Photonics Essentials: Chapter 8 Problems

Source

Thomas P. Pearsall, Photonics Essentials: An Introduction with Experiments (McGraw-Hill, 2003), Chapter 8, Direct Modulation of Laser Diodes, Problems 8.1–8.3, printed page 190.

The book labels both of its final two exercises 8.2. This page calls the last one 8.3.

Problem 8.1: Band filling and wavelength chirp

Let a current pulse be shorter than the recombination time.

Before the pulse

\[E_C:\ \text{few excess electrons}, \qquad E_V:\ \text{few excess holes}.\]

During carrier injection

Carriers enter faster than they recombine. States nearest the band edges fill first; Pauli exclusion forces later electrons to higher conduction-band states and holes deeper into the valence band:

\[E_{\gamma,\mathrm{early}} =(E_C+\Delta E_e)-(E_V-\Delta E_h) =E_g+\Delta E_e+\Delta E_h.\]

After stimulated recombination begins

The highest-energy occupied states recombine and empty first. The carrier distribution retreats toward the band edges:

\[E_{\gamma}(t)\downarrow E_g.\]

Since \(\lambda=hc/E_\gamma\), decreasing photon energy means increasing wavelength:

\[\boxed{\lambda(t)\ \text{chirps from shorter to longer wavelength}}.\]

Problem 8.2: Delay with 90% threshold prebias

Equation 8.8 gives

\[\tau_d=\tau_r \ln\left(\frac{J_2-J_1}{J_2-J_{\mathrm{th}}}\right),\]

where \(J_2>J_{\mathrm{th}}>J_1\). Put \(\tau_r=10^{-10}\ \mathrm s=100\ \mathrm{ps}\) and \(J_1=0.9J_{\mathrm{th}}\):

\(J_2/J_{\mathrm{th}}\)	1.1	5	10	20
\(\tau_d\) (ps)	69.3	2.47	1.10	0.525

For example,

\[\tau_d(1.1J_{\mathrm{th}}) =(100\ \mathrm{ps}) \ln\left(\frac{1.1-0.9}{1.1-1}\right) =69.3\ \mathrm{ps}.\]

Problem 8.3: Other prebias levels

Repeat the same calculation:

Design curves from Equation 8.8. Both axes are logarithmic.

\(J_1/J_{\mathrm{th}}\)	\(1.1J_{\mathrm{th}}\)	\(5J_{\mathrm{th}}\)	\(10J_{\mathrm{th}}\)	\(20J_{\mathrm{th}}\)
0.1	230 ps	20.3 ps	9.53 ps	4.63 ps
0.5	179 ps	11.8 ps	5.41 ps	2.60 ps
0.9	69.3 ps	2.47 ps	1.10 ps	0.525 ps

The design trends are clear:

\[J_1\uparrow\Longrightarrow\tau_d\downarrow, \qquad J_2\uparrow\Longrightarrow\tau_d\downarrow.\]

A negligible delay is obtained by prebiasing close to threshold and driving several times above threshold. Neither variable should be taken to an extreme: high prebias increases off-state light and power, while a large current step increases heating, relaxation oscillations, and wavelength chirp. A practical choice is therefore the smallest \(J_1\) and \(J_2\) that meet the system’s delay, extinction-ratio, thermal, and spectral requirements.