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title: dispersive propagation & Fractional FT
tags: [dispersive propagation & Fractional FT]

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tags: dispersive propagation & Fractional FT
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# dispersive propagation & Fractional FT
### The fractional Fourier transform
Following Victor Namias in J. Inst. Maths Applies (1980) 25, 241-265, the fractional Fourier transform $\mathcal{F}_\alpha(x)$ of $f(x)$ is given by (Eq. 3.6):
$$
\mathcal{F}_\alpha f(x) = \frac{\exp\left[i\left(\frac{\pi}{4} - \frac{\alpha}{2}\right)\right]}{\sqrt{2\pi\sin\alpha}} \exp\left[-\frac{i}{2}\cot\alpha x^2\right] \int\exp\left(-\frac{i}{2}\cot\alpha y^2 +\frac{ixy}{\sin\alpha}\right) f(y) dy
$$
which can also be written as
$$
\mathcal{F}_\alpha f(x)  = \frac{\exp\left[i\left(\frac{\pi}{4} - \frac{\alpha}{2}\right)\right]}{\sqrt{2\pi\sin\alpha}} 
\exp\left[\frac{i}{2}\tan\alpha x^2\right]  \int\exp\left[-\frac{i}{2}\cot\alpha\left(y-\frac{x}{\cos\alpha}\right)^2\right] f(y)dy
$$
The definiition of the inverse FFT is obtained by complex conjugaison of this equation with $\alpha$ real.

### The propagation in a dispersive fiber
Following Keisuke Goda and coll. in PHYSICAL REVIEW A 80, 043821 (2009) (Eq. 14), the field amplitude $u(z,T)$  after a propagation distance $z$, at time
$t = T + \beta_1 z$ (where $\beta_1$ is the inverse of the group velocity) is given by:
$$ 
u(z,T) = \exp\left(-i\frac{T^2}{2\beta_2 z}\right) \int  \exp\left[i\frac{\beta_2 z}{2}\left(\omega - \omega_s -\frac{T}{\beta_2 z}\right)^2\right] \tilde{u}(0,\omega-\omega_s)d\omega
$$
where $\tilde{u}(0,x)$ is the Fourier transform of $u(0,x)$. With the change of variable $y = \omega-\omega_s$ we obtain
$$
u(z,T) = \exp\left(-i\frac{T^2}{2\beta_2 z}\right) \int  \exp\left[i\frac{\beta_2 z}{2}\left(y -\frac{T}{\beta_2 z}\right)^2\right] \tilde{u}(0,y)dy.
$$
Defining 
$$
\beta_2z \equiv \cot\alpha, \quad  T \equiv \frac{x}{\sin\alpha},
$$
we obtain
$$
u(z,T) = \exp\left(-i\frac{x^2}{\sin(2\alpha)}\right) \int  \exp\left[i\frac{\cot \alpha}{2}\left(y -\frac{x}{\cos\alpha}\right)^2\right] \tilde{u}(0,y)dy
$$
comparing with the definition of the fractional Fourier transform, we have,
$$
u(z,T) = \sqrt{2\pi\sin\alpha}\exp\left[i\left(\frac{\pi}{4} - \frac{\alpha}{2}\right)\right]\exp\left[\frac{i}{2}\cot(\alpha) x^2\right] \mathcal{F}_{\alpha}^{-1} \left[\tilde{u}(0,y)\right].
$$


We conclude that the proppagation over a distance $z$ in a dispersive fiber does not correspond exactly to an inverse fractional Fourier transform. There is an additional quadratic phase $\exp\left[\frac{i}{2}\cot(\alpha) x^2\right]$.

The same problem occurs in the case of Fresnel diffraction (see "Fresnel diffraction and the fractional-orderFourier transform" by  Pierre Pellat-Finet in OPTICS LETTERS / Vol. 19, No. 18 / September 15, 1994).
In this reference Pellat-Finet claims that observing the diffraction pattern on a spherical surface instead on a plane solves the problem of the additional quadratic phase. Can we use a similar trick ?

