Fractional Fourier Transform and Fresnel Propagation

Question

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?

Answer 1

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?