# Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media

## Abstract

**:**

## 1. Introduction

## 2. Generic Dispersion Analysis

## 3. Fundamental Dispersion Theorems

#### 3.1. Pulse Boosting and Decaying

#### 3.2. Pulse Chirping

## 4. Gaussian Pulse

#### 4.1. Boosted Gaussian

#### 4.2. Chirped Gaussian

## 5. Singular Pulses

#### 5.1. The Step Function

#### 5.2. Rectangular Pulses

#### 5.3. Chirped Rectangular Pulses

#### 5.4. Exponential Pulse

#### 5.5. Cosine Pulse

#### 5.6. Square Cosine Pulse

#### 5.7. Generalization and Applicable Examples

## 6. Smooth Pulses

#### 6.1. Smooth Step Function

#### 6.2. Smooth Rectangular Pulse

#### 6.3. Relations to Super-Gaussian Pulses

#### 6.4. Chirped Smooth Rectangular Pulse

#### 6.5. Smooth Cosine Pulse

#### 6.6. Smooth Exponential Pulse

## 7. Singular Pulses in the Spectral Domain

#### 7.1. The ideal Nyquist-Sinc Pulse

#### 7.2. Nyquist Sinc Pulse with Smooth Spectrum

## 8. Undistorted Airy Pulses

#### 8.1. Undistorted Ideal Accelerating Pulses

#### 8.2. Physical Accelerating Pulse

#### 8.3. Attenuation Compensating Airy Pulse

#### 8.4. Physical Attenuation Compensating Airy Pulse

## 9. Pulse Broadening Comparison

## 10. Discussion and Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Equation (3)

## Appendix B. Proof of Equation (6)

## Appendix C. Proof of Equation (10)

## References

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**Figure 1.**The real (upper panel) and imaginary (lower panel) components of the pulse (Equation (14)). The dashed curve represents the initial profile (for $\zeta ={\beta}_{2}z/{\theta}^{2}=0$), while the solid curve represents the signal after a distance, which corresponds to $\zeta ={\beta}_{2}z/{\theta}^{2}=2$. Time is measured in units of $\theta $, which corresponds to the pulse’s temporal width.

**Figure 2.**A false-color presentation of the pulse’s intensity ${\left|A\left(t,z\right)\right|}^{2}$(Equation (14)) as a function of the normalized time $\tau \equiv t/\theta $ and the normalized distance $\zeta \equiv {\beta}_{2}z/{\theta}^{2}$.

**Figure 3.**The real (upper panel) and imaginary (lower panel) components of the Gaussian pulse (Equation (16)). The dashed curve represents the initial profile (for $\zeta ={\beta}_{2}z/{\theta}^{2}=0$), while the solid curve represents its final shape (for $\zeta ={\beta}_{2}z/{\theta}^{2}=2$). In this example, the carrier frequency is ${\omega}_{0}=-4/\theta $.

**Figure 4.**Same as Figure 2, but for the intensity (${\left|A\left(t,z\right)\right|}^{2}$) of the pulse presented by Equation (16). (In this example, ${\omega}_{0}=-4/\theta $).

**Figure 5.**Same as Figure 2, but for the intensity (${\left|A\left(t,z\right)\right|}^{2}$) of the pulse presented by Equation (18) (in this example, $q=-2/{\theta}^{2}$).

**Figure 6.**The real (upper panel) and imaginary (lower panel) components of the rectangular pulse (Equation (27)). The dashed curve represents the initial profile (for $\zeta ={\beta}_{2}z/{\theta}^{2}=0$), while the solid curve represents its final shape (for $\zeta ={\beta}_{2}z/{\theta}^{2}=0.01$ on the right and $\zeta ={\beta}_{2}z/{\theta}^{2}=0.3$ on the left).

**Figure 7.**Same as Figure 2, but for the intensity (${\left|A\left(t,z\right)\right|}^{2}$) of the pulse presented by Equation (27).

**Figure 8.**Same as Figure 2, but for the intensity (${\left|A\left(t,z\right)\right|}^{2}$) of the pulse presented by Equation (31). On the left, $q=4/{\theta}^{2}$; and on the right, $q=-4/{\theta}^{2}$. The dashed lines correspond to the pulse’s boundaries ${t}_{B}=\pm \theta \left(1+2{\beta}_{2}qz\right)/2$.

**Figure 9.**Same as Figure 8 but with the initial pulse Equation (34) for the parameter $q=4/{\theta}^{2}$.

**Figure 10.**The real (upper panel) and imaginary (lower panel) components of the exponential-step function pulse (Equation (36)). The dashed curve represents the initial profile (for $\zeta ={\beta}_{2}z/{\theta}^{2}=0$), while the solid curve represents its final shape (for $\zeta ={\beta}_{2}z/{\theta}^{2}=0.01$).

**Figure 11.**Same as Figure 2, but for the intensity (${\left|A\left(t,z\right)\right|}^{2}$) of the pulse presented by Equation (36).

**Figure 12.**Similar to Figure 10, but for the bounded cosine pulse (Equation (39)) and for the final distance of $\zeta ={\beta}_{2}z/{\theta}^{2}=0.3$.

**Figure 13.**Same as Figure 2, but for the intensity (${\left|A\left(t,z\right)\right|}^{2}$) of the pulse presented by Equation (39).

**Figure 14.**Similar to Figure 12, but for the square cosine pulse (Equation (42)).

**Figure 15.**Comparison between the intensities of the three singular pulses represented by Equation (27)—dashed curve, Equation (39)—solid curve, and Equation (42)—dotted curve in a logarithmic scale.

**Figure 16.**Similar to Figure 10, but for the pulse presented by Equation (53). In these plots, $a=1/\theta $ on both, but $b=4/\theta $ on the left figure (final distance corresponds to $\zeta ={\beta}_{2}z/{\theta}^{2}=0.1$) and $b=1/\theta $ on the right one (final distance corresponds to $\zeta ={\beta}_{2}z/{\theta}^{2}=0.2$).

**Figure 18.**A comparison between smooth rectangular pulses (Equation (63), solid curves) and super-Gaussian pulses (Equation (69), dashed curves) for $n=4$ (right) and $n=14$ (left). In these plots, only the real part of the fields is presented. The imaginary part is zero.

**Figure 21.**Same as Figure 10, but for Equation (77) with the transition width $a=1/\theta $ and $\Delta =0.2\theta $ for two final distances $\zeta ={\beta}_{2}z/{\theta}^{2}=0.01$ on the left and $\zeta ={\beta}_{2}z/{\theta}^{2}=0.1$ on the right.

**Figure 22.**Same as Figure 10, but for Equation (80) and for the final distance of $\zeta ={\beta}_{2}z/{\theta}^{2}=0.3$.

**Figure 23.**Same as Figure 2, but for the intensity (${\left|A\left(t,z\right)\right|}^{2}$) of the pulse presented by Equation (80).

**Figure 25.**Presentation of the temporal dynamics of the accelerating Airy pulse (Equation (88)). On the left, the real and imaginary components of the pulse’s field are presented (for the final distance of $\zeta ={\beta}_{2}z/{\theta}^{2}=1$), and on the right, the pulse’s intensity (${\left|A\left(t,z\right)\right|}^{2}$) is presented (for the final distance of $\zeta ={\beta}_{2}z/{\theta}^{2}=2$).

**Figure 26.**Same as Figure 25, but for Equation (90) with $a=0.1/\theta $ (for the final distance of $\zeta ={\beta}_{2}z/{\theta}^{2}=2$).

**Figure 27.**Same as Figure 2, but for the intensity (${\left|A\left(t,z\right)\right|}^{2}$) of the pulse presented by Equation (92).

**Figure 29.**Same as Figure 28, but for the pulse presented by Equation (97), with $\eta /\theta =0.2$ and $a=0.1/\theta $.

**Figure 30.**Same as Figure 2, but for the intensity (${\left|A\left(t,z\right)\right|}^{2}$) of the pulse presented by Equation (97) for $a=0.2/\theta $ and $\eta /\theta =1$. The horizontal line corresponds for the maximum intensity distance $z=\eta /{\beta}_{2}a$.

**Figure 31.**FWHM Comparison between four different pulses: Gaussian (dashed curve), Rectangular (dotted curve), Exponential (solid curve), and Sinc (dot-dash).

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**MDPI and ACS Style**

Granot, E. Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media. *Appl. Sci.* **2019**, *9*, 527.
https://doi.org/10.3390/app9030527

**AMA Style**

Granot E. Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media. *Applied Sciences*. 2019; 9(3):527.
https://doi.org/10.3390/app9030527

**Chicago/Turabian Style**

Granot, Er’el. 2019. "Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media" *Applied Sciences* 9, no. 3: 527.
https://doi.org/10.3390/app9030527