User:Mwphysics/sandbox

This is the first sentence of the lead section, and it should contain the definition of the topic and highlight its notability. It should appear under the title of the article which is set, by default, to "User:Mwphysics/sandbox" for this test article. This section should be quite concise – between one and four paragraphs. It should provide an overview of the following sections in the article.

The overview of the first important point, as it appears in the following sections, may be given in this short paragraph.

The overview of the second important point, as it appears in the following sections, may be given in this short paragraph.

This section should be used to catch the reader's attention and encourage them to read the following sections.

Formal definition of topic or description of problem

For scientific Wikipedia articles, this section may provide the formal definition of the topic using mathematical syntax or relevant diagrams. This section should also define any useful notation that will be used in the remaining sections of the article.

$P=|\psi _{1}+\psi _{2}|^{2}=|\psi _{1}|^{2}+|\psi _{2}|^{2}+\psi _{1}\psi _{2}^{*}+\psi _{1}^{*}\psi _{2}$

Let ${\textstyle P_{1}=|\psi _{1}|^{2}}$ and ${\textstyle P_{2}=|\psi _{2}|^{2}}$, then the following holds,

$P=P_{1}+P_{2}+2{\sqrt {P_{1}P_{2}}}\cos {\left[\arg {(\psi _{1}\psi _{2}^{*})}\right]}$.

Motivation

Main theorem supporting the topic

Here is a formal theorem that supports the main topic defined in the previous section. If the main topic of the article is the theorem itself, then this would likely be moved to the previous section.

Rainbow dove surface hologram used as a security feature on most visa cards.

The compression ratio for the three-lens spaceplate is defined as,

$R={\frac {d_{eff}}{d}}={\frac {f_{ext}}{2f_{mid}}}-1$,

where $f_{ext}$ is the focal length of the external lenses and $f_{mid}$ is the focal length of the middle lens.

Corollary to the main theorem

Here is a supporting statement that is easily deduced from the main theorem and may help motivate the main topic of this article.

The ideal spaceplate phase is given by,

$\varphi _{SP}(k_{x},k_{y},d_{eff})=d_{eff}(|{\textbf {k}}|^{2}-k_{x}^{2}-k_{y}^{2})^{1/2}$.

Another perspective

Here is a classical perspective of the problem and how it contrasts with the quantum perspective. This should highlight the need for a quantum perspective.

Properties

Almost all scientific Wikipedia articles that cover specific mathematical concepts or formulae provide a properties section. This will likely have links to external sources for proofs.

Property 1:

• Definition or equation describing property 1. Example: additivity.
• Example supporting property 1.
  • Additivity:$f(x+y)=f(x)+f(y)$
• Reference to formal proof.

Property 2:

• Definition or equation describing property 2. Example: homogeneity of degree 1.
• Example supporting property 2.
  • Homogeneity of degree 1: $f(\alpha x)=\alpha f(x)$ for all $\alpha$
• Reference to formal proof.

History

Albert Einstein, 1921.

Important background story

If there is an intriguing story behind the main topic or problem, the article will likely reference it in a history section. If the main topic is experimental in itself, this section may be combined with the following section on experimental evidence.

Albert Einstein is credited with this discovery. Explain the history behind the main topic.

Experimental evidence

Experiment 1

Overview

Some articles will provide descriptions of famous experimental results that support the main topic.

An interferometer typically used for demonstration purposes in educational settings.

Setup

Here is the setup of experiment 1. Explain the setup.

The interferometer is seen in the image on the right.

Results

Here are the results of experiment 1.

Experiment 2

Overview

Some articles will provide descriptions of famous experimental results that support the main topic.

Setup

Here is the setup of experiment 2. Explain the setup.

$E=mc^{2}$

Results

Here are the results of experiment 2.

Applications

The interior of LIGO at Hanford, Washington.

Real-world application 1

The interior of LIGO is seen in the image on the right.

Real-world application 2

The Hubble Space Telescope. Explain the application.

See also

• Fourier optics
• Holography
• Quantum information

Further reading

• Savoia, S., Castaldi, G., & Galdi, V. (2013). Optical nonlocality in multilayered hyperbolic metamaterials based on Thue-Morse superlattices. Physical Review B - Condensed Matter and Materials Physics, 87(23). https://doi.org/10.1103/PhysRevB.87.235116
• Zhou, Y., Zheng, H., Kravchenko, I. I., & Valentine, J. (2020). Flat optics for image differentiation. Nature Photonics, 14(5), 316–323. https://doi.org/10.1038/s41566-020-0591-3
• Pagé, J. T. R., Reshef, O., Boyd, R. W., & Lundeen, J. S. (2022). Designing high-performance propagation-compressing spaceplates using thin-film multilayer stacks. Optics Express, 30(2), 2197. https://doi.org/10.1364/oe.443067