Understanding Lasers: Chapter 4 Quiz
Source: Jeff Hecht, Understanding Lasers: An Entry-Level Guide, fourth edition (2019), Chapter 4 quiz, printed pages 123–126. Approximations below follow the conventions used by the book.
Quick answers
Question |
Answer |
|---|---|
1 |
b, \(168\ \mathrm{\mu m}\) |
2 |
e, \(1.5\ \mathrm{km}\) |
3 |
a |
4 |
e |
5 |
c, \(1.58\ \mathrm m\) |
6 |
e, \(76\ \mathrm{km}\) |
7 |
b, \(48\ \mathrm m\) |
8 |
d, \(1.46\ \mathrm m\) |
9 |
c, \(1.3\ \mathrm{mm}\) |
10 |
b, \(5.25\%\) |
11 |
d, \(90.7\%\) |
12 |
a, \(1\ \mathrm W\) |
Worked reasoning
Coherence length from wavelength spread: b. Using the chapter’s convention,
\[L_c\approx\frac{\lambda^2}{2\Delta\lambda} =\frac{(820\ \mathrm{nm})^2}{2(2\ \mathrm{nm})} =168\ \mathrm{\mu m}.\]Coherence length from frequency spread: e.
\[L_c\approx\frac{c}{2\Delta f} =\frac{2.998\times10^8}{2\times10^5} =1.50\times10^3\ \mathrm m.\]Doppler broadening: a. Different line-of-sight atomic velocities produce different Doppler shifts and widen the observed line.
Number of longitudinal modes: e. It is roughly gain bandwidth divided by cavity-mode spacing, so there is no universal fixed count.
Near-field distance: c. The chapter uses the order-of-magnitude Rayleigh-range estimate
\[z_R\approx\frac{D^2}{\lambda} =\frac{(10^{-3}\ \mathrm m)^2}{632.8\times10^{-9}\ \mathrm m} =1.58\ \mathrm m.\]A Gaussian-beam definition using waist radius instead of beam diameter has a different numerical factor, so the convention must be stated.
Unexpanded beam at geosynchronous distance: e. Taking \(1\ \mathrm{mrad}\) as the half-angle,
\[d\approx2L\theta=2(38\times10^6\ \mathrm m)(10^{-3}) =76\ \mathrm{km}.\]One-metre diffraction-limited transmitter: b.
\[\theta\approx\frac{\lambda}{D}=6.328\times10^{-7}\ \mathrm{rad}, \qquad d\approx2L\theta\approx48\ \mathrm m.\]Bare diode spot: d. With a 20-degree half-angle,
\[d=2L\tan20^\circ=2(2\ \mathrm m)\tan20^\circ=1.46\ \mathrm m.\]Lens aperture for a 1-mm spot: c. The quiz uses \(s\approx\lambda L/D\), hence
\[D\approx\frac{\lambda L}{s} =\frac{(650\times10^{-9}\ \mathrm m)(2\ \mathrm m)}{10^{-3}\ \mathrm m} =1.3\ \mathrm{mm}.\]Exact Airy-disk or Gaussian-beam definitions introduce order-one factors.
Maximum wall-plug efficiency: b.
\[\eta=(0.70)(0.25)(0.60)(0.50)=0.0525=5.25\%.\]Ideal pump conversion: d. One pump photon produces at most one laser photon, so the energy ratio is
\[\eta_{\max}=\frac{E_l}{E_p}=\frac{\lambda_p}{\lambda_l} =\frac{980}{1080}=0.907=90.7\%.\]Average pulsed power: a. First find pulse energy and then multiply by repetition rate:
\[E_p=(500\times10^3\ \mathrm W)(10\times10^{-9}\ \mathrm s) =5\times10^{-3}\ \mathrm J,\]\[P_{\mathrm{avg}}=E_p f_r=(5\times10^{-3})(200)=1\ \mathrm W.\]