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Chapter 11: Problem 9

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).

Short Answer

Expert verified
Based on the step by step solution provided, the short answer for this problem is: To find the boundary coordinate values \(u_{1}\) and \(u_{2}\), we need to solve the equations \(I(u_{1}) = 0.99 * I(\infty)\) and \(I(u_{2}) = 1.01 * I(\infty)\), using tables or computer software. Once we have these values, we can calculate the intensity contribution from the region \(\Omega\) between \(u_{1}\) and \(u_{2}\) and, given the distance between the aperture and screen \(R_0=2 m\) and the wavelength \(\lambda = 500 nm\), comment on whether most of the light reaching \(P_0\) comes from the region near \(\Omega\).

Step by step solution

01

Understanding Fresnel Integrals

Fresnel integrals are defined as: \(C(u) = \int_{0}^{u} \cos \left(\frac{\pi}{2} t^2\right) dt\) and \(S(u) = \int_{0}^{u} \sin \left(\frac{\pi}{2} t^2\right) dt\) The Cornu spiral is a graphical representation of Fresnel integrals, where the \(x\) and \(y\) coordinates are \(C(u)\) and \(S(u)\), respectively.
02

Finding the intensity error condition

The intensity \(I(u)\) is given by: \(I(u) = \lim_{|u| \rightarrow \infty} (C(u)^2 + S(u)^2)\) We are looking for the boundary coordinate value such that the intensity error is 1 percent. This means we need to find \(u_{1}\) and \(u_{2}\) values for which: \(I(u_{1}) = 0.99 * I(\infty)\) and \(I(u_{2}) = 1.01 * I(\infty)\)
03

Using tables or software to find the boundary coordinates

Now that we have set up the equations, we can use tables or a computer software specialized for Fresnel integrals to find the corresponding \(u_{1}\) and \(u_{2}\) values.
04

Comment on the statement

Once we have found the boundary coordinate values, we can calculate the contribution to intensity from the region \(\Omega\) between \(u_{1}\) and \(u_{2}\). Given the distance between aperture and screen \(R_0=2 m\) and the wavelength \(\lambda = 500 nm\), we can comment on whether most of the light reaching \(P_0\) comes from the region near \(\Omega\).

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Most popular questions from this chapter

Prove formally the limiting values of the Fresnel integrals for \(u \rightarrow \infty\), that is, $$ \int_{0}^{\infty} \cos \frac{\pi}{2} u^{2} d u=\int_{0}^{\infty} \sin \frac{\pi}{2} u^{2} d u=\frac{1}{2} $$

Show that the Fresnel integrals are related to Bessel functions of half- integral order by $$ \begin{aligned} &C(u)=J_{1 / 2}\left(\frac{\pi}{2} u^{2}\right)+J_{5 / 2}\left(\frac{\pi}{2} u^{2}\right)+J_{9 / 2}\left(\frac{\pi}{2} u^{2}\right)+\cdots \\ &S(u)=J_{3 / 2}\left(\frac{\pi}{2} u^{2}\right)+J_{7 / 2}\left(\frac{\pi}{2} u^{2}\right)+J_{11 / 2}\left(\frac{\pi}{2} u^{2}\right)+\ldots \end{aligned} $$

What is the effect of an obstacle (as opposed to an aperture) in the Fraunhofer diffraction limit?

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?

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.
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