91Ó°ÊÓ

In the context of Jones calculus, a polarization state describes how the electric field of a light wave oscillates. It tells us the direction and orientation of the wave's electric field vector. This is important because light can be polarized in different ways, such as linearly or circularly.
To specify a polarization state, a Jones vector is used. The Jones vector is a complex two-dimensional vector that represents the amplitude and phase of the wave's electric field components in a specific plane.

• Each component of the Jones vector reflects either the horizontal or vertical parts of the electric field.
• These components are complex numbers, allowing them to encapsulate both magnitude and phase information.

The given exercise starts with a polarization state, \(\tilde{\mathbf{E}}_{1} = \begin{bmatrix} 1 \ -2i \end{bmatrix}\), which implies that the wave has equal amplitude along the horizontal and vertical directions but a phase difference between them. Understanding these components is key to manipulating light in optics and engineering applications.

Complex vectors, like the ones used in Jones calculus, are foundational in describing wave properties. These vectors have components that are complex numbers. Complex numbers include a real part and an imaginary part, allowing them to express both magnitude and angle.
This makes complex vectors invaluable in physics and engineering. They are capable of illustrating not only size but also direction and phase shift in waves, which is crucial for tasks like signal processing and wave interference.

• The standard complex vector in two dimensions has a form like \( \begin{bmatrix} a \ b \end{bmatrix} \), where both \(a\) and \(b\) are complex numbers.
• When dealing with complex vectors, special operations like the dot product incorporate complex conjugates to compute meaningful results.

With the task to find an orthogonal complex vector to \( \tilde{\mathbf{E}}_{1} \), we need to use these operations. You start by ensuring that the inner product (including the complex conjugate of one vector) equals zero. It is the essence of orthogonality in the complex domain.

In the realm of vectors, two vectors being orthogonal means they are perpendicular to each other. For complex vectors like Jones vectors, this perpendicularity isn't just spatial; it involves their phases too.
Understanding orthogonal vectors is essential in optics, where polarization states are manipulated. Orthogonality implies no overlap in the "direction" of the vectors in their respective spaces.

• Mathematically, for two complex vectors \( \tilde{\mathbf{E}}_{1} = \begin{bmatrix} a \ b \end{bmatrix} \) and \( \tilde{\mathbf{E}}_{2} = \begin{bmatrix} c \ d \end{bmatrix} \), orthogonality is expressed with the dot product: \( a \cdot \bar{c} + b \cdot \bar{d} = 0 \).
• This condition considers complex conjugates, emphasizing how the phase influences the relationship between vectors.

In the given exercise, after applying this relationship, we derive an orthogonal vector \( \tilde{\mathbf{E}}_{2} = \begin{bmatrix} 2i \ 1 \end{bmatrix} \). This vector is not just numerically different; it represents a polarization state that is entirely distinct from \( \tilde{\mathbf{E}}_{1} \), showing no component in its "direction" or phase orientation.