math lessons - Fresnel integral

In mathematics and optics, the two Fresnel integrals, S(x) and C(x) arise in the description of near field Fresnel diffraction phenomena, and are the integrals defined as follows:

$S(x)=\int_0^x \sin(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+3}}{(4n+3)(2n+1)!}$

$C(x)=\int_0^x \cos(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+1}}{(4n+1)(2n)!}$ .

Some may use π t²/2 instead of t², in which case the S(x) and C(x) above should be multiplied by $\sqrt{\frac{2}{\pi}}$ .

S(x) and C(x) - Note that C(x) does not actually reach 1, as it may appear in the image. The maximum of C(x) is actually about 0.977451424. If πt²/2 was used, instead of t², then the image would be scaled vertically by the factor mentioned above.

The Cornu spiral, a.k.a. clothoid, is the curve generated by a parametric plot of S(x) against C(x). The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering. It is a logical shape with a varying radius, in use for the transition of a straight to a circle curve in roads and railways because a vehicle following the curve at constant speed will have a constant rotational acceleration, reducing lateral stress on the rail tracks, however it may not be the ideal transition spiral, especially at higher speeds, due to other forces acting upon the passengers.

{C(x), S(x)} (Note that the spiral should actually converge on the centre of the holes in the image as x tends to positive or negative infinity) If πt²/2 was used, instead of t², then the image would be scaled by the factor mentioned above.

Following the curve, the length of the curve from {S(0), C(0)} to {S(x), C(x)} must be equal to x, since $S'(x) 2 + C'(x) 2 = 1$ . The total length of the curve (from x=−∞ to ∞) is therefore infinite.

In the domain of complex numbers, the Fresnel integrals can be written using the error function as follows:

$S(x)=\frac{i\sqrt{\pi}}{4} \left( \operatorname{erf}(\sqrt{i}\,x) - \operatorname{erf}(\sqrt{-i}\,x) \right)$

$C(x)=\frac{\sqrt{\pi}}{4} \left( \operatorname{erf}(\sqrt{i}\,x) + \operatorname{erf}(\sqrt{-i}\,x) \right)$ .

It is possible (but not trivial) to evaluate the Fresnel integrals in the limits, we have

$\int_{0}^{\infty} \cos t^2\,dt = \int_{0}^{\infty} \sin t^2\,dt = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}}$

This can be seen by integrating the function

$e^{-\frac{1}{2}t^2}$

Around a pizza-slice shaped area beginning in the point (0, 0) (on the complex plane), then going out to (R, 0), up along the arch of the circle centered in (0, 0) and with radius R to the point $e i π / 4$ and back to (0, 0) in a straight line.

As R goes to infinity, the integral around the line segment on the edge of the circle will tend to 0, the one along the real axis will tend to the well known integral

$\int_{0}^{\infty} e^{-\frac{1}{2}t^2}dt = \frac{\sqrt{2\pi}}{2}$

And the last - along the slope - will evaluate to the Fresnel integrals after some rearangings.

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The Cornu spiral (Uses πt²/2 instead of t².)

Categories: Optics | Special functions

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