Fresnel integral

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:

Some may use π t²/2 instead of t², in which case the S(x) and C(x) above should be multiplied by .

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

The Cornu spiral is the curve generated by a parametric plot of S(x) against C(x). The Cornu spiral was created by Marie Cornu as a nomogram for diffraction computations in science and engineering.

{C(x), S(x)} (Note that the spiral should actually reach 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 . 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:

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(please tidy this up) It is possible (but not trivial) to evaluate the fresnel integrals in the limits, we have

This can be seen by integrating the function

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 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

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

External links

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