Amplification of Radiation in a Doped Glass Fiber

Abstract

We study the dynamics of gain of fiber amplifiers and fiber lasers.

Problems

18.1

Fiber laser. Estimate the efficiency of an erbium-doped fiber laser pumped with a pump power twice the threshold pump power if the laser is pumped with 1480-nm radiation or if it is pumped with 980-nm radiation.

18.2

A glass contains erbium ions of a density $N_0 =7 \times 10^{25}$ m$^{-3}$.

• (a) Determine the average distance $r_0$ between neighboring erbium ions.

• (b) Estimate the number of neighbors of an erbium ion that lie in spherical shells (thickness $r_0$) with the radii $r_0$ and $2 r_0$; $2 r_0$ and $3 r_0$; $3 r_0$ and $4 r_0$; furthermore with the radii $s r_0$ and $s r_0 + r_0$ with s $\gg 1$.

18.3

Occupation number difference in an erbium-doped fiber amplifier.

• (a) Show that the occupation number difference at the energy $E_\mathrm{c}$ for Fermi energies in the vicinity of $E_\mathrm{c}$ is given by $f_2-f_1 =2f(E_\mathrm{c})-1\approx (E_\mathrm{F}-E_\mathrm{c})/2kT$.

• (b) How large is $f_2 -f_1$ if $E_\mathrm{F}=E_\mathrm{c}+k T$?

• (c) Determine the percentage of energy levels of the quasiband lying in the energy range $E_\mathrm{c}-\varDelta E/2, E_\mathrm{c}+\varDelta E/2$.

18.4

Discuss why the following lasers do not belong to the type “quasiband laser”:

• (a) Titanium–sapphire laser, alexandrite laser; and, generally, vibronic lasers.

• (b) Helium–neon laser.

• (c) Continuous wave CO$_2$ laser and TEA CO$_2$.

18.5

Density of states. We consider the following case: the density of states of quasiparticles in an erbium-doped fiber is the sum of two densities of state, $D =D_1 + D_2$; the center frequencies have a frequency distance of 4kT(T $=$  300 K); the halfwidth of both densities of states is 2kT.

• (a) Estimate the maximum gain coefficient $\alpha _\mathrm{max}$.

• (b) Estimate the maximum gain coefficient in the case that the center frequencies have a frequency distance of kT.

18.6

Present arguments that show that it is most likely that the spontaneous lifetime $\tau _\mathrm{s p}$ of the $^4$I$_{13/2}$ level of erbium ions in a glass fiber depend on the quasiband filling factor.

18.7

Einstein coefficients. Consider an impurity-doped fiber with a Gaussian shape of the density of states of quasiparticles.

• (a) Design a dependence $B_{21}(E)$ that leads to a double peak in the gain curve.

• (b) Then discuss the dependence of $\tau _\mathrm{s p}$ on the filling factor.

18.8

Temperature coefficient. Make use of the quasiparticle model to estimate the temperature coefficient (in units of dB/$^{\circ }$C) of an erbium-doped fiber amplifier of 10 m length for the temperature ranges 10–20, −50 to −40 and 50–60 $^{\circ }$C:

• (a) If the frequency of the radiation is equal to the center frequency.

• (b) If the frequency of maximum gain occurs at a filling factor of 0.6.

18.9

Fiber laser and fiber amplifier. Determine the gain of radiation passing through an erbium-doped fiber (length 16 m) pumped at twice the transparency density; for data, see Sect. 18.6.

18.10

Why is the population of the multiplet levels of the ground state of Er$^{3+}$ not in thermal equilibrium during optical pumping?

18.11

Spectral diffusion and quasiband model.

Describe diffusion of excitation energy in an infinitely long rectangular slab of a glass containing a large concentration of Er$^{3+}$ ions. At time t = 0, excitation energy $E_{c}$ at the center of the Gaussian quasiband is homogenously deposited over the slab, with the quasiparticle density $N_{0}$. [Hint: apply the one-dimensional diffusion equation $\partial f/\partial t=D_\text {E} \partial ^{2}f/\partial E^{2},$ where $f(E-E_\text {c} , t)$ is the distribution function and $D_\text {E}$ the spectral diffusion constant, and replace the Gaussian shape of the density of states by a constant.]

• (a) Show that $f(E-E_c , t)=N_0 /(2\sqrt{2D_\text {E} t})\exp (-(E-E_c )^{2}/4D_\text {E} t).$

• (b) Determine the variance and the halfwidth (FWHM) of the distribution at time t.

• (c) Determine the average frequency range over which excitation energy of a two-level system traveled in a random walk after z ($\gg$1) inelastic scattering processes.

• (d) Estimate the spectral diffusion constant, assuming that the average energy transferred in a spectral diffusion process according to (18.2) is $\delta =0.1\hbox { meV.}$ [Hint: the spectral diffusion constant is $D_\text {E} =(1/3)\vee _\text {E}^2 /\tau ,$ where $\vee _\text {E} =\delta /\tau$ is the velocity in the energy space and $\tau \hbox { }(\approx 10^{-13}\hbox {s)}$ the lifetime of an excited state level with respect to an energy transfer process.]

• (e) How many scattering events are necessary to distribute the energy over the whole width of a quasiband of a width of 50 meV? Show that the corresponding time is still much shorter than the spontaneous lifetime of an excited state. (This is an essential condition for the applicability of the quasiband model.) [Hint: neglect the influence of thermal effects.]

• (f) Determine, for the given numbers, the value of the maximum of the distribution function for the case that the energy is distributed over the whole quasiband.

18.12

Range of validity of the quasiband model.

The quasiband model is applicable for glass laser materials at room temperature. Cooling of the material leads to a slowing down of the energy transfer processes. Determine the temperature at which the quasiband model is no longer applicable if the lifetime of an excited state level with respect to an energy transfer process is inversely proportional to temperature. [Hint: make use of data of the preceding problem.]

18.13

Spectral-spatial diffusion in an active glass medium.

In a fiber laser, the laser field as well as the pump field is non-uniform over the cross section. Redistribution occurs by both spatial and spectral diffusion. It is the purpose of this problem to study the speed of redistribution by spectral-spatial diffusion.

• (a) Describe spectral-spatial diffusion by a one-dimensional differential equation for the distribution f(x, E,t).

• (b) Solve the equation for the following case: The fiber has a quadratic cross section. At time t = 0, excitation energy is deposited at the center of a Gaussian quasiband with a homogenous distribution over the cross section at x = 0. The two-dimensional quasiparticle density for $_{ }$ $t$ = 0 and x = 0 is $N_{0}$; x is the direction of the fiber.

18.14

Spectral-spatial diffusion in an erbium-doped glass fiber laser.

Apply the results of the preceding problem to a cw erbium-doped fiber laser assuming that the pump radiation is homogeneously distributed in the fiber. Assume that the laser field has a nearly Gaussian distribution within the fiber.

• (a) How broad is the spatial hole?

• (b) Estimate the time it takes to fill the spatial hole.

• (c) Estimate the time it takes to fill the spectral hole.

• (d) How deep is the spectral hole?

18.15

Formulate the formulas describing spectral-spatial diffusion (a) in dimensionless units and (b) in the frequency space $\nu =E/h$ instead of the energy space.