Cornu Spiral Homework Help, Assignment Help, Online Tutor, Tutoring

Submit Assignment

* *

Payment Online Assignment Help

Thermodynamics

Homework Help

sfsf

Cornu Spiral

To find the effect at a point due to an incident wavefront. Fresnel’s method consists in dividing the wavefront the wavefront into half period strips or half period zones. The path difference between the secondary waves from two corresponding points of neighbouring zones is equal to λ/2.

In fig. S is a point source of light and XY is the incident spherical wavefront. With reference to the point P, O is the pole of the wavefront. Let a and b be the distances of the points S and P from the pole of the wavefront. With P as centre and radius b draw a sphere touching the incident wavefront at O. The path difference between the waves travelling in the directions SAP and SOP is given by

d = SA + AP – SOP

= SA + AP – (SO + OP)

= a + AB + b – (a + b)

= AB

For large distances of a and b, AM and BN can be taken to be approximately equal and the path difference d can be written as

But, from the property of a circle,

If AM happens to be the radius of the n th half period zone, then this path difference is equal to nλ/2 according to Fresnel’s method of constructing the half period zones.

The resultant amplitude at an external point due to the wavefront can be obtained by the following method. Let the first half period strip of the Fresnel’s zones be divided into eight substrips and these vectors are represented from O to M₁. The continuous increase in the obliquity factor from O to M₁. The resultant amplitude at the external point due to the first half period strip is given by OM₁ (=m₁). Similarly if the process is continued, we obtain the vibration curve M₁M₂. The portion M₁M₂ corresponds to the second half period strip. The resultant amplitude at the point due to the first two half period strips is given by OM₂ (=A). If instead of eight substrips, each half period zone is divided into substrips of infinitesimal width, a smooth curve will be obtained. The complete vibration curve for the whole wavefront will be a spiral as shown in fig. X and Y correspond to the two extremities of the wavefront and M₁, M₂ etc. refer to the edges of the first, second, etc. half period strips. Similarly M₁’, ’2 etc. refer to the edges of the first, second etc. half period strips of the lower position of the wavefront. This is called Cornu’s spiral. The characteristic of this curve is that for any point P on the curve, the phase lag δ is directly proportional to the square of the distance v. The distance is measured along the length of the curve from point O. For a path difference λ, the phase difference is 2π. Hence for a path difference of d, the phase difference δ is given by

substituting the value of d from equation (ii)

Now, a new variable v (which is dimensionless) is introduced and the value of v is given by

The Cornu’s spiral can be used for any diffraction problem irrespective of the values of a, b and λ.