We will start our analysis of the effects of dispersion by first considering pulses purely affected by dispersion, i.e. setting $\gamma$ = 0 to ignore the effects of nonlinearities. The normalized amplitude we found in the last section should satisfy the linear partial differential equation given by:

$$i\frac{\partial{U}}{\partial{z}} = \frac{1}{2}\beta_2\frac{\partial^2U}{\partial{T^2}} \label{diff}$$

We can note here that this equation looks a lot like the paraxial wave equation for light diffraction and is identical to the paraxial wave equation if the diffraction is only is one transverse direction and $\beta_2$ is suitably replaced to match the equation. This thus means that temporal dispersion has close analogies with spatial diffraction. The solution to Equation \eqref{diff} in the Fourier domain is given as:

$$U(z,\omega) = U(0,\omega)e^{\frac{i}{2}\beta_2\omega^2{z}}$$

This equation shows that the GVD has the effect of changing the phase of each spectral component, which does not change the form of the spectrum but which can act to distort the pulse shape. We can then convert back to the temporal domain by taking the Fourier transform again to obtain the amplitude of the pulse in the temporal domain:

$$U(z,T) = \frac{1}{2\pi}\int_{-\infty}^{\infty}U(0,\omega)e^{\frac{i}{2}\beta_2\omega^2z-i\omega{T}}d\omega \label{gen_sol}$$

We can then use this general amplitude form for the different pulses we may encounter in the lab.

1. Gaussian Pulse
2. Chirped Gaussian Pulse
3. Hyperbolic-Secant Pulse
4. Super-Gaussian Pulse
5. Experimental Results

GAUSSIAN PULSE

An example solution to Equation \eqref{gen_sol} is for the case of a Gaussian pulse given by the incident field:

$$U(0,T) = e^{\frac{-T^2}{2T_0^2}}$$

where $T_0$ is the half-width point, and is related to the FWHM of a Gaussian pulse by:

$$T_{FWHM} = 2(ln2)^{1/2}T_0 \approx 1.665T_0$$

The integration of the Gaussian field in Equation \eqref{gen_sol} then gives the amplitude for any arbitrary z as:

$$U(z,T) = \frac{T_0}{(T_0^2-i\beta_2z)^{1/2}}e^{-\frac{T^2}{2(T_0^2-i\beta_2z)}}$$

We can thus see from this that the pulse maintains its basic Gaussian shape on propagation but its width increases and becomes:

$$T_1 = T_0[1+(z/L_D)^2]^{1/2}$$

where the dispersion length $L_D = T_0^2/|\beta_2|$ shows that the dispersion length is shorter and the pulse broadens more for shorter pulses. At $z = L_D$, the Gaussian pulse broadens by a factor of $\sqrt{2}$.

By looking at these equations, we can note that although the incident pulse was bandwidth limited and had no chirp, the transmitted pulse becomes chirped. This can be seen by writing the normalized amplitude in the form:

$$U(z,T) = |U(z,T)|e^{i\phi(z,T)}$$

where

$$\phi(z,T) = -\frac{\pm\beta_2(z/L_D)}{1+(z/L_D)^2}\frac{T^2}{T_0^2}+\frac{1}{2}tan^{-1}[\frac{z}{L_D}]$$

Here, the time dependence of the phase implies that the instantaneous frequency differs across the pulse from the central frequency $\omega_0$. What this basically means is that ordinarily, the instantaneous rate of frequency change under the envelope would be given by $\omega_0$.

CHIRPED GAUSSIAN PULSE