# Fractional Fourier Transform and Fresnel Propagation

I am currently trying to wrap my head around Fresnel propagation, and I understand it is mathematically linked to the Fractional Fourier Transform, but I'm having a hard time with the units and the interpretation. We can write out the equation for Fresnel propagation (ignoring the factors that ensure conservation of energy)

$$e^{i\pi f_y^2 \frac{z}{f}}\int{e^{-i 2\pi(f_y y-\frac{1}{2}f\frac{y^2}{z}) }g(y)dy}$$

where $$f$$ is the wavenumber in free space, $$y$$ is the coordinate in the plane we are propagating from, and $$z$$ is the propagation distance, and $$f_y$$ is the wavenumber in the $$y$$-direction.

The Fractional Fourier Transform, in contrast, can be written as (ignoring the thing out front that I believe is a normalization factor analagous to the $$2 \pi$$ in the regular FT):

$$e^{i \pi u^2 cot(\alpha)}\int{e^{-i 2\pi(csc(\alpha)ux-\frac{cot(\alpha)}{2}x^2)}h(x)dx}$$

where I have tried to write the two as similarly to each other as I can. The problem I am having is that when I try to equate the arguments of the exponentials I get a set of equations which appears not to have a solution $$f_y^2 \frac{z}{f}=u^2cot(\alpha)$$ $$f_y y=csc(\alpha)u x$$ $$f \frac{y^2}{z}=cot(\alpha)x^2$$

When I try to solve them I get some nonsense like $$cos^2(\alpha)=1$$. Am I doing something wrong, and how does Fresnel Propagation map onto the Fractional Fourier Transform as it is typically defined?

I had not come across the fractional Fourier transform before but loking at the Wiki article suggests what is going on. Fresnel's transform provides a kind of representation of the action of the non-compact symplectic group that underlies Hamiltonian optics. (The "metaplectic" representation). Non compact means that the propagation distance $$z$$ is not a periodic variable: it can be as large as one likes.
The fractional fourier transform has a parameter $$\alpha$$ instead of $$z$$, and $$\alpha$$ is periodic, so that the fourth power of the usual Fourer is the identity opertor. Thus it is providing a representation of a compact group hence the ordinary $$2\pi$$-periodic trig functions. The algebra is indeed similar but I suspect one needs to make $$\alpha$$ complex to get the mapping between to two, so $$\cot\alpha$$ has to become $$\coth \alpha$$ and $${\rm cosec}\, \alpha$$ becomes $${\rm cosech}\, \alpha$$. Did you try this?
Note added: There is another possibility which I now think more likely. It comes from thinking of Fractional Fourier in terms of the Mehler Kernel. When one takes the coeffcient of the Hermite Polys as a pure phase this coincides with the propagator of the quantum harmonic oscillator. This gives a periodic transform in your $$z$$ direction but in its optics interpretation is is not free wave propagation. Instead it is propagation in an optic fibre with the a quadratic refractive index gradient $$n \propto y^2$$ so that the beam is periodically passes through a focus. This gradient alters the coefficient of the $$y^2$$ term in Fresnel so that it is no longer just $$fy^2/z$$ but is $$fy^2/z+\beta$$ for some parameter $$\beta$$ proprtional to the focusing effect. Perhaps this extra degree of freedom allows you to match all parameters?