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Chapter 11: Problem 5

A zone plate is a screen made by blackening even-numbered (alternatively, oddnumbered) zones whose outer radii are given by \(\rho_{n}{ }^{2}=n \lambda R_{o}(n=1,2, \ldots) .\) Plane monochromatic waves incident on it are in effect focused on a point \(P_{o}\), distant \(R_{o}\) from the zone plate, since the wavefronts arriving at \(P_{0}\) from the odd-numbered zones are all in phase. Show that a sequence of such focus points exist. Show also that the zone plate focuses a point source at a distance \(p\) into a point image at a distance \(q\), where \(p\) and \(q\) are related by the familiar lens equation $$ \frac{1}{p}+\frac{1}{q}=\frac{1}{f}=\frac{\lambda}{\rho_{1}{ }^{2}} $$ Does an equation of this form hold for each of the other images?

Short Answer

Expert verified
Answer: Yes, a zone plate can create a sequence of focus points, and the lens equation can be extended to other images formed by the zone plate.

Step by step solution

01

Understand the problem and given information

The zone plate has even-numbered zones being blackened. The outer radii of those zones are given by \(\rho_{n}{ }^{2}=n \lambda R_{o}\). From this given information, we need to find out if a sequence of focus points exists, and relate the lens equation to the scenario.
02

Find a sequence of focus points

As waves incident on the zone plate from odd-numbered zones are all in phase, the wavefronts will constructively interfere at a common point \(P_0\) at a distance \(R_0\). This means that: $$ (n+1/2)\lambda = \sqrt{n'\lambda R_0} $$ For each new focus point \(n'\), this equation can be rewritten as: $$ n'+1/2 = n \frac{R_0}{R'}, $$ where \(R'\) is the distance of the new focus point. From this, we can see that a sequence of focus points exists, as we can change n or, in other words, increase the distance \(R'\) to form more focus points according to this relationship.
03

Relate the lens equation to the focusing properties of the zone plate

We now proceed to relate the lens equation to the focusing properties of the zone plate. The given lens equation is: $$\frac{1}{p}+\frac{1}{q}=\frac{1}{f}=\frac{\lambda}{\rho_{1}{ }^{2}}$$ Let's consider an incoming wave coming at a distance p and being focused at a distance q from the zone plate. Using the outer radius value for the first zone, \(\rho_{1}{ }^{2}=\lambda R_{o}\), and the equation previously derived in step 2, we get: $$ \frac{1}{p}+\frac{1}{q}=\frac{1}{(n+1/2)\lambda}=\frac{\lambda}{\rho_{1}{ }^{2}} $$
04

Apply the lens equation to other images

Using the lens equation, let's see if the relationship can be applied to the other images. Let's consider an image formed by the incoming wave and the next focus point. The equation derived in step 2 still applies here. This means that for the next focus point \(n'\), we have: $$ \frac{1}{p'}+\frac{1}{q'}=\frac{1}{(n'+1/2)\lambda}=\frac{\lambda}{\rho_{1}{ }^{2}} $$ As \(\rho_{1}{ }^{2}\) is a constant, this equation will hold true for all focus points in the sequence. In conclusion, the exercise shows that a zone plate can create a sequence of focus points, and the lens equation can be extended to other images as well.

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Most popular questions from this chapter

A step function is a function whose value changes abruptly from zero to some constant. In general, this abrupt rise takes place over a small but finite interval of the argument. Also, the function may overshoot before converging to its final constant value. The rise-inlerval (or rise-time if the abscissa happens to be time) is usually defined as the increment in the argument over which the function changes from 10 to 90 percent of its final value. What is the rise-interval \(\delta u\) of the straight-edge intensity diffraction pattern? To what distance in the observation plane does this correspond for the special case of \(R_{\mathrm{s}}=R_{\mathrm{o}}=2 \mathrm{~m}\), \(\lambda=500 \mathrm{~nm}\) ? What is the percentage overshoot? Answer: \(\delta u=1.17, \delta x=1.17 \mathrm{~mm} ; 37\) percent.

Using the Cornu spiral or tables of the Fresnel integrals, investigate how large the boundary coordinate \(u_{1}\) or \(u_{2}\) must be in order that an intensity error of only 1 percent be made by substituting the limit point \((|u| \rightarrow \infty)\). Assuming \(R_{s}=R_{0}=2 \mathrm{~m}, \lambda=500 \mathrm{~nm}\), comment on the statement that most of the light reaching \(P_{0}\) comes from the region of the aperture near \(\Omega\) (Fig. 11.3.1).

Develop the Fresnel diffraction of a slit using a coordinate system fixed to the center of the slit (but source and observation point may be off axis). Show that the phase exponent \(\kappa\left(r_{s}+r_{o}\right)\) in the integrand of the Kirchhoff integral contains both a linear and a quadratic term in the aperture coordinate. (Under what conditions can the quadratic term be neglected, giving the Fraunhofer limit?) By completing the square, show that the "moving" coordinate system used in this section (Fig. 11.2.1) eliminates the linear term, i.e., results in a pure quadratic phase exponent and the conventional Fresnel integrals.

In the paraxial approximation, the Fresnel diffraction pattern of a slit may be specified uniquely by two normalized parameters: (1) the width of the slit measured as $$ \Delta u=\left[\frac{2}{\lambda}\left(\frac{1}{R_{s}}+\frac{1}{R_{0}}\right)\right]^{1 / 2} a $$ or alternatively the number \(n=(\Delta u)^{2} / 8\) of Fresnel linear zones unmasked, and (2) the angle of the observation point off the axis measured as \(\theta / \theta_{0}\) where \(\theta_{0}=\lambda / a\). Compare the diffraction patterns when \((a) a=1 \mathrm{~mm}, R_{s}=R_{0}=1 \mathrm{~m}\) and, \((b) a=2 \mathrm{~mm}, R_{\varepsilon}=R_{0}=4 \mathrm{~m}\).

You wish to construct a pinhole camera (no lens) with \(10 \mathrm{~cm}\) separation between pinhole and film plane. The film is sensitive to visible light \((\lambda \sim 500 \mathrm{~nm}) .\) What pinhole diameter would you choose for optimum resolution, i.e., smallest photographic image of a point object, and what order of magnitude of angular resolution would be obtained?
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