Explorations-02Learning objectives:

a) Describe the nature of diffraction patterns in the near and the far fieldand distinguish between them.

b) Predict the on-axis intensity for different cases of diffraction.c) Analyze complicated wave forms in terms of appropriate sinusoidal

components.

1. A spotless theory! : Find out the intensity distribution at farfield when a plane wave is obstructed by a small opaquedisc of diameter d. What is the most surprising aspect andwhat is the historical significance of this result?

2. Jincle all the way to patterns of current relevance…: Use any simplegraph plotting software tofamiliarize yourself with the sinc andjinc functions (and their squares),paying attention to their positions ofmaxima, zeroes, variation ofamplitudes etc. It is a very goodidea to be in sync with Math, it willhelp you all the way throughout your career!

3. The plot thickens… : Plane waves of wavelength λ are incident on a platewith a circular aperture of diameter d.a) Find the intensity at an on-axial point (x=y=0)

located at a distance z from the origin, asgiven by the direct solution ofFresnel-Huygens equation without Fresnel’sapproximation (worked out in the ebookchapter in Moodle) and obtain the ratio (I/I0)as a function of z.

b) Repeat using Fresnel’s approximation.

c) Now repeat this with Fraunhofer approximation and plot all the threedifferent curves, plots of (I/I0) as a function of z obtained from thethree methods a, b and c, in the same plot, using logarithmic scale(why?).

c) Interpret your result physically.

4. What is the typical distance where Fresneldiffraction changes over to Fraunhofer diffraction?Estimate this for a 1 mm X 5 mm slit when youuse radio waves (100 m), microwaves (1 cm),red light (650 nm) green light (500nm) andGamma rays (1 picometer, i.e. 10-12m). Also foracoustic waves ( 1m) . What would be the typicalslit width to be used for radio waves of 1 km wavelength in order toobserve diffraction?

5. Circling a square ... A rectangular wave is drawn in such a way that( ) 1 for / 8 / 8f u and ( ) 0 for / 8 7 / 8f u ; this

repeats on either side of the origin to form the square wave.

a) Expand this function in a Fourier series.

b) Plot the first several terms (using some software) and observe theemerging wave pattern as we add more and more terms.

c) Plot the first several Fourier coefficients and draw the envelope.What do you conclude?

8. Extreme exploration (out of syllabus for the exams!) : Use any freewareMath package such as Sage or Scilab (open source equivalents ofMathematica and Matlab), or just Excel, if it can be used!) toobtain the Fresnel and Fraunhofer diffraction patterns for a rectangularslit of given dimensions at any distance. Use appropriateapproximations to make your life simple!

Bottom Line:

Optics is NOT light work, but making light work!