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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#2, Rev. B, due Wednesday, 2017-09-13 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
Ch.~1 \#4, %p14, triangle wave
      \#5%p14, exp triangle time-dep
.

{\bf 1.} Optical elements can be analyzed in terms of three independent
principles: surface wave optics, focusing geometry, and geometric ray optics.

{\bf \quad a) Surface optics}---Show using the principle of positive
interference that the same formula
$\Delta(nd\sin\theta)\equiv (n_2\sin\theta_2-n_1\sin\theta_1) d = m \lambda$
applies to both reflection, refraction, and diffraction, where $d$ is spacing
between individual rules of the diffraction grating and $m=0$ (thus $d$ is
irrelevant) for reflection or refraction.  Note that diffraction can be either
reflective or transmissive, with the 0$^{\mathrm{th}}$ order diffraction peak
corresponding to reflection or refraction, respectively.  The modified length
$nd$ is called the optical path length, and is a measure of relative phase.

{\bf \quad b)} The following construction generalizes paraxial reflective and
diffractive focusing all angles: given a 2-d curved mirror, the {\bf Rowland
  Circle} is drawn with its diameter extending from the middle of mirror to
the mirror's center of curvature.  Use the
\href{http://en.wikipedia.org/wiki/Inscribed_angle}{inscibed angle theorem} to
show that all rays originating at the same point on the Rowland circle will
have the same angle of incidence on the mirror, and thus the same angle of
reflection/diffraction.  Working backwards, show that all of these rays will
focus on another point of the Rowland circle.  [Bonus:] How does this
generalize for refraction?

{\bf \quad c)} [bonus] {\bf Geometric optics} involves tracing rays to find
the image point where they intersect.  Show that the main result $1/f=1/i+1/o$
generalizes from paraxial reflective/refractive optics to off-axis and also to
diffraction.  In general, the focal length $f$ is twice the distance from the
center of the optics to the intersection of the ray and the Rowland circle.

{\bf 2. COW Experiment}---A beam of neutrons of de Broglie wavelength 2~\AA\
passes through a Mach-Zehnder interferometer, with lattice spacing
$d$=3.135~\AA\ between the $\langle 111\rangle$ planes of Si.  These planes
are alligned perpendicular to the blade faces, so that diffracted neutrons
pass through the blade and are bent in the opposite direction as undiffracted
neutrons, but at the same angle of incidence about the normal, as shown in the
figure.

{\bf \quad a)} Calculate the Bragg scattering angle $\theta$, which equals the
angle \mbox{of incidence and diffraction.}

{\bf \quad b)} As neutrons rise and lose kinetic energy to the gravitational
potential, their wavelength expands.  Assuming an $L=30$~mm separation between
each of the three blades, calculate the phase difference between two
inteferring paths.  (see
\href{http://www.phys.utk.edu/neutron-summer-school/lectures/wietfeldt.pdf}{Neutron
  interferometry}, page 41).

{\bf 3.} Normalize the {\bf Gaussian Wave Packet}
$A(k)=Ne^{-\frac14 \left(\frac{k-k_0}{\Delta k}\right)^{2}+ikx_0+i\phi_{0}}$ and
calculate it's Fourier transform to determine the coresponding wave function
in position space.  Identify the features corresponding to the carrier wave,
packet envelope, and global phase in both frequency and position
representations.
% calculate <k>,<k^2> also

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\includegraphics[width=4in]{H01_fig1}
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