Photonics Essentials: Chapter 6 Problems

Source

Thomas P. Pearsall, Photonics Essentials: An Introduction with Experiments (McGraw-Hill, 2003), Chapter 6, Light-Emitting Diodes, Problems 6.1–6.6, printed pages 139–141.

Problem 6.1: Red LED turn-on

Figure 6.4 peaks at approximately \(700\ \mathrm{nm}\). Therefore,

\[E_\gamma=\frac{1239.84}{700} =\boxed{1.77\ \mathrm{eV}}.\]

Visible emission begins around \(1.4\ \mathrm V\), but the applied voltage is not a strict per-electron energy ceiling. Carriers have a thermal energy distribution, the junction has a built-in potential and band bending, and recombining electrons and holes already occupy states in the conduction and valence bands. The photon energy is set principally by the band-to-band energy separation, not simply by \(qV\).

At \(77\ \mathrm K\):

  • the thermal tail becomes narrower, so the onset should be sharper;

  • the semiconductor band gap normally increases, shifting emission to shorter wavelength and requiring a somewhat larger forward voltage;

  • nonradiative processes and series resistance may also change.

Problem 6.2: Why forward bias emits efficiently

Forward bias does two essential things before substantial light and current appear:

  1. It narrows the depletion barrier and brings injected electrons and holes into the same physical region.

  2. It creates large nonequilibrium carrier populations, including occupied conduction-band states and empty valence-band states at compatible energy and momentum.

These conditions produce a high radiative recombination rate. Reverse bias separates and removes carriers, which is why it is useful for photodetection rather than efficient LED emission.

Problem 6.3: Accepting an LED shipment

Values must be read from the printed spectrum, so use appropriate precision:

\[\lambda_{\mathrm{peak}}\approx480\ \mathrm{nm},\qquad P_{\mathrm{peak}}\approx0.095\ \mathrm{mW}.\]

The half-power crossings are about \(442\) and \(521\ \mathrm{nm}\). The wavelength FWHM is therefore about \(79\ \mathrm{nm}\). Convert the two endpoints to energy rather than treating the conversion as exactly linear:

\[\begin{split}\begin{aligned} \Delta E_{\mathrm{FWHM}} &=\frac{1239.84}{442}-\frac{1239.84}{521}\\ &\approx\boxed{0.42\ \mathrm{eV}}. \end{aligned}\end{split}\]

For a \(20\ \mathrm{mA}\) drive, the injected electron rate is \(I/q\). Treating the graph’s peak optical-power value as the emitted power specified by the exercise,

\[\eta =\frac{P_{\mathrm{opt}}/(hc/\lambda)}{I/q} =\frac{P_{\mathrm{opt}}q\lambda}{Ihc} \approx1.84\times10^{-3}.\]

Thus

\[\boxed{\eta\approx0.184\%>0.1\%},\]

so the sample passes and the shipment is accepted, subject to a statistically adequate sampling plan. One sample cannot establish the defect rate of 500,000 devices.

Problem 6.4: Traffic-light profitability

This problem requires local quotations; the following is a reusable model and an illustrative calculation, not a claim about current prices in any particular city.

Let \(P_i\) and \(P_L\) be incandescent and LED powers, \(t_y\) the energized hours per year, \(c_e\) the electricity price, \(C_m\) the annual maintenance saving, and \(C_0\) the installed conversion cost. Annual saving is

\[S_y=(P_i-P_L)t_yc_e+C_m.\]

For an illustrative \(70\ \mathrm W\) lamp, \(10\ \mathrm W\) LED module, one-third duty cycle, \(c_e=\$0.25/\mathrm{kWh}\), \(C_m=\$40/\mathrm{yr}\), and \(C_0=\$500\),

\[t_y=\frac{8760}{3}=2920\ \mathrm h,\]
\[S_y=(0.070-0.010)(2920)(0.25)+40 =\$83.80/\mathrm{yr}.\]

The simple payback is

\[\boxed{C_0/S_y\approx6.0\ \text{years}}.\]

For a ten-year life and 5% discount rate,

\[\mathrm{NPV}_{\mathrm{savings}} =S_y\frac{1-(1.05)^{-10}}{0.05} \approx\$647.\]

Since \(\$647>\$500\), the illustrative conversion is profitable. A real study should replace every assumed value and include failure rate, cleaning, driver replacement, traffic-control labor, and residual value. The book’s Example 6.3 obtained an eight-year affordable conversion price of about \(\$480\) using its older energy and labor assumptions.

Problem 6.5: LED bandwidth

Equation 6.40 is

\[\frac{R(f)}{R(0)} =\frac{1}{\sqrt{1+(2\pi f\tau)^2}}.\]

The chapter defines bandwidth at half amplitude, not at the conventional \(1/\sqrt2\) amplitude point. Set \(R(f)/R(0)=1/2\):

\[1+(2\pi f\tau)^2=4,\]

so

\[\boxed{f_{\mathrm{BW}}=\frac{\sqrt3}{2\pi\tau}}.\]

For high injection, the chapter gives

\[\frac1{\tau_{\mathrm{ac}}} =\left(\frac{BJ}{3qd}\right)^{1/2}.\]

Therefore,

\[\boxed{ f_{\mathrm{BW,high}} =\frac{\sqrt3}{2\pi} \left(\frac{BJ}{3qd}\right)^{1/2} =\frac{1}{2\pi}\sqrt{\frac{BJ}{qd}} }.\]

For low injection,

\[\frac1{\tau_{\mathrm{ac}}}=Bn_D+\frac1{\tau_{n-r}},\]

which gives

\[\boxed{ f_{\mathrm{BW,low}} =\frac{\sqrt3}{2\pi} \left(Bn_D+\frac1{\tau_{n-r}}\right) }.\]

Thus high-injection bandwidth scales as \(\sqrt{BJ}\), while the low-injection result is linear in \(B\) and independent of drive current within that approximation.

Problem 6.6: Green and amber traffic signals

This question reflects the economics and LED technology at the time the book was written.

Green. Efficient wide-band-gap green emitters were historically harder and more expensive to manufacture than mature red devices. Because the green indication often has a substantial duty cycle, its energy and maintenance savings can still be large. The original barrier was therefore mainly device technology and purchase price, rather than an inability to save operating cost.

Amber. Amber devices also had a cost and efficiency disadvantage, but an amber traffic phase is usually brief. Its low duty cycle means fewer saved kilowatt-hours and fewer avoided lamp-hours, so the payback is weaker. That makes the barrier more strongly economic. Other amber-LED uses include turn signals, hazard flashers, warning beacons, construction signs, and status indicators, where visibility, ruggedness, and fast switching can justify the device even without long daily operating time.

The chapter’s own summary notes that efficient red, green, and blue LEDs were already commercially available, so the problem should be read as a source-era engineering comparison rather than a timeless statement of market availability.