Exercises

1. 1. Consider the geometric series
      $$S= \sum_{n=0,N-1} z^{\,n},$$
      where $z$ is a complex number. Demonstrate that
      $$S = \frac{1-z^{\,N}}{1-z}.$$

   2. Suppose that $z={\rm e}^{\,{\rm i}\,\theta}$, where $\theta$ is real. Employing the well-known identity
      $$\sin\theta \equiv \frac{1}{2\,{\rm i}}\left({\rm e}^{\,{\rm i}\,\theta} - {\rm e}^{-{\rm i}\,\theta}\right),$$
      show that
      $$S = {\rm e}^{\,{\rm i}\,(N-1)\,\theta/2}\,\frac{\sin(N\,\theta/2)}{\sin(\theta/2)}.$$

   3. Finally, making use of Euler's theorem,
      $${\rm e}^{\,{\rm i}\,n\,\theta} \equiv \cos(n\,\theta) + {\rm i}\,\sin(n\,\theta),$$
      demonstrate that
      $$C=\sum_{n=1,N} \cos(\alpha\,x_n),$$
      where
      $$x_n = [n-(N+1)/2]\,d,$$
      evaluates to
      $$C = \frac{\sin(N\,\alpha\,d/2)}{\sin(\alpha\,d/2)}.$$

2. Derive Equation (10.30).

3. An interference experiment employs two narrow parallel slits of separation $0.25\,{\rm mm}$, and monochromatic light of wavelength $\lambda=500\,{\rm nm}$. Estimate the minimum distance that the projection screen must be placed behind the slits in order to obtain a far-field interference pattern.

4. A double-slit of slit separation $0.5\,{\rm mm}$ is illuminated at normal incidence by a parallel beam from a helium-neon laser that emits monochromatic light of wavelength $632.8\,{\rm nm}$. A projection screen is located $5\,{\rm m}$ behind the slit. What is the separation of the central interference fringes on the screen? [From Crawford 1968.]

5. Consider a double-slit interference experiment in which the slit spacing is $0.1\,{\rm mm}$, and the projection screen is located $50\,{\rm cm}$ behind the slits. Assuming monochromatic illumination at normal incidence, if the observed separation between neighboring interference maxima at the center of the projection screen is $2.5\,{\rm mm}$, what is the wavelength of the light illuminating the slits?

6. What is the mean length of the classical wavetrain (wave packet) corresponding to the light emitted by an atom whose excited state has a mean lifetime $\tau\sim 10^{\,-8}\,{\rm s}$? In an ordinary gas-discharge source, the excited atomic states do not decay freely, but instead have an effective lifetime $\tau\sim 10^{\,-9}\,{\rm s}$, due to collisions and Doppler effects. What is the length of the corresponding classical wavetrain? [From Crawford 1968.]

7. If a “monochromatic” incoherent “line” source of visible light is not really a line, but has a finite width of $1\,{\rm mm}$, estimate the minimum distance it can be placed in front of a double-slit, of slit separation $0.5\,{\rm mm}$, if the light from the slit is to generate a clear interference pattern.

8. The visible emission spectrum of a sodium atom is dominated by a yellow line which actually consists of two closely spaced lines of wavelength $589.0\,{\rm nm}$ and $589.6\,{\rm nm}$. Demonstrate that a diffraction grating must have at least 328 lines in order to resolve this doublet at the third spectral order.

9. Consider a diffraction grating having 5000 lines per centimeter. Find the angular locations of the principal maxima when the grating is illuminated at normal incidence by (a) red light of wavelength 700 nm, and (b) violet light of wavelength 400 nm.

10. A soap bubble 250 nm thick is illuminated by white light. The index of refraction of the soap film is $1.36$. Which colors are not seen in the reflected light? Which colors appear bright in the reflected light? What color does the soap film appear at normal incidence?

11. Suppose that a monochromatic laser of wavelength $632.8\,{\rm nm}$ emits a diffraction-limited beam of initial diameter 2 mm. Estimate how large a light spot the beam would produce on the surface of the Moon (which is a mean distance $3.76\times 10^{\,5}\,{\rm km}$ from the surface of the Earth). Neglect any effects of the Earth's atmosphere. [From Hecht and Zajac 1974.]

12. Estimate how far away an automobile is when you can only just barely resolve the two headlights with your eyes. [From Crawford 1968.]

13. Venus has a diameter of about 8000 miles. When it is prominently visible in the sky, in the early morning or late evening, it is about as far away as the Sun; that is, about 93 million miles away. Venus commonly appears larger than a point to the unaided eye. Are we seeing the true size of Venus? [From Crawford 1968.]

14. The world's largest steerable radio telescope, at the National Radio Astronomy Observatory, Green Bank, West Virginia, consists of a parabolic disk that is 300 ft in diameter. Estimate the angular resolution (in minutes of an arc) of the telescope when it is observing the well-known 21-cm radiation of hydrogen. [From Crawford 1968.]

15. Estimate how large the lens of a camera carried by an artificial satellite orbiting the Earth at an altitude of 150 miles would have to be in order to resolve features on the Earth's surface a foot in diameter.

16. Demonstrate that the secondary maxima in the far-field interference pattern generated by three identical, equally spaced, parallel slits of negligible width are nine times less intense than the principal maxima.

17. Consider a double-slit interference/diffraction experiment in which the slit spacing is $d$, and the slit width $\delta$. Show that the intensity of the far-field interference pattern, assuming normal incidence by monochromatic light of wavelength $\lambda$, is
    $${\cal I}(\theta)\propto \cos^2\left(\pi\,\frac{d}{\lambda}\,\sin\theta\right){\rm sinc}^2\left(\pi\,\frac{\delta}{\lambda}\,\sin\theta\right).$$
    Plot the intensity pattern for $d/\lambda=8$ and $\delta /\lambda =2$.

18. What is the theoretical maximum angular resolving power (in arc seconds) of a conventional reflecting telescope with a 12 in main mirror?

19. 1. Demonstrate that, in the far-field limit, Equation (10.110) reduces to
       $$\frac{{\cal I}(u',v')}{{\cal I}_0} = F_c^{\,2}(u',v')+F_s^{\,2}(u',v'),$$
       where

       $$F_c(u',v') =\frac{1}{2}\int_{-\infty}^\infty\int_{-\infty}^\infty f(u,v)\,\cos\left[\pi\,(u\,u'+v\,v')\right]\,du\,dv,$$

       $$F_s(u',v') =\frac{1}{2}\int_{-\infty}^\infty\int_{-\infty}^\infty f(u,v)\,\sin\left[\pi\,(u\,u'+v\,v')\right]\,du\,dv.$$

    2. Hence, deduce that
       $$\frac{{\cal I}(0,0)}{{\cal I}_0} = \left(\frac{A}{\lambda\,R}\right)^{\,2},$$
       where $A$ is the area of the aperture, $R$ the distance of the projection screen behind the aperture, and $\lambda$ the wavelength.

20. Consider the far-field diffraction pattern of a circular aperture of radius $a$, normally illuminated by monochromatic light of wavelength $\lambda$. Let $\theta$ denote the angular distance from the optic axis (i.e., the line that passes through the center of the aperture, and is perpendicular to the plane of the aperture and the projection screen) on the projection screen. Demonstrate that the mean energy flux that falls within the region $0<\theta<\theta_0$ is
    $${\cal I}_0\,\pi\,a^{\,2}\left[1-J_0^{\,2}(k\,a\,\theta_0)-J_1^{\,2}(k\,a\,\theta_0)\right],$$
    where ${\cal I}_0$ is the intensity of the incident wave, and $k=2\pi/\lambda$. [Hint: $J_1'(z)=J_0(z)-J_1(z)/z$ and $J_0'(z)=-J_1(z)$, where $'$ denotes a derivative with respect to argument.] Hence, deduce that the total energy flux that illuminates the projection screen is the same as that predicted by geometric optics. Finally, show that the Airy disk contains approximately $84\%$ of the total energy flux.

21. The Fresnel integrals have the asymptotic forms (Abramowitz and Stegun 1965)

    $$C(z) \simeq \frac{1}{2}+\frac{1}{\pi\,z}\,\sin\left(\frac{\pi\,z^{\,2}}{2}\right),$$

    $$S(z)\simeq \frac{1}{2}-\frac{1}{\pi\,z}\,\cos\left(\frac{\pi\,z^{\,2}}{2}\right)$$

    in the large-argument limit, $z\gg 1$. Use these forms to demonstrate that the intensity of a diffraction pattern of a semi-infinite opaque plane bounded by a sharp straight edge, which is specified in Equation (10.132), reduces to
    $$\frac{{\cal I}(u') }{{\cal I}_0}\simeq \frac{1}{2\,\pi^{\,2}\,\vert u'\vert^{\,2}}$$
    in the extreme shadow region (i.e., $u'<0$ and $\vert u'\vert\gg 1$). This implies that the intensity in the shadow region attenuates as the inverse-square of the distance from the straight edge.

22. 1. The $J_n$ Bessel functions satisfy the recursion relation (Abramowitz and Stegun 1965)
       $$\frac{d}{dz}\left[z^{\,n+1}\,J_{n+1}(z)\right]=z^{\,n+1}\,J_n(z),$$
       where $n$ is a non-negative integer. By repeatedly integrating by parts, demonstrate that the functions

       $${\cal C}(u,v) = u\int_0^1\cos\left(\frac{u\,z^{\,2}}{2}\right) J_0(v\,z)\,z\,dz,$$

       $${\cal S}(u,v) = u\int_0^1\sin\left(\frac{u\,z^{\,2}}{2}\right) J_0(v\,z)\,z\,dz$$

       can be expanded in the forms

       $${\cal C}(u,v) =\cos\left(\frac{u}{2}\right)U_1(u,v)+\sin\left(\frac{u}{2}\right)U_2(u,v),$$

       $${\cal S}(u,v) =\sin\left(\frac{u}{2}\right)U_1(u,v)-\cos\left(\frac{u}{2}\right)U_2(u,v),$$

       where
       $$U_n(u,v)=\sum_{s=0,\infty} (-1)^s\left(\frac{u}{v}\right)^{n+2s}J_{n+2s}(v).$$
       is a Lommel function.

    2. The $J_n$ Bessel functions also satisfy the recursion relation (Abramowitz and Stegun 1965)
       $$\frac{d}{dz}\left[z^{-n}\,J_{n}(z)\right]=-z^{-n}\,J_{n+1}(z),$$
       where $n$ is a non-negative integer. By repeatedly integrating by parts, demonstrate that

       $${\cal C}(u,v) =\sin\left(\frac{v^{\,2}}{2\,u}\right)+\sin\left(\frac{u}{2}\right)V_0(u,v)-\cos\left(\frac{u}{2}\right)V_1(u,v),$$

       $${\cal S}(u,v) =\cos\left(\frac{v^{\,2}}{2\,u}\right)-\cos\left(\frac{u}{2}\right)V_0(u,v)-\sin\left(\frac{u}{2}\right)V_1(u,v),$$

       where
       $$V_n(u,v)=\sum_{s=0,\infty} (-1)^s\left(\frac{v}{u}\right)^{n+2s}J_{n+2s}(v).$$
       is a Lommel function. Note that $\lim_{\,z\rightarrow 0} J_n(z)/z^{\,n} = 1/(2^{\,n}\,n!)$ (Abramowitz and Stegun 1965).

23. Consider the near-field diffraction pattern a circular aperture of radius $a$, normally illuminated by monochromatic light of wavelength $\lambda$.
    1. Show that the illumination intensity at the point where the optic axis (see Exercise 20) meets the projection screen is
       $${\cal I} = 4\,{\cal I}_0\,\sin^{2}\left(\frac{\pi}{2}\,\frac{a^{\,2}}{\lambda\,R}\right),$$
       where ${\cal I}_0$ is the intensity of the incident wave, and $R$ is the distance of the projection screen behind the aperture. Hence, deduce that if the so-called Fresnel number,
       $$N = \frac{a^{\,2}}{\lambda\,R},$$
       takes an even integer value then ${\cal I}=0$, and if the Fresnel number takes an odd-integer value then ${\cal I}=4\,{\cal I}_0$.

    2. Show that the illumination intensity a radial distance $r'$ from the optic axis asymptotes to
       $${\cal I}= {\cal I}_0\left[\frac{a}{r'}\,J_1\left(2\pi\,\frac{a\,r'}{\lambda\,R}\right)\right]^{\,2}$$
       in the limit $r'\gg a$.

24. Consider the near-field diffraction pattern a circular disk of radius $a$, normally illuminated by monochromatic light of wavelength $\lambda$.
    1. Show that the illumination intensity at the point where the optic axis (see Exercise 20) meets the projection screen is
       $${\cal I} = {\cal I}_0,$$
       where ${\cal I}_0$ is the intensity of the incident wave.
    2. Show that the illumination intensity a radial distance $r'$ from the optic axis asymptotes to
       $${\cal I}= {\cal I}_0\left[1-\frac{2\,a}{r'}\,J_1\left(2\pi\,\frac... ...r'}{\lambda\,R}\right)\sin\left(\pi\,\frac{r'^{\,2}}{\lambda\,R}\right)\right]$$
       in the limit $r'\gg a$. Here, $R$ is the distance of the projection screen behind the aperture.