization states ensures accurate color reproduction and contrast.

From equations (1.28) and (1.30), we can also derive:

tanφ = S₃/S₂ · sin2θ = tan2β / tanθ = -S₃/S₂ · sin2θ (1.33)

On the other hand, applying the basic formulas of spherical trigonometry to the right triangle AMC on the Poincare sphere (see Figure 1.14), we can obtain:

sin2β = sin2α · sin∠MAC (1.34)

cos∠MAC = cot2α · tan2θ (1.35)

cos2α = cos2β · cos2θ (1.36)

Figure 1.14: Phase difference in elliptically polarized light on the Poincare sphere

Eliminating 2α from these three equations, we get:

tanφ = tan2β / tanθ (1.37)

Comparing equations (1.33) and (1.37), we see that ∠MAC = φ, which proves that the angle ∠MAC on the Poincare sphere is the phase difference φ between the two vibration components in the Ox and Oy directions that constitute the elliptical polarization state represented by point M.

For points M in the northern hemisphere, φ satisfies:

k = 0, 1, 2, 3, … 2kπ ≤ φ ≤ (2k+1)π

This means that the vibration in the x direction leads the vibration in the y direction, so the elliptical polarization states they represent are all left-handed. Conversely, points M in the southern hemisphere represent right-handed elliptical polarization states (see Figure 1.13). This handedness distinction is critical in applications from lcd monitor technology to quantum computing, where polarization state manipulation is essential.

Interactive Poincare Sphere Visualization

Explore the relationship between polarization states and their representation on the Poincare sphere. This interactive model demonstrates how different polarization states map to specific points, a concept fundamental to understanding lcd monitor technology and other polarization-based systems.

This visualization shows how various polarization states map to specific locations on the Poincare sphere. The equator represents all possible linear polarizations, while points moving toward the poles represent increasingly elliptical polarizations, culminating in circular polarization at the poles themselves. This spatial representation is particularly valuable in designing optical systems like lcd monitors, where precise control over polarization states directly impacts display quality and performance.

Conclusion

The Poincare sphere remains an indispensable tool in optics and photonics, providing an elegant geometric framework for understanding and manipulating polarization states. From its theoretical foundations in Stokes parameters to its practical applications in lcd monitor technology and beyond, the Poincare sphere simplifies complex polarization phenomena into an intuitive spatial representation.

As optical technologies continue to advance, from high-resolution displays to quantum communication systems, the Poincare sphere will undoubtedly remain a cornerstone of polarization analysis and design. Its ability to transform abstract mathematical relationships into concrete geometric representations makes it invaluable for both education and innovation in the field of optics.