# Space-Time Coupling: Current Concept and Two Examples from Ultrafast Optics Studied Using Exact Solution of EM Equations

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. What Does the Concept of Spatio-Temporal Couplings Mean in the Theory of EM Waves?

#### 2.2. Exact Solutions of Free Space Electromagnetic Wave Equations with Finite Total Energy

#### 2.3. Maximum Energy Density of a Collapsing EM Beam

#### 2.4. Collapsing Shell: “Conventional or Strange Wave”?

#### 2.5. Excitation of a Two-Level Atom Placed at the Center of a Collapsing Beam

## 3. Results

- (a)
- The concept of space-time couplings of electromagnetic pulses is complemented by the important requirement of finiteness of total pulse energy.
- (b)
- The field of a collapsing electromagnetic beam is found in space and time basing on the exact solution of Maxwell’s equations in terms of the total energy, the spectrum and number of cycles in the incident pulse.
- (c)
- The excitation efficiency of a two-level quantum system placed in the center of a collapsing beam is found with a full account for space-time couplings.
- (d)
- The analysis showed that electromagnetic field distributions originated by solutions of scalar wave equation cannot be single sign (unipolar).
- (e)
- The method to experimentally distinguish between conventional and unipolar pulses is suggested.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Spectral Analysis of a Gaussian Pulse (20) Based on a “Real Signal” Definitions (14)–(19)

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**Figure 1.**(

**a**) is the function $u\left(r=0,t\right)$ and (

**b**) is the function $u\left(r=5,t\right)$, if $f\left(x\right)=exp\left(-{x}^{2}\right),\text{}a=1$. Horizontally, the value of $ct$, vertically—$u$ in arbitrary units.

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**MDPI and ACS Style**

Popov, N.L.; Vinogradov, A.V.
Space-Time Coupling: Current Concept and Two Examples from Ultrafast Optics Studied Using Exact Solution of EM Equations. *Symmetry* **2021**, *13*, 529.
https://doi.org/10.3390/sym13040529

**AMA Style**

Popov NL, Vinogradov AV.
Space-Time Coupling: Current Concept and Two Examples from Ultrafast Optics Studied Using Exact Solution of EM Equations. *Symmetry*. 2021; 13(4):529.
https://doi.org/10.3390/sym13040529

**Chicago/Turabian Style**

Popov, Nikolay L., and Alexander V. Vinogradov.
2021. "Space-Time Coupling: Current Concept and Two Examples from Ultrafast Optics Studied Using Exact Solution of EM Equations" *Symmetry* 13, no. 4: 529.
https://doi.org/10.3390/sym13040529