JEE Main & Advanced Physics Wave Optics / तरंग प्रकाशिकी Fresnel's Half Period Zone (HPZ)

According to Fresnel's the entire wave front can be divided into a large number of parts of zones which are known as Fresnel's half period zones (HPZ's). The resultant effect at any point on screen is due to the combined effect of all the secondary waves from the various zones.

Suppose ABCD is a plane wave front. We desire to find it's effect at point P consider a sphere of radius \[\left( d+\frac{\lambda }{2} \right)\] with centre at P, then this sphere will cut the wave front in a circle (circle 1). This circular zone is called Fresnel's first (I) HPZ.

A sphere of radius \[b+2\left( \frac{\lambda }{2} \right)\] with centre at P will cut the wave front in circle 2, the annular region between circle 2 and circle 1 is called second (II) HPZ.

The peripheral area enclosed between the nth circle and \[{{(n-1)}^{\text{th}}}\]circle is defined as nth HPZ.

(1) Radius of HPZ : For nth HPZ, it is given by

\[{{r}_{n}}=\sqrt{nd\lambda }\,\,\,\Rightarrow {{r}_{n}}\propto \sqrt{\lambda }\]

(2) Area of HPZ : Area of nth HPZ is given by

\[{{A}_{n}}=\]Area of nth circle - Area of \[{{(n-1)}^{\text{th}}}\] circle

\[=\pi (r_{n}^{2}-r_{n-1}^{2})=\pi d\lambda \]

(3) Mean distance of the observation point P from nth HPZ : \[{{d}_{n}}=\frac{{{r}_{n}}+{{r}_{n-1}}}{2}=b+\frac{(2n-1)\lambda }{4}\]

(4) Phase difference between the HPZ : phase difference between the wavelets originating from two consecutive HPZ's and reaching the point P is \[\pi \](or path difference is \[\frac{\lambda }{2},\] time difference is \[\frac{T}{2}\]).

The phase difference between any two even or old number HPZ is \[2\pi \].

(5) Amplitude of HPZ : The amplitude of light at point P due to nth HPZ is \[{{R}_{n}}\propto \frac{{{A}_{n}}}{{{d}_{n}}}(1+\cos {{\theta }_{n}})\]; where \[{{A}_{n}}=\] Area of nth HPZ, \[{{d}_{n}}=\] Mean distance of nth HPZ

\[(1+\cos {{\theta }_{n}})\]= Obliquity factor.

On increasing the value of n, the value of \[{{R}_{n}}\] gradually goes on decreasing i.e. \[{{R}_{1}}>{{R}_{2}}>{{R}_{3}}>{{R}_{4}}>............>{{R}_{n-1}}>{{R}_{n}}\]

(6) Resultant Amplitude : The wavelets from two consecutive HPZ's meets in opposite phase at P.

Hence Resultant amplitude at P

\[R={{R}_{1}}-{{R}_{2}}+{{R}_{3}}-{{R}_{4}}+.........{{(-1)}^{\text{n-1}}}{{R}_{n}}\]

When \[n=\infty \], then \[{{R}_{n-1}}={{R}_{n}}=0,\]therefore \[R=\frac{{{R}_{1}}}{2}\]

i.e. For large number of HPZ, the amplitude of light at point P due to whole wave front is half the amplitude due to first HPZ.

The ratio of amplitudes due to consecutive HPZ's is constant and is less than 1

\[\frac{{{R}_{n}}}{{{R}_{n-1}}}........\frac{{{R}_{5}}}{{{R}_{4}}}=\frac{{{R}_{4}}}{{{R}_{3}}}=\frac{{{R}_{3}}}{{{R}_{2}}}=\frac{{{R}_{2}}}{{{R}_{1}}}=k\]    (where \[k<1\])

(7) Resultant Intensity : Intensity \[\propto {{(\text{amplitude})}^{2}}\]

For \[n=\infty ,\]\[I\propto \frac{R_{1}^{2}}{4}\,\propto \,\frac{{{I}_{1}}}{4}\] 

i.e. the resultant intensity due to whole wave front is \[\frac{1}{4}th\] the intensity due to first HPZ.