Difference between the paraxial approximation and the Fresnel approximation

Question

I'm currently reading literature about diffraction (especially Rayleigh/Sommerfeld-Diffraction, or the equivalent fourier-method), and I'm stumbling across the terms "paraxial approximation" and "Fresnel approximation" all the time. What is the exact difference between these two terms, or, what is the exact definition of these terms? What mathematical expressions are approximated, and in what cases?

I'm currently calculating the Fourier transform of the free space transfer function

$$ H(x, y, z) = \mathcal{F} _{(x, y)}\left( e^{iz\sqrt{k^2-k_x^2-k_y^2}}\right) $$

Using the assumption that the field (which undergoes diffraction) contains only such spectral components with $k^2 \gg k_x^2$ (that would be a constraint that you apply to the whole diffracted field), you can approximate the exponential function to be

$$ e^{iz\sqrt{k^2-k_x^2-k_y^2}} \approx e^{ikz - \frac{i}{2 k z}(k_x^2 + k_y^2)} $$

If you then calculate the impulse response (the Fourier transform), you arrive at parabolic waves that propagate from each point of the surface that contains the diffractingstructur. What approximation did I use here? Fresnel approximation or the paraxial approximation?

Answer 1

Briefly speaking, from Helmholtz equation you can deduce the Rayleigh-Sommerfeld diffraction formula. Further, with paraxial approximation you deduce Fresnel diffraction formula.

In math, for wave $u_0(\mathbf{r})$ propagating distance $z$ as $u_z(\mathbf{r})$, by Rayleigh-Sommerfeld equation:

$$u_z(\mathbf{r}) = -\frac{1}{2\pi}u_0(\mathbf{r}) \star \frac{1}{2\pi}\frac{z e^{j kR}}{R^3}(1 - jkR),$$

where $\mathbf{r} = (x,y)$, $R = \sqrt{x^2 + y^2 + z^2}$, $k$ the wave number, $\star$ denotes convolution.

To get Fresnel formula, you need:

• $z \gg \|\mathbf{r}\|$. And thus: $R = \sqrt{\|\mathbf{r}\|^2 + z^2} \approx z + \frac{\|\mathbf{r}\|^2}{2z}$.

• $z \gg \lambda$. This leads to: $kR \gg 1$.

With these two aproximations, you yield the Frenel formula: $$u_z(\mathbf{r}) = u_0(\mathbf{r}) \star \frac{e^{j kz}}{j\lambda z} e^{j \frac{k \|\mathbf{r}\|^2}{2z}}.$$

Answer 2

The paraxial approximation and the Fresnel approximation are in essence the same thing. It is just that the paraxial approximation tends to be used in connection with the differential equations while the Fresnel approximation is used in the context of the integral expression.

Starting from the Helmholtz equation, one can apply the paraxial approximation to obtain the paraxial wave equation. (Upon request I can insert the relevant expressions.) The solutions of the paraxial wave equation are in the form of Gaussian beams (Laguerre-Gaussian, Hermite Gaussian, etc.). Gaussian beams are not solutions of the Helmholtz equation.

On the other hand, one can use an integral equation approach, as you have done, and then apply the Fresnel approximation to modify the argument in the exponential of the kernel. The result is the Fresnel propagation integral.

To check that these two approaches give the same result. Start with the two dimensional Gaussian shape of a Gaussian beam in its waist. Then apply Fresnel propagation over an arbitrary distance $z$. You'll find an expression that is the same as the solution of the paraxial wave equation.